jeffrey juiius macfarlane b.sc. (hons
TRANSCRIPT
Diffusion, Boundarv Lavers and the Uptake of
Nutrients bv Aquatic Macrophvtes
Jeffrey JuIius MacFarlane B.Sc. (Hons. )
Botany Department
The University of Adelaiile.
Submitted for the Degree
Doctor of Philosophy, June,
of
1985.
flnrocbd tlsl¡tt
To my Father
and Master
: Friend
Teacher
CONTENTS
Summary
Dec Iaration
Acknowl edgement s
INTRODUCTION
I.The Kinetics of Heterogeneous Reactions
(i) Nernst's TheorY
( ii ) Modifications of Nernst's Theory
(iii) Physical basis for kt
(iv) FIuid mechanical description of kT
(v) Conclusions
II.Diffusion and Simultaneous Chemical Reaction
(i) Equations for diffusion-reaction( ii ) Determination of the extent of
internal diffusion limitations
III. Diffusion Boundary Layers and Nutrient
Uptake in Aquatic PIants
(i) Previous studies
(ii) This work
MATERIALS AND METHODS
( i ) PIants
( ii ) solutions
( iii ) ResPiration
( iv ) Photosynthesis
(v) Uptake of methylamine and phosphate
(vi) Stirring gradient tower
vrt
tx
x
1
2
2
6
9
13
20
2I
2L
25
2'7
2B
35
37
37
3B
40
42
4B
49
(iíi)
MEMBRANE TRANSPORT
r . uptake of I l ac
] uethylamine by UIva qrgrqa
( i ) Results
(ii) Boundary layer Iimitations and V
(iii) The effect of stirring at high
methylamine concentrations
(iv) The saturation of influx with stirring(v) Comparison of observed with predicted kinet¡cs
(vi) other ctnolyses
rr.uptake of l32plehosphate by ulva rrglda
rrr.uptake of 1,32p lPhosphate and I r4c]uetrrylamine
by Vallisneria spiralis( i ) Results
( ii ) Discussion
RESPIRATION
I.Kinetics of OxYgen Reduction
II . Respiration in Ulva rl-grqq
( i ) Results
( ii ) Discussion
III. Resp iration of Vallisneria sPiralis
PHOTOSYNTHESIS
I.Photosynthesis of Ulva rfg-Lqq
( i ) Results
( ii ) A note on the meaning of KM
for photosynthetic CO2 fixation
(iii) Photosynthesis at low PH
(iv) Oxygen inhibition of photosynthesis
53
53
53
54
54
56
57
60
62
7I
7l
72
75
BO
80
BI
89
94
94
94
99
100
104
(iv¡
(v) Photosynthesis at higher PH's
(vi) Mechanisms of HCOã use
Il.Photosynthesis of Amphibol is antarctica
and Vallisneria spiralis
( i ) Results
( ii ) C supply for photosynthesis of
A. antarctica
( iii ) C supply for photosynthesis of
V. spiralis
107
II4
r20
t20
r23
r26
CONCLUSIONS
Appendix I
Appendix II
Appendix III
Appendix IV
Appendix V
133
I4IFick's Laws
: Origin of the Quadratic Describing
an Enzyme-Catalysed Reaction in
Series with a Diffusion Resistance
: The Meaning of the APParent Kt
for an Enzyme-Catalysed Reaction
in Series with a TransPort
Proce s s
: Relaxation of Diffusion to
a FIat Plate
Diffusion and Reaction in
ParaI I e I
(i) The diffusion-reaction
equation
(ii) First-order kinetics
( iii ) Zeroth-order kinetics
(v)
143
145
I47
r50
ts0
I52
I57
SUMMARY
Diffusion limitations on the influxes
(vii)
of tI4 C]
respi ratorymethylamine and l32pl phosphate, and oî the
-1
uptake of o2r have been examined in the marine macroalga
Ulva rlgida and the freshwater angiospe rm Vallisneria
spiralis; photosynthetic 02 evolution (and fixation ofl4c-1abe11ed inorganic carbon) has also been studied in
these two plants and in the marine angiosperm Amphibol is
antarctica. In Ulva, the influx of methylamine and the
apparent "KM" of the process are greatly affected by the
rate of stirring of the bulk medium. The kinetics of
the influx are directly predicted by an equation of the
Michaelis-Menten form which includes a term for the rate
of transport of methylamine through the boundary layer
(the Briggs-MaskeIl equation). Transport coefficients
range f rom ,.U ft" =-I in a barely-moving solution to
in a well stirred one, with a
corresponding change in Ku f rom 20 to tO Æ.
The
equation is also directly applicable to the
photosynthesis of the alga at low pH provided that O2
inhibition can be neglected. The equation is not
directly appticable to H2PO4 influx, respiratory O2
nearly 40 ¡rm s/
H2PO 4')_
HPoî
H2PO 4
at the plasmalemma. 02 uptake in dark respiration has a
complex relationship with the bulk concentration of o2,
uptake or photosynthetic CO2 fixation at high pH.
influx is complicated by the concomitant flux of
which effectively increases the concentration of
and the kinetics are not necessarily first-order
Michaelis-Menten; consequently the Briggs-MaskelI
equation is invalid. Rates of photosynthesis that are
observed at high pH could notbe attained if CO2 was the
only inorganic carbon species transporting carbon
through the unstirred Iayer. The alga probably uses
HCOJ ions directly from the bulk solution as a carbon
source and so again the Briggs-Maskell equation, in
terms of CO2 fixation, does not hold. The
photosynthesis of A[phibolis has a similar response to
CO2 and pH as that of UIva - again the Briggs-MaskeIl
equation is not directly applicable. Vallisneria leaves
have a cuticle which has such a high resistance to the
movement of solutes that boundary Iayer Iimitations are
negligible. The Ieaves also have a substantial supply
of endogenous COZ. Here the cuticle is a distinct
advantage because it increases the probabitity of a COZ
molecule being assimilated rather than escaping into the
bulk medium; this source of CO2 (originally derived from
the sediment? ) is probably much more important than the
COZ in the surrounding water.
(vii i )
ACKNOWLEDGEMENTS
This work was carried out under the supervision of
Dr. F.A. Smith, who was the ever-encouraging driving
force for this rather unsteady flux. I should also
I ike to thank others in the Botany Department - Drs.
J.T. Wiskich and G.G. Ganf for many varied and
stimulating discussions, Prof - H-8.S. Womersley, Dr R.
sinclair and B.c. Rowland for advice, Dr. E. Robertson
for culture room space, Ian Dry and Patrick Hone for
some good talks, and Jane Gibson and Anthony Fox for
company. Outwith the Botany Department, I acknowledge
the help given by B.C. van Wageningen in translating
some German scientific papers, the computer time allowed
by Prof. N.A. Wa1ker and the patience of Prof. J.A.
Raven. FinaIIy, to my wife, Carol, for typing the
manuscript, for numerous calculations, fot support and
for a sma1I portion of her indomitable spirit.
(x)
INTRODUCTION
The growth of a p'Iant involves a host of chemical
reactions,, the reactants which sustai,n them being
substances that occur in the plant's environment; I
shall refer to these as nutrients. This thesis has to
do with the transport of these n,utrients up to and into
the cells of aquatic plants. (The word transport is
used merely in the sense of movement from A to B'
regardless of the energetics of the movement.)
Transport is intimately linked with chemical reaction
(and vice versa), simply because reactants must first
meet before they can hope to become a product' or
products. This applies at the microscopic level (for
reactions between individual atoms or molecules) but is
much more obvious at the macroscopic Ievel, i.e. when
reactants are in distinct phases. A living plant in its
environment constitutes such a heterogeneous system - as
does, at a much Iower leveI of complexity, a lead plate
in a car battery.
For these sorts of reactions, transport by diffusion
may play a crucial, rate-Iimiting role. Although a
number of reports imply that this is the case for plants
( e. g. instances of increased growth or nutrient uptake
accompanying the stirring of the medium around the
plant) there have been few quantitative studies. such
studies are important, however, as there are several
1
instances in plant physiology where major conclusions
have been called into question because of "unstirred
layer" effects. Examples are the presence of pores in
the plasmalemma for water transport (see DaintY, I963)
and the presence of C4 photosynthesis in aquatic
macrophytes (see Smith and Walker, I9B0).
In this thesis, the aim is to quantify the
diffusional limitations to reactions involving a variety
of nutrients in a variety of aquatic macrophytes.
However, before considering the relationship between
transport and reaction kinetics in plants, it is useful
to start on a simpler leveI with heterogeneous reactions
of the car battery type.
I The Kinetics of Heteroqeneous Reactions
( i ) Nernst's theory
In 1900 Walther Nernst, in the thírd edition of his
text book on theoretical chemistry (see Nernst, L9I6),
suggested that in certain circumstances the rates of
heterogeneous chemical reactions were entirely due to
the rates at which reactants could diffuse to the
interface and the products diffuse away. To a certain
extent, the idea was based on some earlier work by
Schükarev (1891), Noyes and Whitney (1897) and Bruner
and St. ToIIoczko (1900) on the dissolution of various
solid bodies in water. Noyes and Whitney (1897)
suggested that a layer of saturated solution rapídly
2
formed around the surface of the so1id, from which the
solute particles diffused into the bulk solution. The
rate of this diffusion at any one time (t) would be
proportional to the difference between the concentration
of the solute in the saturated layer (c=) and that in
the bulk solution (c5). (rrris f ol lows f rom Fick's f irst
Iaw which is discussed in Appendix I). Hence, the rate
equation could be written
dck(c Þ̂ t¡) (1)
which was verified experimentally. Bruner and St.ToIloczko
(Ì900) showed that the veloci$l constant, k, \^zas dependent
upon the surface area of the solid, as would be
expected.
Nernst (I904) extended these ideas to heterogeneous
chemical reactions. He assumed that at every boundarv
between two phases, equilibrium is established with a
actical I infinite velocity.* For example, in the
case of magnesia (MgO) in acid, the solution adjacent to
the magnesia would be saturated by vtg2* and therefore
weakly alkaline and the acid at the interface would be
completely neutralized. Granted this assumption' the
rate of dissolution of the magnesia wilI simply depend
on the rate at which the acid reaches the interface by
diffusion. For well-stirred systems, the bulk of the
dr
* Incorrectly based on the fact thatchemical potential at the interface
the difference inis extremely Iarge.
3
solution may be considered as practically homogeneous
and the diffusion zone witl be confined to a thin Iayer
(thickness 6) adhering to the surface of the so1id.
Nernst assumed that the concentration of the acid
varied linearly across the thin layer, i.e. the
concentration gradient v/as given by (.n c")/b where
cb is the concentration of the acid in the bulk solution
and cs that at the surface of the magnesia (Nernst
assumed cs was zero in this particular case)- From
Fick's first law (Appendix I) the equation for the flux'
J, of acid across the thin layer could be written
cb 5 (2)
is the diffusion coefficient of the acid.
J=D I
where D
terms of the rate of
in the bulk solution,
change of the
equation (2)
concentration of
In
acid
becomes
dc cb CSAD(3)
dt vol 6
where A is the surface area
the volume of the solution.
equation (l), with
k
of the magnesia and "vol" is
This is of the same form as
DA
4
vol(4)
If k is measured per unit area at unit volume, then we
may write
kvol D
kT (s)
which has the dimensions of a velocity.
Bircumshaw and Riddiford (L952) t in their review,
show that Nernst's theory has been confirmed for many
heterogeneous reactions. Thus, many different reactions
carried out under the same stirring conditions yield the
same values for k, and ð. Reaction rates have also
been shown to increase with increasing D, although it
would appear the relationship is not a direct one
Eucken (Ig32) found kr * D0'66 and King and Cathcart
(Ig37) found kT * D0'7, while under turbulent flow
conditions the suggested power on the diffusion
coefficient is 0.75 (xing, l-948). kr frequently, but by
no means always, has a similar temperature coefficient
to D. Increased stirring, which would decrease the
thickness, E , of the Nernst layer, has also been shown
to increase reaction rates. Most workers find 6
proportional to 1/uQ where U is the velocity of the
moving liquid. Values for the exponent q usually range
from 0.5 to 1.
Nevertheless, it is now weIl known that there are
heterogeneous reactions which are chemically controlled,
6A
5
i.e. the rate of the reaction does not depend on
diffusion rates but on the physico-chemical properties
of the reactants. Nernst's basic assumption that
equilibrium is always attained virtually instantaneously
at the interface of two phases is, then, not valid for
aIl heterogeneous systems; for the particular cases
where the assumption holds, the reaction rate will
almost certainly be transport limited by diffusion.
The in-between case is where the concentration of
reactant at the interface is neither the equilibrium
concentration (pure transport control) nor that of the
bulk sol-ution (pure chemical control). Nernst's theory
must be modified to take into account such cases.
(ii ) Modifications of Nernst's theory
The rate equation for the chemical process may be
written
dckc
(6)dr
where kc is the velocity constant for the chemical
process and Aa
available for
the transport
be written
the true surface area
chemical reaction. The
of the so1 id
equation forrate(3)process, from equations and (5)' can
dckT
A
6
dr vol-(.n cs) (71
where A is the apparent surface area. In the steady
state, equations (6) and (7) are equal, i.e. kr A (cb
"=) = kC AC (c=)n and for first order reactions (n = l),
cs = kT A c6/ (k, e + k" Ag) = kT cy/(kT + kC (ACIA)).
Substituting in equation (7) we obtain
dc kc kr Acb (8)
dt kc + kT (n/t") vol
and
kc kT(e)
kc + kr (A/Ac)
where krpp is the apparent velocity constant for the
reaction per unit (apparent) area at unit volume. When
kc >>
kc <<
reaction, there may be a change in control if the
experimental conditions change such that kg' kt or the
ratio n/AC are affected.
For many enzyme-catalysed reactions, the rate
equation for the chemical process can be written
5 *VC (r0)
krpp
VKn+ cs
where rC is the overall velocity of the chemical
reaction, V the maximum possible velocity and Kt the
Michaelis constant. If v has the units of a flux
7
-2 -I the rate of the transPort
= kT (.n c=), i.e.(e.9. moI
process, v
Lt Kukr
) then
be vtm S
wiIlTt
c5csVT
kT
Kl,lkt + cbkr + V)
(II )
4c5ktV )
In the steady state, VT = VC = v and. substituting
equation (11) into equation (f0) Ieads to a quadratic
equation in v for which the solution is
V +c ¡kr+v(12)
This equation was first derived by G. E. Briggs (pers-
comm. to F. A. Smith and N. A. Walker; see Appendix II)* Strictly, this equation should be written in terms ofthe activity, rather than the concentratj-on, at thesurface of the enzyme, i.e.
YcsVC V
K, + yc=
where y is the molar activity coefficient of thesubstrate. For a number of substances (ions inparticular) y is often significantly different fromunity, even in quite dilute solutions. In sea water'the term may be very important. The effect on theMichaelis-Ménten equation is to change the meaning of K¡,1
s ince
Ycs csvc V V
K, + yc= (K¡4lV) + cs
I shall simply refer to K¡1,which can be regarded as.thetrue Kr expreésed in terniå of activity; this quantitymay be'Lig;iticantly different from the K¡4 measured in a
solution of different ionic strength.
B
and E. J. Maskell (1928). It has been rederived many
times (e.g. Lommen, SchwinLzer, Yocum and Gates, I97L¡
Winne, L973¡ Wilson and Dietschy, I974¡ MärkI , L9'77¡
Dromgoole, IgTB; WheeIer, 1980; Lívanskiz, L982)- It can
be shown that when kr )> v/2KM, the kinetics are
enzymically controlled and v = V c5,/(K¡4 + "n). When
kT << v/2KM, the kinetics are transport controlled up to
saturation and either v = k1c5 or v = V (eppendices II
and III, HiIl and Whittingham' 1955). Figure I shows
some kinetic curves for various values of kT-
The modified model for heterogeneous reaction
kinetics includes both the chemically and transport
controlled cases; although Nernst's predictions with
regard to the latter have been verified on the whole'
there are deficiencies. The relationship between kt
and D (page 5) is not predicted by Nernst's theory and
the relationship between k, and the fluid velocity is
not we1I defined. The next two sections are an outline
of a more quantitative definition of kT.
(iii) Physical basis for kT
Nernst assumed that the diffusion of solute from the
bulk phase to the interface (and back) was via a thin,
stationary Iayer (thickness 30 ¡m or more) adhering to
the surface of the solid. Does this hypothetical
"unstirred layer" have any physical basis?
9
hrt oo
60
2þ
kr =l:72 ><1è.5vn s-l
^6ob''\c\, /1O
\v
Ki{' 10 Kffi
cs (fM,2D 30
0 KM
FIGIIRE 1. Velocity versus concentration curves for the Sriggs-lvfaskellequation (equatlon 12)r.wittr various velues of k1, K¡4 = 1¡Man¿ V - 517 nmol n-2 g-1;for snall k1, the inltlal slope íeapproxinátely k1. \r"nn = fu + |v/t1-(ece Appenttix III).
--hu--
krrn s-t
= 3'44x lo-5
By studying the dynamics of fluid flow through pipes
and around solid objects it has been found that fluid
motion persists down to at least 0.6 ¡.l* from the surface
(Fage and Townend, 1932) if not closer (RoIIer, 1935).
Nernst's stationary Iayer is, therefore, a fiction.
However, another fact which emerges from the study of
fluid flow around solid bodies is that, even in very
turbulent systems, there is a region adjacent to the
solid surface where the flow is Iaminar (scrrtichting'
1968). In laminar flow, the viscous forces of the fluid
are important, so that slower moving layers of the fluid
can be thought of as retarding the flow of faster moving
layers. When the fluid is flowing over a surface, the
fluid in contact with the surface is stationary which
retards the fluid in contact with it, which in turn
retards the next layer and so on. The essential feature
of Iaminar flow is that the component of the fluid
velocity normal to the surface (i.e. between the
laminae) is very small. The tangential component of the
fluid velocityr âs we have seen, depends on the distance
from the surface. Figure 2a shows the velocity profile
of pure laminar flow over a flat plate.
When the f low in the bulk medium is turbulent (r'ig.
2b) the f luid motion j-s irregular; the arro\^/s therefore
represent the average fluid velocity. Compared with
Fig. 2a, this velocity remains quite large as the
surface of the plate is approached and the region of
IO
u Lhve__*
(ù) (b)
ft
FIçURE Z. VelocLty (") proflles for (a) laminar and (b) tnrtulentflor aläni r if"t plate. II 1s the flui¿ veloolty ln thcbulk meclfi¡m.
laminar flow is much thinner.
Where the surface is reacting with substances
dissolved in the fluid, the fluid dynamics have
important implications. For a solution flowing over a
flat plate, at the leading edge the concentration of
solute will be the same as that in the bulk solution.
As the solution flows over the plate, however, solute
will be used up from the layers immediately adjacent to
the surface. Since convection between these Iayers is
extremely smaIl (laminar flow), diffusion is virtually
the only means by which fresh solute can be supplied to
the surface. It is cl-ear, then, that there is a region
(the extent of which depends on the flow conditions)
where diffusive transport is important. This is the
essential part of Nernst's theory, and in this respect
his ideas are supPorted.
Before going on to discuss the hydrodynamíc/mass-
transport relationships more futIy, I will briefly
mention relaxation of the diffusion process. This
phenomenon illustrates the chief difference between the
hydrodynamic description of the diffusion region and
that of Nernst; namely that the fluid in the diffusion
region is not an "unstirred layer", but is in fact
continually mixing with the bulk solution.
as an example,Again using the reacting
suppose the average flux of
flat plate
reactant to
1I
the surface is
JI. Now part of the leading section is coated so that
it can no longer react, and the average flux of the
substance (per unit area of the uncoated surface) is
again measured. It is found that the new average flux,
J2, is significantly larger than the original. It can
be shown (Appendix IV) that the ratio of J2 to JI is
given by
J2 t(xo/x) 3/c ?/s (13)
xo/x)
If
trJ1 (1
where xo/x is the ratio of the length of the coated
section (Xo) to the length of the whole plate (X)-
is nearly twice Jl when 90U of the plate is coated.
99so of the plate is coated, J2 is nearly four times JI.
The enhancement occurs because the solution streaming
over the coated plate is not depleted of reactant; hence
for a considerable distance downstream of this section
the flux is greater than it would have been were the
plate not coated. At large distances from the coated
region, the two fluxes are similar. The extent of the
relaxation region (i.e. where the new flux is
significantly larger than the original ) is of the order
of Xo (Levicht 1962t p 106). By comparison, the Nernst
picture of the diffusion zone would predict a relaxation
area (due to diffusion from points tangential, as well
as normal, to the surface) of much smaller dimensions,
j2
T2
and the area would not be affected by changes in Xo.
The hydrodynamic picture shows that the tangential
velocity component of the liquid is very important for
mass transport.
(iv) Fluid lrìechanical description of kT
The quantitative interaction between the momentum
transfer occuring in fluid flow and transfer of matter
due to chemical reaction involves solving the diffusion-
convection equation
ò" ò2"= D(-
àt òxz àv'
ò2"
àr2'(t* +
à2" ò"tyr +t.z
aY
ò.
ò"
ò.
t'+ +
(14 )
where üx, uO and u, are the velocity components parallel
to the x, y, and, z axes at a particular point in the
fluid. The first part of equation (I4) is the rate of
change of concentration ( * ) due to diffusion (Fick's
second law for the three-dimensional case) while the
second part deals with changes in concentration due to
convection. Equation (14) can only be solved exactly
where the equations for the fluid motion have been
worked out, which is usually in systems of fairly simple
geometry. However, a number of initial approximations
can be made.
The thickness of the hydrodynamic bounda ry layer, 6o,
can be defined as the distance from the surface of the
I3
solid to the point where the tangential velocity
component is 99eà of the velocity of the main stream
(Scfrtichting, 1968). For liquids,6o is not equival-ent to
the thickness of the diffusion boundar y layer. This isbecause diffusion is so slow in liquids that even
relatively slow rates of convection are able to
transport mass faster than diffusion. For gases, in
which diffusion is comparatively fast, 6o and å are
similar. In non-turbulent systems, Levich (L962) has
shown that 6 - ó1 *n..e Pr (the Prandtl number) isPr'/6
equal to |, V being the kinematic viscosity of the
solvent. (por gases , Pr ru ]. For water, ì) ^' 10-6 m2 s-l
and D for smaII molecules in water is about 10-9 *2 =-1;Pr is therefore about 103, i.e. å- 0.1 6o). rn the case
of Iaminar f low over a semi-inf inite plate, Eo- 5,tff(glasius, I90B; Schlichting, I968) where x is the
distance along the plate f rom the Ieading edge; \^re can
therefore write
6- (rs )
Levich (L942) has solved equation (14) exactly for
the case of a rotating disk electrode, based on
Cochran's (I934) solution of the hydrodynamic equations.
In this instance the flux of mass, J, to the surface of
the disk rotating with an angular velocity of û) (radians
per unit time) is given by
.ñ5 D'Ê v/"
t4
J (0.62 )cb . (16 )o% ¡-k ¡6rz-
(The concentration gradient is simply given by c5r as cs
is assumed to be zero. J is therefore the limiting
ftux.) comparing equation (16) with equation (2), it is
readily seen that the thickness of the Nernst diffusion
Iayer must be
5 L .62 (17 )oß l/' ti-Yz
which is comparable with equation (15). The term rtxrr
does not appear because in this particular case the
diffusion boundary layer is the same thickness over the
surface of the disk. Levich's equation has been tested
(Hogge and Kraichman, L954¡ Kraichman and Hogge' 1955;
Gregory and Riddiford, L956) with very good agreement
between theory and experiment.
For f Iat plates, where the length and width are
large compared with åo, Levich found that the flux is
given by
J - 0.34 #u-'n,4 cb (18)
again with c= 0. Hence
Dcb
J5
15
3 DáV,, l"V;
(re )
In this case 6, and the flux J, are a function of the
distance from the leading edge of the plate. b
increases and the flux, therefore, decreases'
proportionally with ^F.
It can be easily shown that
the average unstirred layer thickness over a plate of
Iength X is simply one half of ó*, where 6" i" tft"
value of equation (19) at x = X.
The preceding expressions for 6 apply where the flow
of fluid throughout is laminar. Where flow is
turbulent, the diffusion-convection equation usually
cannot be solved; in this case, dimensional analysis is
useful (Bircumshaw and Riddiford , 1952). For the
rotating disk and for Iaminar flow over a flat plate'
the flux of reactant, J, depends on the diffusion
coefficient, D, the relative fluid velocity, U, the
kinematic viscosity,V rthe concentration difference,
(c5-c=) = Ac and, where applicable, the distance from
the leading edge of the plate, x. It is reasonable,
therefore, to write for the flux in turbulent flow J =
f (DrU,X,lrAc); we could further assume that f is a power
function of the form
J B DP uQ xr vs act (20)
where B is a number. Because the dímensions of J (U f,-2
t-1¡ must equal the overall dimensions of the right hand
side of equation (20), the powers p' q' T, s and t can be
I6
expressed in terms of any two of them, e.9
J B DP uQ x(q-I) V(I-P-q) ¡. (2r)
By comparison of this equation with that of Nernst (J =
(D,/á )Ac) it can be shown that
ol-p u-e xl-A y9- (1-p¡ (22)b IB
The power on U may be evaluated by the variation of J
with the fluid velocity. As was mentioned on page 5, q
usually varies between 0.5 and L¡ for the rotating disk
and laminar flovù over a plate, I = 0.5. As a rule' q
increases with increasing turbulence. The other power,
p, can be determined by the variation of J with V or D'
assuming the interdependence of D and V is known. The
theoretical value of I - p is * a" ¿, probably dependent
on the degree of turbulence (Levich, 1962) , and this is
generally found to be the case experimentally (cf. p 5)-
From the empirical relationship between J, D' U, x and \),
the value of B can be found. Using this sort of
analysis, many workers have derived an expression of the
form
6 I/e x0'2 U-0'8 90'5 D0'3 (23)
for turbulent flow over a flat p1ate, with B about 0-03
L7
(Rohsenow and Choi' 1961). The thickness of
layer depends on x to only the 0.2th po\^ier'
similar over most of the length of the plate.
average effective Nernst layer thicknes" t 5 I
whol-e plate is given bY
5 0.8 óx
the Nernst
i.e. it isThe
for the
(24)
where 6x is the layer thickness at the trailing edge of
the plate. In fact the power on U (and hence on x) may
be even smaller. Vielstich (1953) suggested 6
proportional to *0'1 u-0'9, in which case 5 would be
O.O6x (Vetter, L967, p f93). As already mentioned, there
is also uncertainty as to the power on D (and hence
on V); it is difficult to distinguish experimentally
between the theoretical "Iimits" of 0.25 and 0.333.
Figures 3a and b (from Vielstich, 1953) summarize the
relationship between the fluid dynamics and mass
transport in laminar and turbulent flow. The purely
operational nature of the value of 5 i= clearly seen; it
is only an approximation to the actual thickness of the
region of changing concentration which is decidedly
Iarger (Dainty, I963). Generally it is a useful
approximation because the concentration gradient is
mostly linear and of the same form for a range of
systems.
18
/7
L,Cø
vVæ
1
v(b)
Ya
(ø)
;T
óN
óx óo
I
óPr
L.Cøo
1 T
05
í¿ ì,
I'IGURE 3. Concentration (c) and veloa¡ra (¡ ) turuulent flor ovedietance nomol to the surthÍclmess of the Nernst 1
dpa the thiclmees of the PFor la¡ninar flov, 6P= = 6o
Fron vielstich (t95¡).
FinaIIy, the transport of mass to a surface in a
completely stagnant solution will be considered. In
theory this is the simplest case, but in practice a
perfectly stagnant solution is not normally attained.
The main reason is the density changes which occur close
to the surface due to the reaction itself. This results
in natural convection currents, equivalent to slow
stirring (Karaoglanoff' 1906). The concentration
gradient that develops with natural convection has been
photographed by Ibl and Muller (1955) using an
interferometric method. The flow conditions were also
photographed and it is clearly seen that, while the bulk
of the solution is still, there is a net flow near the
reacting surface. This is shown diagrammatically in
Fig. 4 (see also Skelland, I974, Fig. 5.2)-
Numerous workers (see Vetter, 1967, p L94) have
derived equations to describe the Nernst Iayer thickness
in stagnant solutions for a vertical plate- The
equations are all of the form
5 = 11""" f' (25)v B\o(g nic)
where o( is the density coef ficient ,,þ *" , P being the
density) and g the acceleration due to gravity.
Estimates of the constant B lie between 0.5I and 0.73.
The value of 5 trrrrs depends on L"-11. From equation (2),
the flux of reactant, J, is then proportional to Àcå
T9
sol,,.tion
reo.fi^g./ surfo ce.
-lmm
FIGIIRE 4. l,tratr¡ral conveetlon neer thc surface of a reaoting, vcrticalplate. Arrors lltustrate flor veloclty. (.4,t tfrc surface,flor is zero) Aftsr IbL endt }î¡1ler (lg>5),
(not directly proportional to Ac) and 6 i= proportional
to I/¡0'2. The interesting result, therefore, is that
the larger the flux, the thinner is the Nernst diffusion
Iayer. This has been verified experimentally (Ibl,
Barrada and Trümp1er, 1954).
In closing, it should be noted that the
hydrodynamical descriptions of kt which I have
considered are for relatively simple systems, i.e. a
rotating disk or a f lat plate, in a f 1o\^/ of f luid which
is either laminar or turloulent. More complex systems
are intractabte theoretically; besides irregular
geometry, the flow may be Iaminar in some places and
turbulent in others and even the scale and intensity of
turbulence may differ in space (and time). However,
dimensional analysis is stiIl useful in these situations
and it is usually possible to assign an avera9e
unstirred layer thickness (based on an average flux) to
the object in question, using a "characteristic"
dimension which becomes I'xrr in equation (22). Thus,
equations of the form of (f9) and (23) can be
constructed.
(v) Conclusions
Transport processes can be important in determining
heterogeneous reaction kinetics. In these cases' the
kinetics can be treated by the simple Nernst theory'
provided the real nature of the region represented by
rr6tr is kept in mind. Where the f lux can be predicted
20
exactly from the hydrodynamics, ô represents the
thickness of an hypothetical, stationary unstirred layer
through which an equivalent flux would occur, assuming a
Iinear concentration gradient throughout- When 6 is
determined experimentally (from a measured fIux,
concentration gradient and diffusion coefficient) it
must be realized that the value is at best only an
approximation to the real extent of the region of
concentration change. At worst, in complicated
hydrodynamic systems 6 may be, physically, quite
meaningless.
It must also be remembered that 6 depends on the
diffusion coefficient, so that in the same system there
can be several different Nernst layers for different
solutes and that, in stagnant solutions, 6 also depends
on the concentration gradient itself.
In this section I have only discussed reactions on a
solid surface. The question of what happens when the
solid is permeable to the reactants is relevant to
living systems and will be briefly considered in the
next section.
II. Diffusion and Simultaneous Chemical Reaction
(i) Equations for dif fusion-lîeaction
In section I, transport and chemical- reaction were
will also be considered inIinked in series. Here they
2T
paralIel, with substance diffusing through a solid body
and being consumed at the same time. Such problems have
been tackled from two quite different points of view:
that of the physiologist and that of the chemical
engineer (Weisz, l-973).
The physiologists were interested in the O2 supply
to respiring tissues. Krogh (1919)' with the
mathematician Erlang, developed an equation relating the
respiration of tissue to the supply of oxygen from a
capillary. Warburg (I92ó) examined respiration and
oxygen diffusion in tissue slices and similar equations
\rzere worked out for cylinders of tissue (penn, 192'7¡
Gerard, 1927¡ HilI, L929) and for spheres (Gerard,1931).
The same question was of great interest to chemical
engineers with regard to optimizing yields from
catalytic reactors, and it led to the concept of
catalyst "effectiveness". The effectiveness factor' f[ t
was defined as the ratio of the actuaf reaction rate to
that which would occur if aII the surface throughout the
inside of the porous catalyst particle were exposed to
reactant of the same concentration as that existing in
the bulk medium (Satterfie]d, I98l' p 130).
In solving the equations that describe reaction rates
under these conditions, it is found that the solutions
are functions of a characteristic dimension' R' of the
(porous) solid body (e.9. the radius of a sphere or the
22
thickness of a plate), the rate
reaction, kt , and the effective
the reactants within the solid,
quantities can be combined into
constant for the
diffusion coefficient of
Deff. These three
a dimensionless
which became known, amongparameter of the form R
chemical engineers, as
E.W. Thiele ( I939 ) and
iele modulus, f (after
see Aris (I975), p. 40).
The importance of the Thiele modulus becomes clear in
Appendix V, where expression for ry are derived forreactions having first-order and zeroth-order kinetics.These ì.l's set lower and upper bounds on the
v
effectiveness factor for reactions with Michaelis-Menten
(Briggs-HaIdane) kinetics.
For a first-order reaction, the expression for the
effectiveness factor is
tr (26)
if the boundary layer resistance is negligible. If this
is not the case,
2 -I
ønhta
ø
+ø
Bi(27 )
where k1 has been expressed in terms of the
dimensionress Biot number, Bi = * ot
Deff
23
For a zeroth-order reaction,
I (ø <4r)
Qo (v. 17 )
for k'' + oê
important
When
2(Ø .<
1 + 2/BLQo
(v.18 )
I 2+ (Ø >,
Bi2 ø2 Bi r + 2/BI
Yamané (l9BI) developed an empirical equation for \for the full range of Michaelis-Menten kinetics, which
is a \^/eighted arithmetic mean o¡ ìtlf and Qg
rl0 (2.6 4o.B) nr
4' (ó >z|2)ø
the external resistance is
I
21
+
1 (v. 21 )
The
Qr l-s
1+ 2.ø40'a
4 i" the dimensionless Michaelis constant, Ky1/c6.
ThieIe modulus to be used in calculating Qg .tta
V
Deff (x* + cb)l=n
This is an "overaIl"
first-order process
processitisY/c6
reaction).
(v .22)
modulus, since k' for the strictly
is vrlxt while for the zeroth-order
(v is the maximum rate of the
24
Fig. 5 shows some rate versus concentration curves
based on Yamané's expression for 1, with various values
of kT. The curves are for a permeabi-e flat plate, of a
definite thickness, containing enzyme. Note that when
internal transport limitations are insignificant
compared with external ones, the kinetics approximate to
the Briggs-Maskell type.
(ii) Determination of the extent of internal
diffusion Iimitations
The extent of internal resistances to mass transfer(e.9. within a piece of plant tissue) can be gauged by
repeatedly subdividing the tissue until there is no
change in the rate (assuming the damage due to cut cellsis accounted for). This method is not appropriate forplants sensitive to being cut into little pieces nor, of
course, to single plant cells (e.g. Characean internodol eells).
The best method would be to isolate the enzyme
catalysing the reaction in question. This assumes the
kinetics of the enzyme in vivo match those found
in vitro, which ma y not be the case (e.9. Jensen and
Bahr, L977 ).
If no data are available about the variation of rate
with the size of the tissue section r oy about the
kinetic properties of the enzyme, it may sti11 be
possible to estimate the effectiveness factor. If the
flow conditions are known, kT can be estimated by the
theory outlined in section I. If there are other
25
lo
I
'/) 6o
þoçÈ4
þ
6
2
6i(Ç)+.o
(k, = õxto-5 m
BL=c
(Rt ^2xlo-5 ø sþL= Z
eL-- 1
(kr = lX fO.S m s-l
2D10 30cb q'À4 )
FIGURE 5o Concentration oureres for the rate of an enzJ¡me-catalyseclreactj.on in a slabr 1OO¡n thick. Deff = 5 x 10-10 ln2
"-1,K¡¡ = 1 ¡U, V = 10 mnol ú-J s-'1. Dotterl line - Michaelis-Menten'kinetics with no tra^nsport LinÍtations (i.e. Deff, kT - - ).
significant external resistances, then these must be
included in kt to
coefficient t e.g.
kr(.tr ) kr(dbr )
where the abbreviations in
diffusion boundary layer'
outer celI wall.
form an effective mass transfer
1
+ (28)kr(wart
)
refer to the
(if present) and an
(2e )
1 II+
kr(cut)
brackets
a cuticle
The only other parameter needed to determine the
extent of diffusion Iimitations is the Thiele modulus.
This contains K¡1, and the measurement of Kvl is subject
to the very diffusion l-imitations whose extent we wish
to ascertain. A way out of this quandary r¡ras f irst
proposed by Wagner (1943) and developed by Weisz and his
school (see Weisz and Hicks, L962). They defined a new
modulus, 0 , based
rate, defined by
on the actual, observed reaction
Þ
R2r(-
Deff voI
dnl-)-dt cb
in which çfu Ê€ is the actual reaction rate (amount of
substrate or product, n, per unit time' t) measured per
unit volume, at a substrate concentration of cb. R isthe ratio of the volume to the surface area of the body
(cf. Figs V.I and V.2). Graphs of q against Q are of
the same shape as those of t against the Thiele modu.lus
26
(fi-g. V.B) and, while the curves vary dependì-ng on K¡4,
aII fal1 between the limits set by a first-order(K¡a/c5 > oo ) and z eroth-order (K¡,1/ cO -> 0 ) reaction.
Hence even if KM is unknoçn' t can be estimated to good
accuracy for Q l-ess than I or greater than 10. For I< 0(10, the estimate of 4 will be uncertain by no more
than 33%. Graphs of n versus þ ate given in Roberts
and Satterfield (f965) and Knudsen, Roberts and
Satterfield (1966). From these graphs it can be seen
that for aII values of KM, if 0 i" less than 0.f,
catalyst effectiveness is nearly 100% while for Q ) 5,
there are very significant internal diffusion
limitations ( fL < 0.5). These values assume there is no
external resistance; if this is significant, the limits
are alI decreased in proportion to kt (cf. Fig. V.4).
III. Diffusion Boundary Layers and Nutrient Uptake in
Aquatic PIants
A plant tiving and growing in its environment
represents a heterogeneous reaction system, albeit a
highly complex one. It is reasonable to suppose, then,
that diffusion may be important in limiting the rates of
various reactions in the plant, where the reactants are
supplied from the environment or where the products are
expelled thereto. Compared with terrestrial plants,
boundary layer restrictions may be quite severe foraquatic plants because diffusion coefficients in water
27
are reduced some I0,0OO-fold compared with those in arr.
(rhat is not to say the problems are 10'000 times worse;
ô itsetf depends on D (equation (15)) and also there isa dependence on the kinematic viscosity of the fluid, l,which is I0 times smaller in water cf. air).
Internal diffusion limitations are of a similar
nature for plants in either habitat. A considerable
amount of work has been done on the gas exchange of
Ieaves of terrestrial plants, where internal diffusion
has generally been assigned a fixed resistance in an
"electrical analogue" model (Gaastra, 1959¡ Lommen
et âI., I97I; Jones and Slatyer, L972). In other cases,
the diffusion-reaction equation has been solved as in
Appendix V, making various assumptions (e.9. Briggs and
Robertson, 1948; Briggs, 1959; Parkhurst, I911).
(i) Previous studies
There is a Iarge body of evidence which gives
qualitative support to the idea that transport
restrictions due to boundary layers can be significant
for the uptake of nutrients by aquatic plants.
Increasing fluid velocity, whether by shaking, stirring
or increased flow down a channel, has been found to
increase growth rates and nutrient fluxes of single-
celled algae (Myers, 1944¡ Falco, Kerr, Barron and
Brockway, I975¡ Pasciak and Gavis, I915), colonies or
compacted communities of microalgae and bacteria
2B
(McIntire, L966 a,bi Rof f , Rough, Cummins and Cof fman'
1966; Hartmann, L967; Sperling and Grunewald, 1969¡
Sperling and HaIe, L973), marine macroalgae (Gessner,
1940; PrinLz, 1942; Matsumoto, 1959; Jones, L959¡
Conover, 1968¡ Doty, I97L; Dromgoole, L91B; Littler,
1979; WheeJ-er, I9B0; Parker' 198I; Gerard, L9B2)
freshwater macroalgae (Whitford and Schumacher,
1961 ,1964; Schumacher and Whitford, 1965¡ Pfeifer and
McDiffet, I975; Thirb and Benson-Evans, I9B2) and
aquatic bryophytes (James, I92B¡ Bain I L9B2) and
angiosperms (Darwin and Pertz, 1896; Gessner, 1938;
Owens and Maris, 1964; WestIake, 1967; Buesa, L977¡
Werner and Weise, L9B2¡ Madsen and Spndergaard, l9B3).
Some of the increases have been quite spectacular;
for example, Schumacher and Whitford (1965) observed a
43-fold stimulation in the rate o f 32p uptake by the
freshwater red alga Audouinella violacea in stirred (fB0
mm "-I)
versus unstirred solutions. The effect of water
velocity is greatest when the maximum rate of the
metabolic process is high - for instance, stirring has
only a minor effect on photosynthesis at low Iight
intensities (WheeIer, I9B0) or where maximum rates are
reduced by seasonal or genetic variation (McIntire,
L966C;Westlake, L967). Such a result is in line with
the predictions of the Briggs-Maskel I equation.
The increased metabolic rates accompanying stirring
also appear to be dependent upon the form of the plant
29
or plant part i-n question. Gessner (I938) found that
stirring increased photosynthesis of entire Ieaves of
Proserpinaca oa lustri s by 22eo (i.e. compared with
photosynthesis in stagnant water), but that the increase
was only 5% for the featherlike leaves of the same
plant. Similar results have been found when different
species of plants, having either entire or highly
dissected leaves, are compared (see Hutchinson, 1975).
Some workers have plotted rate versus velocity curves
(James, I92B; Pasciak and Gavis, L975; WheeeIer, I9B0;
Bain, 1982¡ Madsen and S@ndergaard, 1983) which usually
show some saturation of rate with water veloci-ty' U-
The hydrodynamic description of kt in laminar flow
predicts kT o ^Æ
(p 15 ) which resembles the beginning
of a saturation type curve; generally, however, the
experimental curves have been taken to denote increasing
rate control by the biochemical reaction itself. It has
sometimes been found that there is an adverse effect of
water velocity when this is too high.
Rate versus concentration curves of the type expected
for Michaelis-Menten kinetics in series with a thick
unstirred Iayer (i.e. virtually Iinear at first with a
sharp transition to the saturated rate, Fig. I ) also
frequently occur in the literature (e.9. Hanisak and
Harlin, L97B; Kautsky, I9B2) and are suggestive of
diffusional limitations (Smitn and WaIker' I9B0; Madsen'
30
I984). Indeed Mclntire and Phinney (I965) found that
photosynthetic oxygen evolution increased Iinearly with
ICO2] up to the highest concentration studied (- I mM),
and Werner and Weise (L982), found the same result for
phosphate uptake by Ranunculus penicillatus. Phosphate
and sulphate absorption by Elodea densa leaves was found
to have a complex relationship to concentration by
Jeschke and Simonis (1965); however it was linear from
0.01 to 3.2 ltM and the authors concluded that absorptionI
\^ras timited by the boundary layer in this region.
Similarly, respiratory oxygen consumption by various
aquatic plants has been studied by Gessner and Pannier
(1958a,b) in relation to the external oxygen
concentration and in many plants (including
phytoplankton) respiration rates varied linearly with
concentration and were stimulated by stirring. In other
plants respiration seemed to depend on I02] raised to
some polrer (<1): this may have reflected internal
diffusion Iimitations to oxygen transport (Owens and
Maris, L964t and see Longmuir, L966). Dromgoole (1978)
observed a hyperbolj-c relationship between respiration
rate and oxygen concentration in a number of marine
macro-algae and showed how the curves became more
oblique as the water velocity decreased.
Studies of a more quantitative nature on the
relationship between transport through boundary layers
and the absorption or loss of matter by plants are few
31
and far between. Apart from Wheel-er (1980), they have
been mainly concerned with phytoplankton. Munk and
Riley (L952) used heat transfer theory to predict the
effect of fluid velocity on mass transfer (i.e. nutrient
absorption). Apart from changes in the meaning (and
value) of the symbols, the equations for heat and mass
transfer are virtually identical (Bird, Stewart and
Lightfoot 1960; Thom, I96B). Munk and Riley took
phosphate to be the Iimiting nutrient for phytoplankton
growth, and its absorption rate to be completely
diffusion limited (i.e. cs = 0). Their calculations
point out the advantages of small síze for mass
transfer, the importance of sinking as a means of
increasing it and the relative effects for cyl-inders'
spheres, disks and plates. Some of their predictions
are borne out in natural phytoplankton assemblages-
They also give an equation for the effect of water
veI ocity on mass transfer to an attached plant in the
form of a plate, which is analogous to Levich's equation
(equation (IB)). Based on this equation, using
reasonable values of the inorganic P concentration of
sea water and rates of P uptake by plants, Munk and
RiIey concluded that attached plants would need to
inhabit regions of quite high water velocity. The
calculated water velocity depends upon the phosphate
influx squaredr so that a somewhat lower influx would
make a considerable difference to the answer. Munk and
Riley usedan influx of some 300 nmol m-2 t-I which is
32
large.
Pasciak and Gavis (I91 4 ) derived the quadratic
equation relating c" in the Michaelis-Menten equation to
cb for a sphere in an infinite, perfectly stagnant
continuum. Under these conditions, kT = D/p where R is
the radius of the sphere and D the diffusion coefficient
(see Appendix V, p156). They then incorporated motion by
employing the expression for k, used by Munk and RiIey
(1952) f or sinking spheres, k, = D,/R (f + /z(p/n) u)
where U is the relative water velocity of the bulk
medium and the sphere. They showed how the familiar
Michaelis-Menten hyperbolic kinetic curve became more
and more obtique depending on the value of a
dimensionless parameter, P'I equivalent to kT KM/v.
(Pasciak and Gavis (I97 4) actually def ined p' = (L4.4
n2 )X,, KM/v; however, if consistent units are used and V
is expressed as a flux (e.g. nmol m-2 =-l), the
parameter becomes simply kr KM/v.) rfris dependence
on P' follows directly from the Briggs-Maskell equation,
which predicts severe transport Iimitations through the
boundary layer íf kr (< v/2KM, i.e. kT KM,/V << 0-5-
Pasciak and Gavis (I974) arbitrarily chose kT KM/V <
0.56 as indicative of limitations imposed by the
diffusion boundary layer, and concluded that a number of
species of phytoplankton (both motile and non-motile)
could suffer unstirred layer problems.
For shapes other than spheres, Pasciak and Gavis
33
(1975) incorporated a shape factor into the equation for
P'. They predicted P' values for nitrate and nitrite
uptake by the marine diatom Ditylum brightwellii
(roughly cylindrical cells, about 150 /^ Iong and SO¡m
in diameter) in a stagnant solution and found reasonable
agreement between theory and experiment. They
demonstrated that diffusional constraints were lessened
in shaking solutions, or in solutions subjected to known
rates of shear; however, they did not compare observed
with predicted P' values under these conditions.
The effect of transport through the boundary layer of
a macrophyte (Macrocystis pyrifera) was examined by
Wheeler(1980). In this case, the fluid velocity was
carefully measured and it was also deduced that flow
became turbulent at quite low velocities (10 ,n. "-1)because of the roughness of the Macrocvstis blade.
Based on calculated average unstirred layer thicknesses
( see p 20 ) and using another variation on the Briggs-
Maskell equation, Wheeler obtained reasonable agreement
between predicted rates of photosynthsis and those
actually observed.
Diffusion boundary layers are also important in
animal physiology and in the study of membrane
permeability, but these aspects will not be discussed in
detail. The many articles by SaIIee' Thomson and
Dietschy (see Thomson, L979þ) illustrate the point
for uptake in the intestine. Barry and Diamond (1984)
34
have recently reviewed many of the studies on membrane
permeability where unstirred layers are critical.
(ii) This work
This thesis is an attempt to quantify the limitations
imposed by mass transport from the bulk medium to
reactions involving phosphate, methylamine*, 02 and Co2
in a few aquatic macrophytes. In aII cases, the
geometry is that of a more or Iess flat plate,which
conveniently gives rise to the simplest equations. For
phosphate and methylamine, the "reaction" step wilI be
assumed to be the membrane transport reaction
Rout:Rin, in which the rate of the forward (and back)
reaction can be studied at equilibrium using tracers.
Such reactions frequently display first-order Michaelis-
Menten kinetics, although this gives 1itt1e information
about the mechanism of the process (Stein, 19BI). Each
mechanism gives rise to its o$7n definition of "KM"; here
it wilI simply be taken to mean the intrinsic half-
saturation constant for transport across the membrane in
the absence of other transport limitations. In the
case of 02 and CO2, the reactions (of respiration and
photosynthesis respectively) are more complicated and
the kinetics may not be simple first-order, or even
*tut"thylamine is not a nutrient, as def ined on p.1,but itis an anologue of ammonia in so far as it probablyshares the membrane porter for ammonia (see MacFarlaneand Smith, l-9B2).
35
pseudo first-order, Michaelis-Menten.
however, wilI be crossed laterr âs the
These bridges,
need arises.
36
MATERIALS AND METHODS
(i ) Plants
A number of aquatic plants, both marine and
freshwater, were studied. Many of the experiments were
done using the marine alga UIva ra ida c. Agardh (Fig.6)
collected from pontoons at West Lakes, South Australia.
The plants were stored in natural ' aerated sea water in
a constant temperature room (l6oc) with four cool white
fluorescent tubes for illumination. These provided a
photon flux density at the surface of the sea water of
70 umol (400 - 700 nm) m-2 s-I and were set on aL2]nI
on/ l-2h off cycle. Under these conditions the plants
remained healthy for six weeks or longer. Clumps of the
sea grass Amphibolis antarctica (LabiII.) Sond. &
Aschers ex Aschers (Fig.7) were stored in a similar way
but were generally used soon after collecting. These
were clumps that had been washed up on beaches at Grange
and Semaphore, South Australia, after storms. The
freshwater macrophyt e Vallisneria spiralis L. (Fig.B)
was grown in Iarge plastic tanks in the Botany
Department. The tanks contained about 50 I of deionized
water, to which was added KCI' CaSO4 ¡ KH2PO4 and NaHCO3
to give final concentrations of 1 mM' 0.05 ilM,0.6 mM
and 1 mM respectively. The plants were rooted in a 70 -
B0 mm layer of mud, collected from the River Torrens.
Light was provided by two "Gro-Lux" fluorescent tubes
giving a photon flux density of about 40 ¡mol
(400 - 700
31
(a)
mucl inous c,¡ti¿le
(b)
5ol^
celI wolIcytoT with o sìngle chl"roplost-
XIGURE 6. (a) Stetch of habit of U'¡feldA, about half natural size. Atintervals along the nargin of the Íntact thallus there arenicroscopic spines.
(t) l.s. of thallus (aiagra^matic). Íhe chloroplasts areconvolutecl and so appear in several. places in each celIin transverse section.
(al(b)
(e)4^^
t_9o
(d)o¡( jØce
ç \orcr^nø)
VascL^ Iarbundle,wÍth4ibres
qcunqe.
5o¡n
eoidernl,ql'¿e[(s
Pal.enchqóaI celtgJ
vascularbundle
@
Ð@ @ @@@@ Þ ,@@ Þ @a
+h,¿k out-er.=11 \^/a(t
FIGURE ?. (a) stcetch of habit of antarctica - natural size.A
(¡) ¡ single leaf, ^'twice natural size.(") t.S. of leaf ; zone diagr¿unneo
(a) l.S. of leaf showing cell d.etail. The epidernal cells arectensely cytoplasrnic with many chloroplasts. The parench¡rma cel1shave a very thin layer of cytoplasm, containing chÌoroplasts,and have thin walls.
te2..- ,- t-.t..l¡
rD- 'î -a..r
I
1
II
It,,,
(d)
a¡'í=
(b)mê-5ê ¡hy(( with
lqct4nae
2æ rw
+h¡n cutic\e
*hick oqferce.t[ wqll
ai( s?ac€([qe,.inq)
@>
ll þr*FIGURE 8. (a) ttalit sketch of V.spiralis. Leaves grew 1 - 1.5 m long in the
culture ta¡rks and. upto 20 mn broad.(t) t.S. of teaf; zone diagramme.(c) Ce11 detail fT,.S.). Th" epiclermal cells contain tlense,streaming cytopÌabm ána nany chloroplasts'
IIt
ttI
nm) m-2 s-I at the surface of the water. The lights
were again set on a 1-21n on/ 12h off cycle.
In the experiments with Vallisneria' segments or
disks were cut from the mid-section of healthy leaves,
i. e. avoiding the region near the base or the apex.
When the leaves were in poor condition, the bathing
solution infiltrated the air spaces of the Ieaf when it .
was cut (van Lookeren Campagne, I955); these leaves were.'
discarded. In Ulva, tissue was also cut from the mid-
region of the frond, avoiding the margin and the
rhizoidal region at the base of the thallus. Individual
leaves vüere used in experiments with aqphibolis; younger
leaves were usually chosen as older leaves were often
covered with epiphytes.
( ii ) Solutions
Experiments with marine plants were done using either
natural, filtered sea water (FSW) or an artificial sea
water (ASw). FSW was prepared by filtering sea water
collected at Grange, S.4., through a Whatman GF/C
filter. This removed any particles larger than 1.2 ¡tnwhich includes all but the smal-Iest bacteria. The FSW
was stored at AoC in the dark. Chlorinity \^/as 22.Beoo ,
determined by the Mohr titration (Strickland and
Parsons, 1965). Total inorganic carbon was estimated
from the alkalinity (Strickland and Parsons' 1965) to be
2.2 mM. ASW was deionized water containing 49O mM NaCl,
3B
10 mM KCl, II.5 mM CaCl2, 25 mM MgCl2, 25 mM M9SO4 and
2.5 mM NaHCO3 (Findlay, Hope, Pitman, Smith and Walker'
1971 ) . The chlorinity is 20.3e"o. The medium f or
Vallisneria was an artificial pond water (APW) as used
by Smith (1980) in his experiments with Elodea (and
Chara). It contained I mM NaCl' 0.1 mM K2SO4 and 0.5 mM
CaSO4, plus NaHCO3 to give a final concentration of
2 mM. UsualIy the ASW or APW was buffered with a
zwitterionic buffer, at a pH no more than 0.6 pH units
away from the pKa value of the buffer.
For measurements of photosynthesis and its
relationship to the inorganic carbon concentration, ASW
or AP-I^J was made up with no NaäCO3. To ensure that the
solution was completely carbon-free, the pH of the
solution was taken down to 4 or lower where virtually
atl of the inorganic carbon is in the form of COZ. The
solution was then flushed for at Ieast 30 minutes with
CO;-free nitrogen after which the pH of the solution
(stilI under nitrogen) was adjusted to its proper value
with a freshly prepared NaOH solution. The nitrogen was
scrubbed of COZ by passing it through soda 1ime,
concentrated NaOH and deionized water in that order.
The solution vvas kept under nitrogen until it was ready
to be used; sometimes this was with the solution
illuminated and containing pieces of plant material, a
system which proved to be a very effective "COZ -
scrubbert'.
39
( iii ) Respiration
The uptake of O2 during dark respiration was measured
polarographically using a Rank Bros. O2 electrode
connected to a Rikadenki chart recorder. A full scale
deflection from zero on the chart recorder corresponded
to a solution in equilibrium with air i.e., at 25oC,
236 uM Oc for f resh water, I93 üM f or ASW and IBB ¡rrM f orl'//
FSW (Table 6 in Riley and Skirrow, L975). Zero 02 was
obtained by adding a smal I amount of sodium dithionite
to the solution. Temperature was controlled (usually at
25oC) by pumping water from a thermostatted water bath
through the water jacket around the electrode.
Plant tissue was placed in the electrode chamber as
slices, small pieces or in special holders (see Fig.9).
When small pieces of plant material, or slices of
Vallisneria, were used, â disk of coarse nylon mesh was
positioned inside the chamber to prevent the pieces
touching the stirring flea.
The holders, shown in Fig.9' allowed large (about 20
x 15 mm) "slabs" of leaf or thallus to be placed
vertically in the electrode chamber. For the type A
holder, the tissue was simply cut to size and placed
between the wire supports. In type B holders, the
tissue was cut to a sLze sfightly smaller than that of
the holder, sandwiched between the two sides and the
edges sealed with stop-cock grease or dental wax. The
40
þ rs'"1
sto¡nless steet wire svTpc¡rîs
zo rYìrn
nylon rnesh
wir€ " le3s" # ?"Y!de e¡ace'-Por stirrin3 tteq
b, -7O -tôO t'{ m-r áiqme+er ho(es
t",f-s ff rubÞr þon"ls
vinSf shee? or Þe.rs9'ex rvarious +hicKnesses
g(o) and Bß) z o'5J rn'^"'
ts(|") : L?3 ñF1
b(7) 1 ['q] mm
O(a) ' 7'oa At-.
l5 mm
A m,q
sPqce +ors¡'ring fteq
tr'IGURE !. Hold.ers for plant tissue in the oxygen electrode chamber.
whole assembly was held f irmly together by two smal-I
rubber bands. Usually one side of the holder consisted
of a greased piece of solid vinyl sheet; in some
experiments the two sides \^rere identical. For type B
holders, the surface area available for diffusion in the
boundary layer is less than that of the piece of tissue.
In the type A holder, they are assumed to be the same.
Type B holders were a means of varying the thickness of
the hydrodynamic (and diffusion) boundary layer without
changing the stirring conditions for the O2 electrode.
The thickness of the unstirred layer was also varied
by taying sheets of lens tissue or filter paper over
both faces of a piece of Ulva. The assembly was held
together using two type B(O) holders. The thickness of
the Iayer was estimated by adding together the
thicknesses of the single sheets, as measured under the
microscope. The value agreed tolerably well with the
thickness obtained using outside calipers and/or the
displacement of water. The actual length of the
diffusion pathway will be longer than the thickness of
the layer(s) because of tortuosity. The diffusional
flux wilI also be decreased because the pathway is not
entirely aqueous.
To measure respiration as a function of the external
02 concentration, the solution in the 02 electrode
chamber was allowed to equilibrate with air and then the
plant tissue \^/as added. The chamber was sealed, and the
4L
oxygen concentration recorded as a function of time.
The flux of oxygen hlas determined from the slope of the
recorder trace at various points.
Sometimes bacteria contaminated the solution in the
chamber, and their respiration could be quite
signiiicant at the end of prolonged runs. To correct
for this, ât various stages the plant tissue was removed
and the respiration of the solution alone was measured.
Then the solution was replaced with fresh solution which
was bubbled with N2 gas to Iower the O2 concentration to
the required 1eveI. The plant tissue was then replaced
and the run continued. The respiration of bacteria on
the surface of the tissue itself, however, would not be
corrected for by this method. The resul-ts, after
correcting for 02 uptake by the solution alone, are
expressed as nmoL oz m-2 (tissue) s-I.
(iv) PhotosyntLresis
Photosynthesis vvas measured bot-h by OZ evolution,
using an 02 electrode, and by IAc fixation. For the 02
evolution measurements, the set-up was the same as for
respiration. Light was from a slide projector fitted
with a I50 W quarLz todide bulb and the plant tissue was
either oriented face-on or edge-on to the light beam
with a curved aluminium reflector placed behind the
electrode chamber. In aII cases bar onef the liqht
intensity was saturating for photosynthesis; the
42
exception was
did shade the
the thickest perspex holder, B(3), which
tissue quite markedly.
To measure photosynthesis as a function of the
inorganic carbon concentration, the experimental
solution vvas scrubbed of Co2 by the method described and
an aliquot (: - 5 mI) pipetted into the 02 electrode
chamber. The plant tissue v/as added and the chamber
sealed. UsuaIIy a 5 - I0 min equilibration time in the
dark was allowed before the light was switched on. In
the light there \^/as often a burst of photosynthesis
probably due to inorganic carbon remaining in the free
space of the plant tissue. The rate of 02 evolution
then, usuaIly, declined to zero' corresponding to the
inorganic carbon compensation point.
There were a number of exceptions. Sections of
Vallisneria leaf continued to evolve O2 rn the chamber
for six hours or longer, the rate decreasing only
slightty with time. In some experiments with Ulva'
which were carried out in buffered ASW containing
sulphanilamide, (an inhibitor of carbonic anhydrase-
Mann and KeiIin, 1940) and para-amino benzoic acid (laee¡,
there was net 02 uptake in the light. Otherwise, 02
uptake in the light was never observed with Ulva, but it
sometimes occufred with pieces of Vallisneria epidermis
and Amphibotis leaves. The only other exception was in
one experiment with Ulva swarmers. These were zoospores
43
or gametes (not identified) that were released in the
aquarium sea water two days after collecting the alga.
Batches totalling about 250 mI of sea water were
centrifuged in a bench centrifuge at 3r000 r.p.m. for
5 min. The pellets were gently resuspended and
combined, giving 3 ml of "swarmer concentrate". 400 /t,of this concentrate was added to 3 mI of buffered ASW
(pH^5.5) in the 02 electrode chamber; under these
conditions photosynthesis was completely saturated by
inorganic carbon because of its high concentration in
the sea water of the swarmer concentrate.
Subsequently the concentrate was purged of COZby
taking the pH down to 5.5 and bulobling with CO2-free
N2 under light. Samples of this suspension then behaved
as described in the previous paragraph, even though
rates of respiration were extremely high (cf. Haxo and
Clendenning, 1953) averaging nearly 702 of net
photosynthesis.
With net photosynthesis at zero (or below) a small
volume of fresh NaHCO3 solution (I' l0 or I00 mM) was
added using an automatic pipette, and O2 evolution
quickly reached a steady rate. The concentration of 02
in the solution in the electrode chamber was not allowed
to go much above 602 of the air equilibrated value¡ for
any particular run this generally enabled two
photosynthesis measurements to be made at two different
inorganic carbon concentrations.
44
After the photosynthesis measurements, the light was
switched off so that respiration could be measured.
Respiratory O2 uptake in UIva sometimes showed a post -
illumination burst while Vallisneria displayed the
opposite effect 02 uptake after switching off the
light was zero at first but gradually increased to a
steady ( low) value. In the type B holders there was a
slow (depending on the thickness of the holder)
transition to net OZ uptake, i.e. net O2 evolution
continued for some time af ter s\^Iitching of f the light.
For these last two cases, then, the respiration rate
immediately upon switching off the light was impossible
to determine because of the slow transition to a steady
rate; this h/as probably because of the large volume of
stagnant solution associated with the holders or the
thicker tissue of VaIlisneria, compared with UIva.
It is important, however, to know the rate of O2
uptake during net 02 evolution so that it can be
corrected for (i.e. so that true' or gross,
photosynthesis can be determined as opposed to the
apparentt or net, rate). It is a matter of debate as to
whether or not mitochondrial respiration is affected by
the events surrounding photosynthesis (Graham, L9l9¡
Singh and Naik, 1984). The problem is also complicated
by the fact that there may be recycling of the CO2
produced in respiration (and photorespiration). The
efficiency of recycling wiIl itself depend on the
45
unstirred Iayer thickness. Complete recycling of
respired ano photorespired CO2 would mean a CO2
compensation point of zero and net and gross
photosynthesis, for aIl intents and purposes, would be
the same" Partial recycling would mean a net rate of
photosynthesis somewhat smaller than the true rate' but
not as small as one would expect, based on the rate of
dark respiration, photorespiration, and no recycling.
In the results, I have assumed that the initial,
steady rate of 02 exchange in the light is indeeo for a
[cO2J of zero. Where net 02 exchange v/as zero (the
majority of cases) complete recycling of respired and
ichotorespired CO2 L s irnpl ied.
The pH of the solution was determined at the
beginning and end of a run, using a Philips CAII
combination glass electrode connected to an Orion 70lA
digital pli meter. The pH change was generaIly small
and maintlz due to the NaHCO3 additions. The final pH
was therefore the best measure of the pH during the
period of photosynthesis.
Results are expressed as nmol A2 evolved m-2 (tissue)
s-I or as inol 02 (g chlorophyrl )-I s-l ot, in some
experiments with slices where neither surface area nor
chlorophyll were measuredn as a percentage of the
maximum rate. Normally only one photosynthesis
46
measurement was done at any one inorganic carbon
concentration. Sometimes more than one determination
\^/as made and error bars represent n"'aximum and minimum
values if the number of measurements was two, or the
standard error of the mean if the number was greater.
Photosynthesis was also measured by the fixation of14c-Iabelled inorganic carbon. For experiments
involving the "stirring gradient tower" (see later),
disks of U1va were used, 7.5 rnm in diameter cut with a
cork borer. I{ith Vallisneria, oblong pieces 15 x B mm
were used. The disks or pieces \^/ere left aerating
overnight in ASW or APhl. The following day, a sol-ution
with the required concentration of inorganic carbon was
prepared and gently decanted into the tower which was
then loaded up with the sieves and plant tissue. There
were L2 disks of Ulva or six pieces of Va11is4çr:þ to
each level. The tower was sealed and the whole allowed
to equilibrate in the J-ight for 50 60 min. Af ter this
pretreatment, the solution was pou-red off and replaced
by a solution labelled with IAc (- 0.1 /LtCi mI-I;I
saulnao, supplied by Arnersham). This was done as
quickly and gently as possible, to minimise exposure to
atnrospheric COZ. The plant tissue vvas allowed to f ix
the \Ac-Iabelled inorganic carbon for t0 min, and then
the sieves were taken out and placed immediately in 100
mM nitric acíd to prevent further carbon fixation.
Batches of four disks (or two
47
VaI I i sneria pieces) were
subsequently removed to scintillation vials containing I
mI of 100 mM ÍINO3. A sample of the experimental
solution was taken for a final pH measurement, and then
the whole apparatus was thoroughly rinsed and the
experiment repeated at a different inorganic carbon
concentration or pH.
The scintillation vials containing the plant material
h/ere Ieft open in a fume hood overnight to aIlow the
release of volatile carloon. The next dty, 10 r¡I of
scintillation cocktail (PPo/PoPoP,/toluene/detergent) was
added and the vials counted in a Packard Tri-Carb
Scintillation Spectrorneter. Quenching b)t chlorophyll
was corrected for by the channels-ratio method
( Herberg,1965 ). After each experiment, 50
were taken from the radioactive solutions
specific activities (cts min-L/moL) could be calculated.
Results are expressed as nmol carbon fixed m-2
(tissue surface area) "-1. Error bars in the figures
represent the standard error of the nean of three
samples. usually, tissue of known surface area was afso
weighed so that results could be expressed in terms of
fresh weight. Sornetimes, chlorophyll was also extracted
(Arnon, L949) .
(v) uptake of Selhylamine a.nd phosphate
rnfluxes of IIac]methylamine and l32plphosphate were
determined in the "stj-rring gradient tower" (see later)
TLsamp Ies
thatso
4B
in a similar \,,ray to inorganic IIaC]carbon f ixation.
Methylamine was added as CH3NH3CI and phosphate as
KH2POA. At the end of a I0 - 15 min period in
radioactive solution (^' 0.Iyci/nl-), the plant tissue was
washed for 2 - 5 min in non-radioactive solution to
remove label from the free space. After rinsing, the
tissue was lightly blotted, divided into subsamples and
placed into scintillation vials containing 10 ml of
scintillation cocktail, plus I mI 100 mM HNO3. Radio
nuclides \^/ere supplied by Amersham.
In some cases, slices of plant material vüere used.
These vüere treated in the same way as disks or Ieaves
and were divided into three subsamples at the end of the
experiment. The subsamples were placed into weighed
scintillation vials and the results expressed as nmol
g-I (fresh weight) s-1. Knowing the weight of a piece
of tissue of known surface area, the results could be
converted to nmol m-2 s-I.
(vi) Stirring gradient tower
This is illustrated in Fig.I0. The different levels
of the "tower" had different rates of water movement due
to the stainless steel mesh sieves which decreased the
water velocity of the section immediately above them.
Fig.Il shows the stirring gradient that is set up in
the tower, with the magnetic stirrer on its maximum
49
qfqraíniqrn
o-ri
195 mI^
persFx !.Ín9
l_r ol^l(síphon)
"síevest'
waterjocket
lSnm
ln
ÞefsNX'c¿tinder
(on-coor{-ed stírrîa3 bar
t4o3nefrL slírrer
I'IGURE 10. Apparatus for obtaining a stirring gratlient in a solution'liäfttì"" providecl from ttre sicle by a slide projector witha fó w q"ãrtu iodi<te bulb, 81vinç a photon flux densityof ,^\,6?0 ,1,¡no} (4oo-7oo rrm) ¡-2 s-1 at the centre of thetower. /
5l mrn
Lei¡ef I
Larcl 2
L-evel s
l-:vel 4
Level 6
t ¡Av¿l 6"
r\fn
setting ("8.4"). The measurements were made by timing
the rotation of a match, using a stop watch for Ievels 3
and 4 and a stroboscope for 5 and 6. The water velocity
in levels I and 2 was too low to be measured. As a
check on the accuracy of the method, the angular
velocity of the vortex hlas also measured for each level
when there $/ere no other sieves above it. This produced
a curve of the same shape as that shown in Fig.lIa
although aIt the velocities \^/ere about I/3 again higher.
Fig.ltb is a plot of log (r.p.m. ) against log
(Ievel). The points are a good fit to a straight line
which suggests the empirical relationship
O.TT3 L 4.5 (2el
where L is the level number. This equation is
represented by the Iines in Figs.lla and tIb.
In the Introduction (section I), it was shown that 6
is related to the inverse of the fluid velocity raised
to some po\^Ier, usually between 0.5 and 1. To get an
idea of the actual values of 5 in the various levels,
13 mm diameter disks were punched from ztnc foil (380 ¡mthick) and their rate of dissolution in HCI was measured
at each level of the tower (3 disks,/Iever ). The acid
solution also contained 50 mM KNO3 to act as a
depol arizer and prevent H2 evolution. King and
Braverman (L932) showed that the dissolution of rotating
r. p.m
50
(ø)4æ
300
7,5
L,O
tu)
èiLlr"S
tloo
o 65óLeVel ( L)
ê
t
Sù\o'\ì
t'5
t.o o'5 o'6 o.7 o.z
,'bg L
FIGURE 11. (a),qngufar velocity, U, of water in rlifferent levels, L, ofthe stirring gradi,ent tower.
(t) efot of logU against 1og L. ttre slope of the line is {.1.
zinc cylinders in HCI solutions is rate limited by the
diffusion of HCI in the boundary layer at speeds upto
5r600 r.p.m. Before the experiment, the disks were
etched for I0 s in 5 M HCI' washed in deionízed, water,
then in ethanol and finally blotted and weighed.
Batches of disks not being weighed were kept in
deionized water; the rate of dissolution was
immeasurably small under these conditions. The disks
were removed at various intervals and the change in
weight plotted against time. The change in surface area
was negligible.
Results are shown in Fig.L2 a, b and c. The weight
changes were non-Iinear with time, which may have been
due to impurities in the zínc foil reacting with the
HCI/KNO3 solutions and producing an impervious coating.
However, the initial rates allow kt to be calculated for
each level of the tower, fot the three concentrations of
HCI used. These calculations are shown in Table L,
together with the inferred unstirred layer thicknesses
(both sides of the disk). Fig.I3 shows a plot of log kt
against the log of the water velocity, U (calculated
from equation (29) for levels 1 and 2). The points fit
well to a straight line of slope 0.35, which suggests
kT- U0'35. Low powers on U (i.e.< 0.5) have been f ound
by some workers (Sackur, 1906; Eucken, L932¡ Trümpler
and Zeller, 1951); they would appear to indicate "sub-
Iaminar" flow. Perhaps in my case the interstices of
5I
the stainless steel mesh trap pockets of fluid which do
not mix with the bulk solution, giving rise to a weaker
dependence of k, on fluid velocity than is predicted for
pure laminar fIow.
TABLE 1
Values of the transport coefficient, kT, from theinitial slopes of Figs. l2a, b and c. The lastentry in each column is the mean value, + thestandard error of the mean. Also shown are thethicknesses of the equivalent Nernst layers for HCIassuming, at the ionic strength of theseexperiments, a meRn ioniç diffusion coefficient forHCI of 3.07 x I0-v m¿ s-r (nobinson and Stokes,1959 ) .
x 105 (m s-1)
IHCI I (mM) Level I Level 2 Level 3 Level 4 Level 5
5
10
20
0.275
r.35
1.16
L.92
2 .63
r.94
3.75
5.28
3 .97
5.17
8.39
4.89
7 .40
10.5
7 .6r
0.928+ 0.21
2.L6t 0. r9
4.33t 0.39
6.15+ 0.92
8.50+0.82
b (each side of the disk) for HCl ( um)
33t+ 75
L42t II
70.9+ 5.9
49.9t 6.s
36.1+ 3.2
52
5o
4o
þ
b) a (c)
o
Aa
I
a
^
A80
(a)q)Ë
-ìJ
.ÞIqJ(lrõ
-drJ
30o
rc20
6o
1o
?þ
A
¡A
Ia a
Â
^
A
le-ao
--oo oto
0
loI o
o-o a^A
lo ?o lo õo lo 2o
FIGIIRE 12. Rate of d.issolution of zinc disks in the five levels of the stirring g.radient tower, (e) tevel 1, (o) level 2,-- (.j låve1 l, (a) Ievel 4 and (o) rever 5, ín (") 5 ru, (¡) to mM a¡rd (c) zo mM Hcl . 25oc.
^
1oôoo 2ötin-e @îru)
I
\--a
o.5
SIGURE 13. Relatlonship beùreen t1 (Utfe 1) anil U (ftg. 1.t) in tt¡estlrring graclient torer; the slope of the 11ne is 0.35.
ÐY
o LILþs
MEMBRANE TRANSPORT
I. Uptake of 14cl Methvlamine bv Iva riqida
(i) Re sul ts
Figures L4 - I7 present the results of a number of
experiments on the kinetics of II4c]methylamine* influx
and the effects of stirring. In Fig. IAa, the
experiment was conducted by the method described in
MacFarlane and Smith (L982), with and without shaking.
For comparison, Fig. 7 of MacFarlane (1979) is shown in
an inset. The shaking rate was such that additional
hand swirling made Iittle difference to the rates of
uptake. In Fig. 14b the second, Iinear phase of the
"stirred" curve (shown in Fig. I4a as a dashed Iine) has
been subtracted; the sotid Iine represents the
Michaelis-Menten equation with KM = 20 ¡M and
v - 7O nmol *-2 "-1.
Figures 15, 16 and I7 represent a number of
experíments using the stirring gradient tower. These
experiments were done at pH 1.3 - 7.4 so that CH3NH2
influx would be negligible. At high concentrations of
methylamine it appears that stirring sometimes reduces
the rate of uptake. At low concentrations, CH3NH3+
influx is markedly greater from a stirred compared with
* "Methylamine"base (CH:NHe )
distincti.on'i=written.
refers non specifically, to both freeand conjugate a'cia (CH?NH;+). Where theimportant, chemical fórmúlae wilI be
53
¿.- 1.¿öl
.' ø-lara
¿
lto
@.)
(b)
,'/¿r¿l¿0.lal
loo
8o
6o
1o
þ
oö
a¿â.-+
a 6û0oo t
þarl
o
to'
It',se¡.
1.5
from stirred (o) anttvarious concentrations
Þ2ßI
r)
sË
\¡oË
€
x5
$:'c
rr¡
o
0.
0.5
l.oo
I
6o
6o
40
20
+o
/
nefhyhninel'o
.c,n.uÀ+rofioa (mM)1,5
FIGURE 144 antt Inset: Methylanine infh¡< in U'ri€id'?r¡nstirretl (o) asw (pE B '2 - 8'3, TAPS) atof methYlarnÍne.
b: ftre 'rstirred'f curve above with the second, linearr.ph?""^^ ,subtracted. T = 27oC. ([ = 20'c fOr results shown In Inse!./
10u^
II
ö
+
t+
+
-Ir/^)
{Ë
ìÉe
XèTl>,s
300
I+
o
+o
fl
^EOto^^
o 5onetþghnine co\&rttrat',c.rv (fM)
læ 150
FIGIIRE.'.I]*]ä"rTlÏ'å"tåii'irITîi"ts"Jå*ä"tö'$ï:":ilfIil""I:Ë,P*'"=:'ìHÍ,i;,;H3,""i=î:3'Ïå#¡.'-2)i"Stirrer oû no. J.
+
¿
t
?o
^
o
?
oA
t 5oo
rYt
É
o\(
x-- 7oo
XJ*F=.
loo
++
ßI
goII
5o loomeAhylarncne conçe-rctra'tioru (yUl
FTcIIRE 16. f 4clnetbvra,nine infrux vs. concentration for II.TI€id+1!isk9 -Q\'L: 2.1 I fresh leight t-2) in ttre five levels oft¡e stirring grartientiorã" ("yrtãr"-as in rie-i5)-. rw + 10 nil'tttES, pE 7.3o, 25"c, stirrer on no. 5.
t5ç
+
o
¿
îto,1, 4cþt
E!oEÈ
I
+
o5zØ
-l\-.J
^
Â
I
o
^
ÀI
oA
5o too f5o z.00
and slices in thein level 5. ASW +
neehylarníne- ænæntrø.tLoru (ffl
FIGURE 17. F4C]n.tnyfamine inftux vs. concentration for V.rigida disks (¡8.4 t 3.1 g fresh weight n-f)stirring graclient torå" i"yr¡ãi" as in tr'ig. 15)-iG'çere in leve1s 1 - 4, slices (fnset)10 mM TES, ptrf 7.41, 25oC, stirrer on no. J.
o
200
Ìvlõrlc'.g)g.Ê-ùbÌSto5s
5
2
I
t+
tæ
!sot
a nearly stagnant solution. The difference in the
influx between leveIs 3, 4 and 5 is comparatively
slight.
For the experiments shown in Fig. 15 and I7 the
magnetic stirrer was set on "7" ( rr 420 r.p.m.). In
Fig. 16 it was set on 'r5rr (N 250 r.p.m.).
( ii ) Boundary layer
Maximum rates of CH3NH3+ uptake by U. rigida v\¡ere
found to be quite variable by MacFarlane and Smith
(L982), and possibly affected by the overall nitrogen
status of the tissue (cf.Wallentinus, 1984). In Fig.
LAa if the second, linear phase of the curve represents
CH3NH2 influx then the maximum rate of CH3NH3+ uptake is
very low (cf. inset and Figs. 15,I6 and t7). Inf1ux is
scarcely affected by stirring in this case which is in
agreement with the Briggs-Maskel I equation. This
equation predicts that the extent of diffusion
Iimitations in the boundary Iayer depends on the
relative s tze of v/KM and 2kt (p 9 ); these two
ât€ presumably similar in this case. If diffusion
limitations are slight, then Fig. IAa allows an initial
estimate of 20 /t for the true K, of CH3NH3* transport
(see Fig. I4b).
(iii) fhre effect of stirring at high methylamine
concentrations
Fig. I5 in particular implies that CH3NH3+ influx is
Iimitations and V
54
adversely affected by stirring at high methylamine
concentrations. MacFarlane and Smith (198+) suggested
that this reflected a decrease in the observed V due to
CH3NH2 efflux, which would become more rapid as the
unstirred layer thickness decreased. They corrected for
this by bringing aII estimated V's to the V estimated
for the least-welI-stirred solution (Ieve1 1).
In Fig. L7 increased stirring does not lower influx
at high external concentrations which suggests that the
effect may be associated with my experimental technique.
This probably has to do with the order in which the
disks were placed into their scintillation viaIs. After
the five minute rinse in non-radioactive solution, the
disks were Ieft in their sieves resting on damp paper
towelling, and removed thence to their vials. In Fig.
15, the disks from leveI I were removed first and those
from level 5 last. In Fig. L7 the order was reversed
f or al l concentrations, i.e. disks f rom level 5 \^rere
removed first, level I disks last. AIso in this
experiment the sieves \,vere not lef t resting on the damp
towelling. In Fig. L6, Lhe order of removal varied..
The time delay between first and Iast was of the order
of five minutes and so a significant amount of
intracellular tracer could have been lost to the paper
towelling in that time. If the amount Iost were a
constant fraction of the amount of tracer present, then
it could be corrected for by assuming that the true V in
55
Fig. 15 is the same for each level and that this V is
given by the disks which were first removed to their
vials. This correction would be the same as the one
carried out by MacFarlane and Smith (1984), but for a
different reason. However, in view of the uncertainty
associated with these results, I wiII tend to ignore
them in subsequent discussion.
(iv) The saturation of ilnflux with stirring
CH3NH3+ influx is not greatly increased in Ievel 5 as
compared with level 4 or even level 3 (Fig. 16, I7), yet
the experiment with zinc disks (Materials and ttlethods)
shows that the thickness of the unstirred layer is
significantly reduced. The saturation must be due to
rate limitation by some factor other than transport
through the external boundary layer. The obvious
candidate is transport through the membrane, and this is
suggested by Fig. 16 where the Kfrpp for the most well
stirred solution is about 20 ilM (cf. the estimate from
Fig. 14b). In Fig. 17 ho\,vever' KftPP in level 4 is30 pM or more and influx stilI appears to be linearlydependent on concentration up to near saturation values,
when there is a sharp transition to the maximum rate.It is possible, then, that transport through the celI
walls, which also wilI not be affected by stirring, is
limiting here. This idea is supported by the data for
cH3NH3+ uptake by slices (inset, Fig. I7) in which Kftpp
56
is signif icantly reduced (xftPn ,.., 1B /^M : inset, Fig I9 )./
Compared with Fig. L6, V is some 50% larqer in Fig. L7,
and the fresh weight per unit area (which quantitatively
is related to thallus thickness) is higher by 54>"; both
of these would Iead to more significant diffusional
limitations due to cell waII.
(v) Comparison of observed with predicted kinetics
The experiment with ztnc disks (Materials and
Methods) gives values for the mass transfer coefficient,
kT, ivith the magnetic stirrer on its maximum setting
("8.4" -= 6I0 r.p.m.). For the experiments shown in
Figs. 15 and 17, the stirrer was on tt7tl (- 420 r.p.m.).
Since k,.* U0'35, all the values for kT will be
decreaseo by (420/610¡0'35 = 0.878. In Fig. 16, the
stirrer was set on 5 (= 250 r.P.m.), giving predicted
kT'= 0.732 times their value in the ztnc disk
experiment. kT wiIl be further decreased because of its
dependence on the diffusion coefficient. For the ionic
strengths used in the zínc disk experiment, D for HCI is
about 3.07 x 10-9 m2 s-l (Robinson and Stokes, 1959)
while ¡CHcxHäin sea water is about 1.16 x r0-9 m2 s-l
(Tanaka and Hashitani, I97L, and using D oc molecularI
weight -z). Thus, kT will be reduced in proportion to(I.Ì6 x I0-9/2.07 x rO-9)å (equations (I6) and (IB))
which is equal to 0.523. The predicted kt's f or Fig. 15
and I I are the n 0.426, O.gg2, L.g9, 2.82 and 3.90 x 10-5
m s-I and for Fig. 16,0.355, O.B2'l , L.66, 2.35 and
57
+
t 60o
þo
loo
o
rlI
É
oE(
:.-, 7OO
XJ-¡\F=.
o
ol
A
5o loomeahyl arnLne conce.&Ta'tí oru (yut
ggs-L[askell equation (equation 12). Irorn-Ieft to right, values of5"r
"-1. I(¡¡ = ,Up, i = +ZO mrol'n-2 "-1. ftre s¡rmbols are the
t5o
FIGURE 18. Ir¡flux against concentration curves for the 3rikq are: 3.25, 2.35, 1.66, 0.827 and. 0.355 x 10-d-ata of Tig. 16.
i,nfL
ux (r
vnd
,rt-
^ s-
')
ËÈ
¡
o
oo
nnoL
gnqc
dtw
$ât)
s'I
N(n
.O
tz lo lm l-1
8
I
o
ù
ì o t .s \s ñ= 3 5' o c) b ñ I ct * (-t o P
--ç' ë,
rd H 6) d !d td J vF
9FO
Þ
l-')
1\)t
sØ
O\É
Elx
o'
ctO
Jæ tsos
trr
¡,1P
.r.
¡E0¡
oH
EG
+o
ú!o
<t
loF
f r5
O.
o(a
o '# å
OH
f+¡ ll
At cl-
hd ¿
l-,
'P
.æO
@:-
Þ
.!H
AJç -t ar
r-J
. tr
4È
o ø(* ll
o r-J
o\ (r+
O\ã o
5 Etd
o l-J
PP
.0q 0q a,
rùl
(r¡
g)llo JÞ
I¡G
tsH
F|
i' U' oo c+
'O..É
s)Þ
l<+
ÉP
. .o ùÞ 8ã -P.
50q
ÉÞ
tilÊ -
(D-
oo >g 3 'v <\o ilI w f
\)
Noo
U'J
\):l þr O
oH
\O \o@
rT
l^,ì- F
۟t
t\)
I rst
r5ø
(}l o B N 0
3.25 x I0-5 m s-1 (levels I to 5 respectively).
PIots of the tsriggs-I{askel I equation using these
values of k, and K¡4 = 18 FM are shown in Figs. lB andI
19 with the corresponding data of Figs.16 and L7. In
Fig. lB there is good agreement between theory and
experiment. Fig. l9 shows that the Br:iggs-Maskel I
equation tends to over-estimate the influx at Iow
methylamine concentrations, which might be due to
transport restrictions in the ceIl wall.
If CH3NH3+ porters are distributed evenly over the
surface of the membrane then transport through the cell
walI will be not only in series with membrane transport,
but also in parallel with it. The two situations are
illustrateo. in FiE. 20. The thallus can be thought of
as a reacting slab of the thickness of region II
surrounded b1z the unstirred Iayer of outer cell wa11 and
thin mucilaginous cuticle (region I).
plasmalemmaI
II
Fiq. 20" T.S. UIva thal 1us, diagramatic" I - regionseries with reaction. IIwhere transpo-Es in
transport in paralleI with reaction. Reaction ismembrane transport Ri' s t Rout
5B
The thallus thickness was not measured for the
results shown in Figs. I6 and L7. However, tf the
thickness is linearly related to the fresh weight (and
they are certainly related qualitatively), then the
thickness can be estimated. This gives 51 ¡./m for theI
resufts shown in Fig. l-7 and 33 ¡øn
for Fig. 16. 5 - 7eo
of this consists of outer ce1l wall. In Fig. I1 then,
reqion II is about aB ¡m
thick, and region I about
I.5 ¡,rm. Using a KM for CH3NH3+ influx of 18 þM, V|
'.t J
' /
equal to 27.5 mmol m-3 s-l (660 nmof *-2 =-1) .r,d Deff
in celI wall (and mucilage) half its value in sea
water*, it is possible to use Yamané's equation
(equation V.2I) to calculate ry and thence the expected
influx for various concentrations of methylamine. This
graph is shown in Fig. 2L with the data of Fig. L7.
Biot numbers have been calculated_ by combining the kr'spredicted from the zinc disk experiment with k, for
transport through the two outer celI walls (I.93 x t0-4
m =-I) in the manner of equation (28). The agreement
between theory and experiment is much better when
internal resistances to mass-transport are taken into
account. Yamané's equation is also a good fit to the
* Good measurements of diffusion coefficients in cellwalls are scarce (walker and Pitman (L916). Selfdiffusion coefficients will be reduceo because of thetortuosity of the diffusion pathway and because only afraction of the cell wall is aqueous. A number ofworkers have found a reduction of about 50% for theeffective D in the cell wal1 cf. the bulk medium (xohnand Dainty, L966i Tyree, 1968¡ Pitman, Luttge, I(ramerand 8a11, I974¡ Smith and Fox, L975). There are,however, reports of greater reductions (see Walker andPitman, L976). My estimate of Deff may then be too high.
59
ttod4ûI
É
IFc
!tn!r.J
o
+
o
^
A
5o looneehylarnín¿ ærtælttrv'f,-'wru (f') f50 zPo
FIGüRE 21. Inf}r¡x versus concentration curves bqsed on Ya¡na¡ré's equation for Siot nurrbers of 1.O2, 9.7421 0.391 a¡d 0.173(rert_toright)r.Deff=5.80x10-1012s-1,R=24'ur^,qo=18fla¡rdV=27.5nnolr-3s-1.Syrrbols are-the-daIä'of Fig. 1?. /
I
data of Fig. 16 (again using a stightly
the Briggs-Maskell equation predicts);
internal diffusion Iimitations are not
because of the thinner tissue.
higher V than
in this case,
as severe mainly
(vi) Other analyses
Some of the data h/as analysed using the computer
prograflìme FVKUP developed by N.A. Walker (see Smith and
Wa1ker, 1980; cf. Märkl, L977). This programme fits
the Briggs-Maskell equation to the data by adjusting the
three variables V, KM and k, from initial, guessed
values. A good fit means that the sum of (weighted)
squared differences between observed and calculated
values is very low - this is indicated by a regression
coefficient close to one.
TABLE 2
Parameters in the Briggs-Maskell equation predictedby FVKUP, and regression coefficients, for the dataof Fig. I7 .
Level I Level 2 Level 3 Leve1 4
V(nmol m-2
"-1 )
847 1040 6L7 106
KM(
fM)
-ft* ro-s )
L76 I26 10.6 3I.B
2.L9 6.00 1.39 3.75(m s
Regress ioncoefficient
0.999 r.000 1.000 r.000
60
Table 2 shows some results for the data of Fig. 17;
each of the data points was weighted according to the
inverse of its variance. Calculated KM's vary
considerably and the kt'" do not follow the order
expected in the different levels. In levels 3 and 4,
however, the calculated k,''s are close to those
predicted by the zinc dísk experiment and the K, values
are either side of the 1B gl,t estimated f rom theI
experiment with slices. In levels I and 2, the computer
programme has opted for a relatively Iow boundary layer
resistance and a high K¡4, whereas the opposite is likely
to be the case.
Fig. 22 shows the graphical analysis of Gains (1980).
Fíg 22a is simply an EadieHofstee plot of the data of
Fig. L7, with the estimated V for slices made equal to
that for disks. The deviations from Iinearity brought
about by the unstirred layer are clearly seen (Thomson,
L979a'i cf . Winne, I973). The intercepts on the J/c5
axis are referred to as (l/cr,o'=O; they are the
initial slopes of the J versus c5 hyperbolae ("n refers
to the bulk concentration of methylamine). The intercepts
of the curves in Fig. 22a with various values of J
have been divided by their respective (l/cr,"O=O values
to yield the set of lines shown in Fig. 22b. Eight
values of J \^/ere chosen, ranging from 444 (the Iowest
line), to 55.6 (the uppermost line) nmol m-2 "-I, in
steps of 55.6 nmol m-2 s-1. The points are very
61
aCoo
\o
^
IIrr)
"fS ¡4oo
oE
Eo
At
o
o
\l
\A
!i
\X5
\r-
'Èf
O
(b)
o.3 o'1
^¡
I
o.
\r
(a)
(c)
6
t
6æ
A
o.9
o
a
zr/"ø>< þá (ru s'')
O
oo Ol
õ
2
I
o
AIa)
JÈ
tf)
aXs
\ÙÈ
ñI{l
izoìEa_
o
$rodÈ\it-.;
L
ooo
I
L
o.E o
ooA
tr
to
o
0 70.Ó 4æÍ (nnol m-7 s-()
FIGURE 22. AxøLysfs of the data of Fig. '17 by the method of Gains (lgBO);the symbols in (a) trave the sane meaning as in Fig. 1J, but in(b) ana (o) tney represent different values of J. tr'or furtherexplanation, see text.
scattered, and the straight lines drawn in Fig. 22b are
hardly justified; they are based almost exclusively on
the data for slices and the two most werr stirred levelsof the tower, besides the fact that the quotient of J/cband (J/cb,"r=O at J/c5 = 6 should be unity. The slopes
of the lines have been mu1_tiplied by their respective
J's and plotted against J to generate Fig. 22c. The
intercept on the abscissa is V (653 nmol m-2 =-1)while that on the ordinate (23.7 pmol m-l =-2) is
)V'/KM. The predicted Kpl is then f A¡U. The reciprocal
of kT (tfre boundary layer "resistance") is obtained by
subtracting Xr,/V from the reciprocal of the intercepts
on the J/cb axis (fig 22a). This procedure predicts
kr's of 0.418, O.BB0, I.63 and 2.04 x I0-5 m s I fot
Ieve1s I to 4 respectively (tfre predicted kf for slicesis infinity). There is, therefore, very good agreement
with the kt'= derived from the zinc disk experiment.
However, for the analysis to be at aII reliable when
kt'= are small (i.e. when the apparent K* is far removed
from the true K¡q), one would require a lot of very good
data, particularly at cb's less than the KMapp. In Fig.
22b, if the data for slices were absent, âtyjustification for the straight lines drawn would
practically disappear.
II¡ Uptake of t32Pl Phosphate bv U_Iva srqida
Phosphate uptake by plants is generally considered to
membrane porterbe via, and rate-limited by, a
62
(Schwoerbel and TiIlmans, L964; Loneragen and Asher,
L967; Raven, l9B0; Falkner, Horner and Simonis, 1980).
Over a large range of phosphate concentrations, the
influx versus concentration curve is often complex and
could represent two (or more) separate porters having
dif f erent values of KM and V (l,aties , I9G9); it is al-so
possible that the simple diffusion of phosphate across
the plasmalemma becomes important at high concentrations
(l,aties, 1969¡ Edwards, L970¡ Barber, L972¡ cf . Maynard
and Lucas, I9B2). At low external phosphate
concentrations, however, it appears that the influx is
due to the operation of a membrane porter and thekinetics of influx are of the Michaelis-Menten (Briggs-
Haldane) type (Bieleski, I973; Falkner, Werden, Horner
and Heldt, I974).
Fig. 23 supports this notion showing that 32p-
phosphate influx into disks of Ulva rigida in the fiveleveIs of the stirring gradient tower has a typicathyperbolic response to concentration. However, even
though stirring can increase the phosphate influx by
nearly 2.5 f oId at 1ow cohcentrations, inf lux is much less
affected by stirring than methylamine influx.
In this experiment, the stirrer was on its maximum
setting; therefore the k, values from the zinc disk
experiment are directly applicable, allowing for the
dependence on the diffusion coefficient. Measurements
of phosphate diffusion coefficients are scarce, and
63
¿¡f
Lux
Øm
ol m
'Ls-
')
o -o--
-+1¡
u
->+
V
+J{
+
ÞtF
>o
uì +e
>o
N 0 0\ 0
J* ö rL þ' i's Ð o (¡ \ J Ð õ' 8è I o ID + ð (1. Fe
-s t -l oún
differ depending on the ion species in question (i.e.
H2Po4, HPoî- or eol-). Relatively meagre evidence
suggests that, for the same ion concentration, H2PO4 istaken up much faster by plants than HPoî- if the latter
is taken up at all (eieteski, L973). H2PO4 uptake would
certainly be the least expensive energetically for plant
cells with a more negative electrical potential than the
bathing medium. In U. rigida, phosphate influx from sea
water with a high phosphate concentration is relatively
constant from pH 5.6 Lo 1.9 (rig.24); however, both
tHreo¡l and IHPoZ-l ^ay have been high enough over thisrange of pH to saturate an H2POÃ or HPOî- porter.
Although IH2PO;] is decreasing continually, it is still
about 5 ¡¡M at pH 7.9. IHPOrt-1, on the other hand, isI
increasing to about pH 1.3 and thereafter decreasing;
however there would always be enough HPOî- to saturate')_any HPOí- membrane porter over the entire range of pH.
At pH 8.4 there is a sharp rise in the phosphate influx.
It is very unlikely that this merely reflects the
increase in tPoe-l since one would then have to assume
that no?- influx is much greater than HP}T- influx under
conditions of a much steeper uphill gradient in
electrical potential for the triply charged ion. The
effect of pH is probably more indirect, such as a change
in the driving force across the membrane or in the
degree of phosphate binding in celI walIs.
If H2PO4 is the species taken up by U.
64
ra ida, the
2o
1ot.l
'È
oÈÈ
X3
\î'.S
1o
t ttrIt
tto 5-5 6.o e,5 7'o 7.5 8-o 8.5
PH "f Ashr
FICURE 24. Infh¡x of 3l-pnoephate at varìoue pE's of ASIrt.(unbufferect) forü.riglcla disks (30.9 ! O.27 g fresh welght n-z) ehaken in ae[ãEer 6'eth. Tota1 phosphatc concentration - 5OO ¡W T = 25oc.
correction to kT is approximately (6.88 x 10-10/Z.Ol t_o Z10-v)3, or 0.369. The diffusion coefficient of H2PO4 is
estimated as 6.88 x 10-10 *2 s-l by muttiplying the
value at infinite dilution (9.50 x 10-10 .2 "-lr Gros,
MoII, Hoppe and Gros, 1916) by the activity factord lnÚìT-I+I (Robinson and Stokes, L959), where '{ is the
activity coefficient for H2PO4 and I the ionic strength.
The activity factor was estimated as 0.724 at I - 0.7 M
(sea water) f rom the data of l^thitf ield (1975 ). (tn the
same wày, the correction for f$loâ- becomes 0.147).
Since stirring has a relatively slight effect (at
Ieast compared with methylamine influx), and ignoring
internal diffusion in cell walls, the xftPP for phosphate
influx in level 5 (-t.A ¡u) must be close to the true
KMr. Using KM : I.5 fM(total
phosphate), \,2 = 15 nmol m-2
s-l and the corrected ffizeO4'= from the zlnc disk
experiment, the Briggs-Maskell equation generates the
set of curves shown in Fig. 25. [ureO4-] has been taken
at 2.42 of the total phosphate concentration at pH 1.48
using a pKa'for H2PO¿ i" sea water of 5.90 (an average
value of the measurements guoted in Millero, I9B3).
Clearly the theory is inadequate to explain the
observed effects of stirring on phosphate influx; influx
is grossly underestirnated and the differences in influx
between the levels of the stirring gradient tower are
predicted to be much Iargrer
65
ô
f5
lo
IIvìr{tË
os,
è
xs\.-'
I$I+
0
^
â fo
ol
lo 1o+Þtø1, plvosph æte- cþncerLty6¿tíorv
FIGIIRE 25. Infh:x versu concentration curves predicted þV.the Briggs-M skell equation using3.14, ?.21; 1.6O, O.79? and 0.342 x-10-5 n s-1- (fert to right), K¡¡ (total phosphat
¡n-2 c-1. The s¡mbols are those of Fig. 23.
t
2o 6o -fo
+rn)
klts for EIPOì transPort ofe) = 1 .5 yM aña v = 15 nurol
+
A Iikely explanation for the discrepancy lies in the
fact that most (>97å) of the phosphate at the pH of this
experiment is in the form of HPOî- or pO!-. This wiII
effectively enhance the flux of H2PO4 across the
diffusion boundary Iayer because there are now three
species which can carry 32r, not just the one. The
fl-ux, J, of H2PO4 through the unstirred layer will be
(This assumes that equilibrium between H2PO4, Ètp}î- and2_PO¿ is established rapidly which is true in this caseJ
The concentrations (.n and c
expressed in terms of li2eOI b¡K: K.
trol-l = -#rt
tHreo¿l (eurter, Lg64) where Kurand,= (H+)'K1 are the second and the third stoichiometricd.s
dissociation constants of It3PO4 and (H+) is the activity
of H+ ions. Equation (30) thenbecomes
rH.ro;= ¡ffzPoi(cfrzeo4 - c$zeo;,
+ oËoitc|oi
Ju.eo;= rkgzeo¡. oËt"ã# . kEoïffil
( cflzeo¿
+ rf,eoilcf,roft- - cHPoâ-)
.foå-1. (30)-s t'
clzro< ¡ (3r)
which is of the same form as Nernst's expression but
with an overall kT equal to kfizeo;* kfiPo?- "%
+.-K.K;
- r ' (H*)ffioå n'.ot=. Assuming that the pFI gradient within the' 1n*¡2unstirred layer is negligible (i.e. (H+) is everywhere
equal to that in the bulk medium), overall kr's for the
66
experiment shown in
are shown in Table 3
ignored, because the
term due to PO diffusion isdiffusion coefficient is
Fig. 23 can be calculated" These
The
Poi-
small (in fact the simple formula based on the activity
factor predicts a coefficient Iess than zero), and
teo!-l is only 62 of tHPo?-1.
Fig. 26 shows some plots of the Briggs-Maskell
equation using these overall kT's, together with the
data of Fig.23. The agreement between theory and
experiment is better, but influx still tends to be
underestimated and stirring is still predicted to have a
greater effect than it does in practise.
A probable explanation is that the pH within the
unstirred Iayer is not constant. Although the sea water
was buf fered (I0 mivl TES) a pH rise close to the surface
of the thallus is likely in view of the very rapid
34
TABLE 3
overar f k-'s ( - kll zPoZ ¡transport'in the tive Itowerr using Ki-= I.26 x(pH 1'.48). uîi83 of kr a
ntM
Level 1 Level 2 Level 3 Level 4 Level 5
.r-HzPOl^T 0.342 0.797 I.60 2.27 3.r4
r-HPOI 0.136 0.318 0.631 0.904 L.25
overa 1 IkT 5 .52 12 .9 25 .8 36.7 50.7
67
1Ivìc{tË
oI
xs\J-rlq.
+
$I+
f5
+
lo
5
lo zo æ1o-bta"L p[tosphæte (þn (fit
6o -/otuaþç6ttíorv )
othe::wise as in Tí9,. 25.
uptake of COZ during photosynthesis (see later). This
would lead to a greater proportion of HPOî- (and PO?-)
to H2PO[ in the diffusion boundary layer than in the
bulk medium and a consequent increase in the overall kT
for H2PO4. Thus, kT would be a truly "overall" kT'
being the integral of the differential kt'" at each (H+)
of the pH gradient. Qualitatively, however, if the
average pH of the unstirred layer could be considered as
being B, the overall kt'" at low rates of stirring would
be more than sufficient to account for the data.
There are other lines of enquiry in the
interpretation of the results. The first is suggested
by the predictions of FVKUP for the data of Fig. 23-
These are shown in Table 4. Lines of best fit are
obtained with a low Kt for total phosphate
TABLE 4
Parameters in the Briggs-Maskel I equation predictedby FVKUP for the data- êhown in Fig.23, and regressioncoefficients.
Level I Level 2 Level 3 Level 4 Level 5
(
S
nmo-r )
V1m-2 14.3 14.0 1s.4 L5.7 r5.9
riM r
0.447 0.611 0.833 0.821 0.817
k- - 0 .209 0.27 9 0 .642 0.588 0.851(m s-r*ro-5 )
Reg.coefficient
0.999 0.996 0.996 0.979 0.993
6B
(average 0.77 ¡tv) and with a f airly large dif fusion
resistance, where this resistance changes relatively
Iittle from level I to level 5. The suggestion is that
the diffusion of phosphate within the thallus (i.e. in
the cell waIIs) is slow enough to make negligible the
changes in k,, that occur in the stirring gradient tower.
However, using the Blum-Jenden expression for Kftpp (see
Fig. V.9 and equation V.23) it can be readily shown that
even if the true KM was zero and the internal diffusion
resistance (,åR/Def f ) was large enough to enable a f it to
the data of I evel 5 , the external "res istances " ( 1,/kt's )
would sti1l be Iarge and stirring would be predicted to
have much the same effect as shown in Fig. 26.
The second is that HPOî- is the species that is
transported across the membrane. Allowing for the very
slight enhancement due to H2Po4 and Po;-, HPol- inf l-ux
versus concentration curves based on the Briggs-Maskell
equation are virtually identical to those shown in Fig.
26. The unstirred layer pH would have to be a good deal
Iarger than I for there to be significant enhancement
by po!-; the ratio of teofl- I to tHpo?- I is K;/Aú
which is only 0.2 at pH B (using pKa.= 8.69¡ MilIero'
I9B3 ). In addition, kT for PO?- is very low because of
its very low diffusion coefficient. This further
reduces its effectiveness as an "enhancer" of HPOZ-
dif fusion .
The third possibility is that of a relatively low pH
69
in the celI wal1, given an equilibrium distribution of
H+ between the Donnan free space (DFS) and the bulk
medium. Since H2Po4- and HPofr- witl always be in
equilibrium, the concentration of H2Po¡ could be
increased in the cell walI relative to tHPo?-1. whilst
this would lower the ef fective kT for I{2PO4 transport
through the cell waII, it would increase the
concentration of H2PO4 on the "unstirred layer" side of
the walI, and possibly at the plasmalemma also.
Equilibrium between the DFS and the unstirred layer with
regard to H+ is untikely; even so' the overall, steady-
state pHof the wall may still be somewhat lower than in
the adjacent unstirred layer and may contribute
something towards an increased rate of H2POÃ uptake.
It is reasonable to conclude that U. rigida possesses
a membrane porter for HZPO4 ions which, at the pH of
this experiment, is prevented from diffusional
limitations external to the membrane because of the
parallel diffusion of HPO?- and eO!- and a possible
lowering of pH in the celI waII. The KM for total
phosphate ( f.S ¡u)
is quite low; indeed, at zero ionic
strength, the KM could be halved if the activity
coefficient for H2PO4 in sea water is 0.5 (Whitfield,
L915; footnote to equation (f0)). Howevern very low
Ktut's f or phosphate uptake have been measured in many
marine algae, including the related genus Enteromorpha
(Watlentinus, L9B4). The deduced K* for H2PO4 is yg-rj¿
70
Iow (0.036 /"rM in ASW) but again in line with theImeasurements quoted in Wal lentinus (I9B 4) when the pH of
the medium is taken into account. At lower pH's, it
would be predicted that the Kftpp for total phosphate
would decrease, but that diffusional limitations would
become greater. This would be worth testing in future
experiments.
rrr. Uptake of l32P I phosphate and [lac] methvlamine bv
Vallisneria spiralis
( i ) Results
Phosphate influx versus phosphate concentration is
shown in Fig. 27, at three different stirring rates. It
is immediat,.fy apparent that stirring has even less of
an effect on influx than it does with Ulva, and that the
apparent K, is much larger (around 60 fM).
This is in
spite of the fact that at the pH of this experiment
H2PO4 constitutes 742 of the total phosphate, using the
pKa for H2PO4 in distilled water (7.21). The potential
f or enhancement of H2pO4 f luxes by HeOa2-, therefore, is
not large. In Fig. 28, IIaC]methylamine influx is shown
as a function of concentration. Although the data are
not particularly conclusive, stirring appears to have
tittle effect overall and the observed K, is
comparatively high. In each of these experiments, the
stirrer was on its maximum setting. The correction
factors for the kt's measured in the z:-nc disk
7T
I
3
I\t)
JI
€\)
0Ëd,
Xf!Y-\
2
1
IA
o¡A
5o loo lso+ota,L phc, s ¡hfre- cønæ lttratîort- ( N)
FIGUBE 2?. l3blpno"phate infh¡x agalnst concentration for sections of-'- v.tpi-"Jr-"-iãátt"t, in lãvc1s r (r), ¡ (¡) end-5-(') of thcstirrÏffiratlient tower. 'aPW + 10 nM MES, pE 6'72 - 6'77' 25oc'
voo
aI
t
c..1/fp*I\nñtFJÈ
^ôÈc
X loo
J\ìc
,J
+
oI
I
A
lôo ?Æ
frwt| lwúne. concÊntvatørv Çø)
FIGURE 28. [14c]t"tttylanlne lnftr¡x versus concentratileaf pieces. S¡rrrbols as in Fig. 27. APtlV +6.72 - 6.77, 25"c. [re lines represent the
æ loo 5ú Øo -foö
on for V.
51
10 mM I\,ÍES, pE
^BrigSs-MaskelL¿ s-l antl k1 =equation, with K1¡ = 20 yM,8.35, 8.21, and 7.33 x-10 -
29s-
¡rno1 n-V=7n a
experiment are thus (9.5 x I0-I0 /l.l+ * f O-9)å
= 0.432 for H2Po4-, and (1.46 x rO-9 /2.34 x ro-9)ê
= 0.57 6 f or cHruttr+. (A1I dif f usion coef f icients are
those at infinite dilution and are from Gros et al.,I976, Tanaka and Hashitani, L97I and Robinson and
Stokes, I959.)
(ii) Discussion
There are basically three possible explanations for
the results: (a) the kinetics of uptake are determined
by the rate of the membrane transport reaction itself,
(b) internaf diffusion resistances are so large (due
to the thickness of the leaf) that the diffusion
boundary l-ayer constitutes a relatively small
resistancer or
(c) the cuticle is the dominant resistance.
The shape of the methylamine concentration curve at
the highest rate of stirring (nig. 28) is immediately
suggestive of a large in-series resistance, implicating
the cuticle as the'culprit. The solid lines shown in
Fig. 28 are for a cuticle with a permeability to CH3NH3+
of 8.5 x 10-7 m "-I, together with the corrected kr's
for the different levels of the stirring gradient tower
and a K¡,1 f or CH¡NH3* i.rf lux of 2O ft.
Such a low
permeability of the cuticle is not unlikely. McFarlane
and Berry (I974) found that isolated apricot cuticles
had a permeability to Lil of 0.39 - 2.85 x I0-10 * s-1.
72
CH3NH3+ is Iikely to penetrate more rapidly because of
its smaller hydrated radius. Also, the cuticles of
aquatic plant leaves are much thinner than those of
terrestrial plant leaves (Arber, 1920). Schönherr
(1982), for a range of terrestrial plants, showed. that
the permeability of the cuticle to water is 0.I 2.3 x
l0-9 m =-1, while for the aquatic Potomogeton lucens
it is abouL 2.55 x 10-6 ,n =-l
The phosphate concentation curve (F.ig. 27) looks like
a rectangular hyperbola typical of Michaelis-Menten
kinetics; indeed the computer programme FVKUP showed
that the data fitted best to an equation of this type,
with a KM of 40 B0 lM. The other possible
explanation is that diffusion in the cell waIls is
Iimiting, which cqn also lead to a reasonably
rectangular hyperbola when it is in parallel with
reaction (see Fig. 5). With a KM f or H2POi as lo\^r as
ulva, o&Eoã would need to be of the order of 2 x Lo-L2
m2 s-f, if phosphate were diffusing through the faces
of the Vallisneria Ieaf pieces. Such a low effective
diffusion coefficient is unlikely. However, it may be
that the cuticle presents a virtually impenetrable
barrier to phosphate ions, in which case they would
diffuse chiefly via the cut edges. Such a situation
could easily account for the observed uptake kinetics
with reasonable values for oå?Eof in the cel1 walls.
That the cuticle, in general, presents more of a barrier
73
to the passage of anions than of cations \^/as shown by
Yamada, Wittwer and Bukovac (Lg64,1965; see also \r/¡ftwef and
Te,rbner , 1959) who suggested that the more rapid
penetration of cations \^Ias partially due to their
greater binding on the inward-facing side of the cuticle(Yamada, Bukovac and Witt\^zer' 1964). Again, steric
effects wiIl also be important (McFarIane and Berry'
I974)¡ H2POÃ will be considerably less mobile in the
cuticte than CH3NHI simply because of its bulk.
74
RESPIRATION
I Kinetics of Oxvqen Reduction
By far the most important catalyst for the reduction
of O2 in dark respiration is cytochrome oxidase
(Beevers , 196I), which resides in the inner membrane of
the mitochondrion. In the reaction catalysed by
cytochrome oxidase, O2 acts as the final electron
acceptor of the electron transport chain, being reduced
to water. Some plants possess an alternate oxidase,
insensitive to the normal inhibitors of cytochrome
oxidase (e.g. CN-, CO) which has a Iower affinity for O2
than cytochrome oxidase (Sargent and Taylor, L972¡
Solomos, L977) -
The kinetics of oxygen uptake by cells and tissues
have been studied by many workers. The rate of uptake
is often subject to internal, and sometimes external,
diffusion limitations particularly in l-arge cells or
groups of cells or thicker tissue sections (Warburg,
1923; Berry and Norris, 1949; Yocum and Hackett' 1957¡
Longmuir, L966¡ Johnson, 1967i Mueller' BoyIe,and
Lightfoot, lg6B; PooIe, L97B; Raven,1984). In other
cases the kinetics appear to be Michaelis-Menten
(suggesting insignificant transport limitations
Longmuir, lr966) but with unpredictably changeful kinetic
parameters, especially K¡1 (Tang, 1933; Dromgoole, I97B;
Morriset, 1978; KetIy, f9B3). Some workers have found
75
that the Michaelis-Menten equation of an arbitrarily
modified order fits best with the experimental data.
For example, Bänder and Kiese (1955) proposed
v = v "1.4 /{xvl + .l'4 ) while rvanov and Lyabakh (LgB2)
suggested the equation v = v c2 / {x*z * "2).
The problem is almost certainly related to the
complexity of the reaction. The initial reaction of 02
with cytochrome oxidase (here designated cyt.a3) can be
written
aLcyt. a3' ' -l- O2 cyt.r33+{+ (i)2o
(k+t- 5 x I07
the rate of 02
well as on the
volume of the
M-1 s-1 (chance, 1965) at
reduction will depend on
25oc ). Thus,
I cyt.al+ 1
concentration of C_2. In the very small
mitochondrion, the concentration of
AS
the rate
supply by
cyt. u32* wilI be a steady-state determined by
of its consumption in reaction (i) and of its
the reaction (ignoring cytochrome a)
2f,cyt.a3'' + cyt c cyt. u32* * cyt..3*. (ii)
Here again, lcyt.c2+l will be a steady-state
concentration which wiII be determined by the rate of
reaction (ii) and a supply reaction of the form
-_
k+32rAH + cyt. c' '
76
A+H++cyL.c2+ (iii)
hlhere AH represents a reducing agent such as NADH or
succinate and where other components of the electron
transport chain (e.g. cytochrome b, flavoprotein) have
been ignored. In general, the volume of the cell
accessible to AH includes the cytoplasm' so it is more
difficult to disturb the steady-state concentration of
AH than that of the oxidized or reduced form of the
cytochromes in the mitochondria. There is also a much
wider variety of reactions which are capable of
supplying AH. IaH] ' therefore, is a convenient
reference on which to base a prediction of the rate of
electron transport in the mitochondrion.
From reactions (i), (ii) and (iii), Petersen,
Nicholls and Deqn (I974) showed that the rate equation
2
V+A2 V
c_5Oze A3 (32)
can be deduced, where A1r A2 and A3 have the following
values:
k+z k+3 [eu1 eA1
4k+t(k+Z k+3 [ns] )+
kr c k-?_r_ z
4k+t(k+2 + k+3 [eH])
k+z k+3 [eH] e
A2
A3
and
k+2 +
77
k+3 [eu1
The symbol rrerr represents the total concentration of
cytochrome oxidase in the mitochondrion, which is
assumed to approximately equal the total concentration
of cytochrome c (Chance and Williams' 1955; Lance and
Bonner, I968); .=O' is the 02 concentration at the site
of reaction. KM for 02 is equal to ny/e - n2A3/2e and
the maximum rate of electron transport, V, is equal to
43. (The maximum rate of the reduction of 02 to 2H2O rs
one quarter of V.) Both nfi" and V, therefore, will be
sensitive to teHl. Kfi" wilI also be sensitive Lo A2r
which is proportional to the difference between k*2 and
k-2. This arises from a change in the order of the
reaction with 42, illustrated in Fig. 29 a,b and c.
When A2 = O (k+2 : k-2), the kinetics are first-order
Michaelis-Menten (Fig. 29b). For k-2 smaller than k*,
(AZ > 0), the overall reaction order becomes qreater
than one (I¡ig. 29a) and the kinetics are of the form of
Bãnder and Kiese (1955) and Ivanov and Lyabakh (1982).
For k-2) k+2 (42 < 0), the reaction order is less than
one, which was what Petersen et al. observed for fully
energized mitochondria (ni9. 29c). The dependence of
the velocity on the 02 concentration is weakened in this
case and saturation is approached quite siowly.
It can be readily shown
displaying these kinetics
that if mitochondria
were
shape (half-thickness = R) r in
constrained in
which internal
a slab
1B
o2
Itra .ff1
(æl I aaa o
t5 ao
^t?
1'o
a
o
E¡.
AÂ
(bt
al
I a
t
a
I
Â
lo
t\rì (c
t
a
A
f'
a
a
,/
S //
2a
.E
/
/
5 a
'/ Áa
/0,5 Co,J {¡M)
FIGäRE 2). Rzte of electron transport, v, in nitochondria vs. fO2l basecl on equatiott (32); o/" is plottect as this eliminatese from the rest of the "qoátiót. Values of k., , k+lafia tnrþnl ""á fro¡n Petersen et aI .(lgl+) giving A.¡/e = 0.0B/u}Ia¡ñ, Lz/e = 16 s-1. Values of A2are (a) O.OO4 (¡) o and (c) - o.Ot8,AM s. The dotted a¡cl tlashed lines represent 'equatlon (33) with A2= 0.OO4 ancl - O.O18¡M s respectively and e g4 f = 0.03¡M s.
transport
transport
vùas fast, then an external resistance to the
of 02 would change equation (32) to
e"*rr+ArkT+eA3Rv=
2(A2kT + e R)
kt+AlkT+e A:R ) 4(A2kr+eR)ee3cf;kt
Z(AZkt + e R)
(33 )
(.8. is the oxygen concentration in the bulk phase and
kT the transport coefficient between bulk phase and the
surface of the slab). The dotted and dashed lines in
Fig. 29 show the effect of an external transport
resistance on kinetics of the greater-than-first-order
type or less-than-first-order type respectively. (when
A2 = 0, equation (33) becomes equivalent to the Briggs-
MaskeII equation.) Fig. 29 shows that the same
transport resistance wilI have a much larger effect on
kinetics of the type shown in curve (a) compared with
those shown in curve (c).
The differences in the shapes of the various
hyperbolae shown in Fig. 29 are brought out more clearly
in the linear transformations of the Michaelis-Menten
equation. Fig 30 shows the Woolf plot (c/v against c);
the Bänder and Kiese type of kinetics transforms to a
curve which is concave upward while the order ( I type
is concave downwards (cf. Petersen et al., L974)'
Transport Iimitations will always tend to force the
ec
79
loo
15
L5
/
//
A
Â
a (c)
o(b)
(e)/ r/
/
V)
ñìt
Ë
oË.{
/
/ trI
t/
ô
o,5 .t.o
/
À/5o/ a
a
a
//
ta
rIr!r . E. '
A
t
Â
¡
¡a
00 fozf ,fr)
tr'ig. 2). T}l..e sloPe of the line at)-1 ); the intercept on the orclinate'transport limitations (soIlrt lines)m (aotteo and dashed lÍnes).
concavity upward in plots of this type (cf. Winne,
I973), with the result that the concave-downward curve
(c) becomes almost Iinear (i.e. a rectangular hyperbola
for v against c) while the inflection in curve (a) is
augmented. However, aI1 the curves tend towards the
some slope (L/v ) at high substrate concentrations.
II. Resprrataon in Ulva riq-ida
( i ) Results
Figs. 31 36 show a number of experiments on the
relationship between 02 consumption in the dark' the
lozl of the surrounding sea water tcf") and the
thickness of the diffusion boundary 1ayer. I,rioolf plots
of the data are shown as insets. In the presence of the
artificial unstirred layers, the response to "flti=atypical of diffusion in series with first-order
Michaelis-Menten kinetics because the rate at tow cfr" is
so large and is little affected by an increase in the
layer thickness from 0.75 I.B mm (nigs. 32 - 34)-
Incieed the VüooIf plots are all concave downward, the
exact opposite of what would be expected for rate
limitations by diffusion. In the absence of any
artificial- unstirred Iayers, the Woolf plots are mainly
straight (Figs. 32, 33, 36), sometimes concave downward
(nigs.3I, 35) and in one case slightly concave
upward(fig. 34) indicating Michaelis-Menten kinetics of
orders equal to, Iess than and greater tha-n one
BO
t{s
d"ìF
talr{'ùp
a
a
^
o
ao
^t
¡ lo o
loo þ IÑSET ^A
a
5
Lo^J
FIGURE 31. Rate of respiratory oxygenuptake as a function of the oxygen concentration forU.Tisi*a _(it_tyae^Aholder) intr'SW at 25'C-. Inset shors a lÍõotf plot of the data (v refers to the rate of respiration). Results oftro experiments.
Laa
A¡
oo
^
N
Èv)
v)
'eX
Elr
o
0
1æ
f50l@
ao
EÌ
ao
oo
ÒIv)
.¡ts
d'o
è
I
€o
l¡,
þx
to+:ds'ä toov)(u(*
o
o
o
o
o
o
o
2@
6o INSET
2
Þ
tæN
trrGIIRE 32. As in Fig. 31, for Il.riFida in holder ¡(o) + ¡(o) itrr (o) or rithout (o) 10 layers of lens tissue (rotrr sides)¡
- O.75 m thick each side . zO-C.
so c.ozï
15o
resp
iratio
rr"
(n,r
*l /.
íL
-2 "-
')À
) Þ oÕ u o
oa
ao
oo
8
b Þ tJ è
oa
çozl
/uxl
o-s
g n-
'¡N
A o o
o
2 U1 m -t
oo
o
B
oo
a
a
o
oO
o
o o
It.n
ñ!d,I'Ss
Io.J+i(J5a_./)qJs
oo
oa
I NSEÏ
o
o
100
FIGIIRE 34. Äs in FÍg. 33 ritJr (o) or çithout (o) ¡O layers of lens tissue (tottr sides)rn1.8 nn thick each side, antl tissueagetl for 22 hours. 27oC.
o
I\(â
¡(,I
eXÀ.
Sq
2
f50
0 læfo^f N
oo
a
lt 2oo
*(I
Éi leoù
e.9{Jd loot..J
O
a
o
o
o
INsEÏ
o
ræ
o
o -Tr^
|o¡aX
Sd
o
oa_oa 5
o50
Eozf
FIçIIRE 35. As in Fig. l{ rit}r (o) or without (o) ¡ layers of Whatnan No. 1 filter paper (tottr sicles)^, 0.8 nn thickeach side. 25"C.
æ N
09 ooL,,JEZOJoçl Ø¿
o
oooooood(¡
T.I!¡roë
È'IoÞÈ
È
U't
oOO o o
o o oool
o oo
t
I
¡
t I
t I¡
I¡
t
¡
t+
1?
./"oo
o
o
1o
6
2
o
ÒI
Èr
()
uX
[tr
o
o
Io
6
+
o
o ./t¡l
1æ
[o"f N
IIIGIIRE 36b. Woolf plot of the ctata of Tig. 36e.
o
respectively. In all cases' apart from the results
shown in Fig.35, the maximum rate appears to be the
same with or without the artificial unstirred Iayers.
Results for tissue pieces which were aged for 22 hours
in aerated FSW after being cut from the thallus are
shown in Figs. 34 and 35-
(ii) Discus s ion
There are a number of snags in the method used to
determine the kinetics of o2 uptake which may run the
intrepid experimenter aground. I wiII begin by
identifying some of them, but have more-than-Iikely
missed (and consequently hit) plenty of others'
Dromgoole (1978) pointed out that sections of the
thallus of various marine brown algae could display a
rate of 02 uptake that was artefactually large when
transferred from sea water with a low lo2) (72 ¡M) to
sea water at a high I02] (307 /*),
due to the
equilibration of the 02 outside the tissue and that
inside. with lozl in the thallus lower than in the bulk
medium, there is a net Ioss of 02from the latter which
is not associated with respiration.
In my case, the initial I02] of the medium was the
air-saturated l)2l (-200 ¡u);
it is unlikely that the
t02l in the tissue would have been greatly different
since pieces of thal lus \^7ere al ways kept in air-
8l
equilibrated seawater prior to an experiment.
Another artefact may appear for green tissue
displaying rates of dark respiration which are higher
immediately following illumination than after a Ionger
period in darkness. If such tissue was sealed into the
darkened 02 electrode chamber immediately after being in
the Iight, the respiration rate would decline with time
which could be falsely interpreted as a decline with
decreasing lo21 (cf. Dromgoole, 1978). Although
U. riqida did show an occasionally significant post-
illumination respiratory "burst", it was a short-lived
event (< I0 min). For the data shown in Figs. 3t 36,
the tissue had been in the dark for at least this period
of time before the first measurement of the respiration
rate was made. In addition, the alga was previously
exposed to only dim laboratory light, Ieading to
relatively Iow rates of photosynthesis and presumably
Iow rates of whatever process \^/as responsible for the
post-illumination burst (possibly photorespiration) .
With the artificial unstirred Iayers, respiration was
remarkably fast at low "8, t indeed, it woul-d appear,
impossibly fast. In Fig. 34, for instance, with an
unstirred layer on each face of the tissue of at least
1.8 mm (i.e. without taking tortuosity and porosity into
account) tfre highest possible 02 flux by diffusion at
10 ¡,rM Or would be J = (D/ ã )c5 ^"' I0 nmol *-2 "-I; the
*"i.,rt.'u rate \^/as crose to 200 nmol *-2 "-r. rt could
B2
be argued that the cut edges of the tissue, where the
boundary layer would be thin, had a very high rate of
respiration, in which case virtually alI the 02 uptake
at low "8, would be due to the respiration at the edges.
SmaII pieces (nig. 36) of tissue, however, in which the
edge constituted a much bigger proportion of the total
surface area, did not respire any faster than Iarger
pieces. AIso, in the experiments where the tissue was
aged, there were stilt high rates of respiration at 1ow
.82(r'ig=. 34, 35), although the period of aging may not
have been sufficient to alleviate the effects of cutting
(cf. discs of storage tissue; Laties, 1967)-
Per unit area of edge rates of respiration would be
absurdly Ia19ê, e.g. 25 f "1 m-2 s- I in Fig. 35
( 5 ¡M o, ). Edge effects may extend beyond the actual
edge of the tissue, however, since the length of the
diffusion pathway will be Iess for those faces of the
tissue near the edge. Assuming that for a distance
from the edge of the tissue equal to the thickness of
the unstirred layer diffusional Iimitations are
negligible, it can again be calculated what the rate of
respiration must be to account for virtually all the O2
uptake at lot .ff. From Fig. 34, the rate is 450 nmol_a _]m-¿ s-r at 5 ¡M 02. This is a much more reasonable
/number than the one just quoted for the edge alone but
it is based on a very generous assumption. It is
possible that there is an additional contribution from
B3
the respiration of bacteria on the surface of the
thatlus with thick unstirred layers, as these
experiments hlere more prolonged.
Another potential difficulty, which may be relevant
to the magnitude of the rates of 02 uptake, is that the
O2 measurements h/ere not made under strictly steady-
state conditions. Athough the rate of change of
concentration in the bulk medium hras not large, it was
sufficientty Iarge to be measurable within five minutes;
this period of time would probably allow the enzymic
reactions of oxygen reduction to come into an
approximately steady-state, but may not have been
sufficient for steady-state diffusion to be
established. This would mean that the concentration
gradient would not be fully developed at a particular
bulk 02 concentration, and the unstirred layer thinner
than one would have predicted based on the assumption of
a steady-state.
A rough indication of the time required to reach
steady-state for diffusion in one dimension is given
by the equation
^( 0.257 ( 34 )
(Sundaram, Tweedale and Laidler, 1970) in which 'Ú is
the time at which the concentration gradient is within
10? of the steady-state profile. For an unstirred
^2ò
;
B4
layer 1.8 mm thick, A is about 7 minutes, suggesting
that diffusion may have been quite a way from steady-
state throughout the course of the experiment shown in
Fig. 34, and possibly in the other experiments as welI.
Given the complexity of the non-steady-state
equations for diffusion, let alone diffusion in series
with reaction, as well as the uncertainty even as to the
whereabouts of the reaction within (or upon? ) the
thallus, it is not worthwhile analyzrng the absolute
rates of O2 uptake that were found with the thick
unstirred Iayers. Whether any significance can be
attached to the shape of the absorption isotherm will
be discussed later.
rn the belief that the relationships between l}2l and
the rate of O2 uptake f or thallus with no artif icial
unstirred Iayers approach reality, it is clear that the
kinetics are often first-order Michaelis-Menten but that
the order is sometimes greater and sometimes less than
one. In terms of the mechanism proposed by Petersen et
aI. (L974), the forward and backward rate constants for
the reduction of cytochro*" ul+ fry cytochrome "2t change
their relative magnitudes, i.e. the equilibrium constant
changes. The unlikelihood of this event suggests that
the reaction actually involves more chemical species
than just the two cytochromes. Since the reduction of
cyt. u3* is normally coupled to the phosphorylation of
ADP, reaction (ii) could be written
B5
L+3
+
ADP + Pi + cyt. "2r + cyt.a]+ cyt. a + ATP
H2o'
A change in the ratio of [ATP] to [ADP] and [pi] would,
therefore, appear as an apparent change in the
equilibrium constant if adenylates were not taken into
account. SpecificaIly, the rate of the back reaction
would increase if Iefp] were to build up relative to
leop] and Ipi], which would account for the order ( I
kinetics obtained by Petersen et al. (1974) in fully
energized mitochondria.
A simple explanation, then, for the observed
differences in the reaction order is a change in the
energy charge of the tissue. A large number of
processes contribute to the energy charge of tissue,
including photosynthesis, glycolysis, the oxidative
pentose phosphate pathway, the Krebs cycle and oxidative
phosphorylation itself. In my case, there is a positive
correlation between the observed order and the
"freshness" of the tissue; the times when reaction
orders Iess than one \^Iere f ound (f or tissue with no
unstirred layers attached) correspond to fresh tissue
(Iess than I days in the laboratory Figs. 3I and 35)
while the tissue which had been kept quite some time in
the laboratory (fig. 34) shows a reaction order greater
than one, implying a slow back reaction and a low energy
1.^*2t cvt.a3*+T;,-z
B6
charge. Kinetics of order one \^/ere obtained with tissue
of intermediate "freshnessrr -
The addition of an artificial unstirred Iayer appears
to decrease the order of the reaction (in all cases but
one Fig. 35) suggesting that the unstirred layer
somehow increases the energy charge. In Fig. 35 (very
fresh tissue only one day in the laboratory) the order
is virtually unchanged, but the maximum rate is nearly
halved. The latter phenomenon suggests a decrease in
the concentration of reductant. It is hard to see how
the addition of an unstirred layer could bring about
changes like these. The possible immediate effects of
the "unst j-rred layers" (e.g. partial anoxia, partial
darkness) would tend to increase the ratio of reduced to
total pyridine nucleotides and decrease the ratio of ATP
to total adenylate, if they do anything (Kawada and
Kanazawa, Ig82). It is like1y, then, that the shape of
the concentration curves', as well as the absolute rate
at different concentrations, is altered by the factors
previously discussed and that the apparent order < I
kinetics are artefactual.
With regard to the O2 transport resistances (internal
and external ) for U. riqida tissue with no artificial
unstirred layers attached, these are difficult to
determine because the kinetics of the driving reaction
are so complex. If they are first-order "Michaelis-
Menten" (i.e. A2 = 0 in Petersen et q--/s (Ig'7 4)
B7
analysis) then the relationship KM = vrz(4k*1e) holds
(cf. Chance, 1965). The maximum rate of electron
transport, V, is four times the maximum rate of O2
reduction, k+l ,\' 5 x l0 M-I s -1 (Chtrr.., 1965) while e
is uncertain, but probably within the range I0
I80 nmol cyt.c per g of chlorophyll (Raven, L984). For
the results of Fig. 36, then, with BB mg chlorophyll
m-2 in u. rigida (MacFarlane and Smith , LgB2), the
range of Kt could be 0.3 , fr. For A2 ( 0 (i.e. k-2
) k¡2), the apparent Kt would be further increased.
Thus, the Ku's for 02 that were observed (: - 9 ¡Ut)could be easily accounted for by reasonable values of
k+I, k+2, k-2, k+3, IaH] and e, without invoking
Iimitations to the transport of O 2 Lo cytochrome
oxidase.
The similarity between the concentration curve for a
large piece of Ulva and several much smaller pieces (cf-
Fig. 32, for example, and Fig.36) as far as apparent
KM's are concerned, suggests that transport Iimitations
are indeed smalI. Internally, this may be brought about
by the diffusion path for 02 not being confined to the
ce11 watl (cf. CH3NH3+); a mitochondrion positioned
adjacent to the inner periclinal wall of the cell can
receive O2 via cytoplasm and vacuole as weII as via the
cell walI. Although protein and organelles in the
cytoplasm will tend to lower diffusion coefficients
(Wang, Anfinsen, and PoIestra, I954¡ Garrick and
BB
Redwood , Igl'7), there is also the possibility of
enhanced O 2 transport by macromolecular 02 carriers
(Raven, Ig77). In addition, if the apparent kinetics of
respiration are of the Iess-than-first-order type (as
appears to be the case for fresh u. rlqida) the same 02
transport resistance limits respiration less severely
than if the kinetics are first-order or greater-than-
first-order Michaelis-Menten. Fig. 29 illustrates this:
here the o2 transport "resistance" (en,/rt) is 0.03 ¡u s r
corresponding to a k, for O2 of 2'g 53 x I0-5 * =-Idepending on the value of e.
III. ResPiration of Va lisneria sp iralis
In Fig. 37 are sh
experiments with V.
Using the thicker, basal part of the leaf, sealing the
air lacunae with wax or increasing the thickness of the
diffusion boundary Iayer makes no difference to the
initial slope of the respiration versus o2 concentration
curve although the maximum rates vary significantly-
This, combined with the fact that the shape of the
concentration curve is tn. ?ame for the different
treatments, suggests that something other than the
intrinsic kinetic properties of the enzymic reactions is
limiting at Iow .8"-
several facts suggest that this may be o2lLtansport
through an external resistance. FirstIy, the shape of
ovrn the results of a number of
spiralis under various conditions'
B9
Tes
PrØ
tgm
olg
n'^
s-)
îorL
o ¡Þ
o o r oD
a
o
hd H cl g F tst (, j
Par
td
Ft
rà k
Þgä
8 gË
{ or
i..þl
P-t
400t
tÞ
J !t
Þ l
*Ho
o oo
!fl+
JoG
otJ
O
HtO
rìd
ôPf+
¡ ol
¿P
O,Ë
G+
tJ
ÈoF
Ffo
Po
I'f H
O
l-å<
+H
toï
.,o B
--
O O
lr'q
(c+
tùX
gtlO
< i
;E
Ftlc
t O
fd
D'll
*¡u
o +
5
^ll-l
- r
f,llf.
€iv
- È
" Il¡
cl-
. 9r
-, I
t P
X o
O
O H
Oo
FO
O(D
b
I'tct
S
t P
tt rt
t P
. rt
üÐ
c+á{
O
¡-
P.5
p
H:+
c+
|Ú
o !'o
H
'Fà
HË
r*88
o+oo
gåÈ
ã"Ë
r (D
É
O=
AJd
c:tÞ
l-l
giÁ
ob
t-¡o
---5
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l-åO
rPO
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rt1¿
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pp.
È¡<
fÈ
F'
oo Ê¡'
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I
the concentration response curve itself, which shows a
very sharp transition to saturation (cf. methylamine
influx, Fig. 28). Although such a transition also
arises for Michaelis-Menten kinetics of an order greater
than one, Fig. 29 shows that these kinetics are very
susceptible to limitations by transport- It is odd'
then, that the addition of an artificial unstirred layer
is without effect. The apparent xffz is also high (15
20 gM), whereas with a reaction order ) l, Kftpp's tendI
to be very l_ow (Fiq. L9)¡ the actual value wiII also
depend on Ia H], however. Secondly, even though the
basal part of the V. spiralis leaf is more than twice as
thick as the middle part, the initial slope is the same.
If diffusion internally were timiting, the rate should
show a strong dependence on the leaf thickness. For a
typical thickness of the expanded leaves of V- spiral is
of 300 prñ, D"tP" would have to be Iess than 4 x 10-10,19m"=-I to produce rates of the order shown in Fig. 37
(using equation (v.22) , Fig. V.8 and assuming that KIfi"
not more than l0 f*). o.t?z, therefore, would need to
of its value in the bulk medium
8.qz x lO-9 ^2 =-1 rL 25og - Himmelblau, 1964; Hung and
Dinius, 1972)¡ such a low D"¡f is unlikely given that
the cytoplasm in the epidermal cells is streaming and
much of the diffusion pathway in the mesophyll of the
tissue is through vacuolar sap or air spaces. AIso, if
the distribution of the cytoplasm is a guide, the number
of mitochondria may well be greater per unit of volume
IS
be Iess than 17%
90
of epidermis than per unit volume of mesophyll (rig.
Bc); this would have the effect of decreasing R in the
diffusion-reaction equation which would necessitate an
even lower D"rPz.
FinaIIy, there remains the possibility that 02
diffusion within the mitochondria themselves is rate
Iimiting. In vitro rates of 02 uptake by mitochondria
can be 3 mmolg-r s-l (¡ hmol *-3 "-r) or more
(¡. T. Wiskich' pers. comm.). Assuming that the
mitochondrion is a sphere, radius 0.5 ¡m, and that D"¡
is the same in the mitochondrion as in the medium,Q
(equation (29) ) would then be about 1-7 at .82 = 2o ¡rM.
With this value of 0, even if the kinetics of 02 uptake
were zeroth-order, the mitochondrion would be Iitt1e
more than Toeo effective. Having no information about
the rate of 02 uptake of the mitochondria of V. spira I i s
(i.e. in vivo) it is impossible to say whether or not
intra-mitochondrial diffusion limitations can be
significant.
Assuming, albeit teleologically, that mitochondria
are effective in what they do (i.e. they are not
partially anoxic) it is likely that the results of Fig.
37 are due to an external resistance to 02 diffusion-
The respiration experiments were done under well-stirred
conditions in the oxygen electrode chamber and the
diffusion boundary Iayer would therefore be thin. That
O-7t
91
the boundary layer is an insignificant resistance is
conf irmed by the experiment (Fig. 37) in which it I^7as
artif icially increased using holder þ(z) tO-z mm
thick). The results plotted as amount of 02 per m2
leaf are the same as the other results in Fig- 37 at low
.8.. This is in spite of the fact that the surface area
available for diffusion though the holder \^las only 3I.4U
of that of the leaf. That is, calculated on the basis
of the surface area available for diffusion in the
boundary Iayerr o2 uptake would be ovtr 3 times faster
with the holder than without. This is unreasonable and
supports the notion that whatever is limiting 02 uptake
has to do with the leaf (i.e. cuticle and outer cell
waII) and not with the diffusion boundary layer-
comparison of Figs.37 and 28 shows that the rate of
02 transport through the cuticl-e and outer celI wall
must be nearly 5 fold the rate of CH3NHj transport
(Iimiting slopes are 3.82 x 10-6 m s I for 02 and 8.35_1 -lx 10- / m s-r for methylamine at the fastest stirring
rate). This is not unlikely in view of the known
properties of plant cuticles. Cuticle has a Iipid
fraction (Martin and Juniper, 1970) and a polymer matrix
with fixed charges, negative at pH's above about 3
(Schônherr and Huber, L977). The penetration of ions
through the polymer matrix can be quite rapid, depending
upon the ion exchange capacity of the polymer (McFarIane
and Berry, 1974¡ Schönherr and Huber, 1917). The ripid
92
part of the cuticle, however, constitutes a severe
barrier to the movement of ions and polar molecules
(Schönherr and Schmidt, I979). Schönherr (I976) showed
that its removal with chloroform increased water
permeability by a factor of 300 500. More lipid
soluble substances, therefore, are likely to penetrate
the cuticle much faster than ions, although the
difference in permeability wilI be related to the
proportion of Iipid in the cuticle. urea has been found
to cross some cuticles more than I0 times faster than
Rb+ ions (Yamada et aI., 1965) while Lendzian (1982)
showed that the oxygen permeability of isolated cuticles
from various terrestrial plants was 44 Lo nearly 300
times the water permeability. The approximate 02
permeability of the v. spiralis cuticle postulated here
(3.82 x 10-6 m =-1 ) is considerably higher than the
values determined by Lendzian (2.72 x 10-7 i--42 x 10-6
m "-1), as would be expected for the thin cuticle of V.
spiralis and submerged aquatic angiosperms in general
(Arber, L92O; Schönherr, L9B2).
93
PHOTOSYNTHESIS
I. Photosvnthesis of UIva riqida
( i ) Results
Photosynthesis was mainly studied as 02 evolution,
using the 02 electrode. Fig. 3B shows the effect of the
pH of the medium (¡'Sw) on the photosynthesis of U-
rlgida under saturating concentrations of inorganic
carbon (rur 2.2 mM). There is a slight maximum at around
the pH of the alga's native sea water (pH 8.3), but
photosynthesis is not greatly affected by pH at values
Iower than this. At higher values there is a fairly
steep decline in photosynthesis, although at pH I0.2 the
rate is stilI some 30% of the maximum rate.
In Figs. 39 - 43 are shown the relationship between
photosynthesis at 25oC and the calculated
concentration of CO2 in the surrounding sea water, at
four different pH's and with various unstirred layer
thicknesses or degrees of slicing of the tissue. There
is considerable variability in the maximum rates of
photosynthesis that are attainable by tissue of fixed
surface area. Nevertheless, it is clear that the
perspex artificial unstirred layers have a marked effect
on photosynthesis at low pH (fig. 40). Some of this is
due to the fact that the surface area available for
diffusion in the perspex unstirred layers is around 30%
of the surface area available for transport were the
94
l5 o
a
o
oo
LIv)"¡s
NìÈ
{..e(r)0l.c$J
*\D,o+ro
Ë-l-¡
AJ
a
o
a
O
0 6 lo1
PHF5l'/i
I,IGÛRE 38. photos¡mthetlc 02 evoluüion of U.rigjrdg-(two t4 nn ctie¡reterctiscs)-at various- pEts of tr'SW. T = 2O.5oC,
@
40o
Ittì
f{I
È
N.ì
È.
è
\{a$R\t'ì,
+)q)e
o/
s.//
-6
¡ -a-
--
¡
1 e
I
o
ot
ot
I
/.
I
I
Io
;!;I,l
/I
I/
,/,/
a
Á o/
o
oo
.@- /o
¡II
Ecozf ç"a*!ff"a) {¡rltL)
FIGIIRE l!. photosynthesis of. in ASlr + 10 nIú MES, pE 5.45 - 5.92, at various concentrations of co2-for: (o) piece
or trrälus-i" n(õl side only exposett to sol-uiio", 1o) same with both sides ex¡rosed' (a) slices antl
(o) ferer slices. Inset shors photosynthesis of Ulva swarmers. T = 25 Q.
!50 2þ
@IbJt5sìó1oI
!è
S..
r NsÉÏ
l20
x
r4 -*-x
t
A
^
/
/¡ç)
{6æ\
-d.o\.
A
//
-
Âa ./
L
/ ÂAt
I /
II
JI
tr
/
/
I
(nc)
\(\vì)oso
Ë_{Jq)\
Â
A/o
troI
o
/I/
¡A
Âa^
2Ø
tæzl
FIGURE 40. Photos¡mthesis of II.rigida itl êtr.+ 10 nU.lfEp'of thallus in 3(O), (^) in 3(1)' (o) in B(1a)'
(calcutad.) W)pH 5.45 - 6.00, at various eoncentrations of CO2 for: (o) piece(a) in ¡(z). T = 25oc.
þ /./ A
tæ zæ
'9.Ç¿ = iL '(PauTura+eP+ou €ar€ ece¡:ans) sacTTS Jo srsaq+ufsoloqd saot{s +asul '(¿)g uf (V) pu€ (€t)a uf (o)='(O)U uf snTT?q+ Jo
acald (o) :to¡: Zg3 Jo suor+€r+uocuoc sïì-orre +€ 'Ol'L - tE'9 Hd tSgW ytt OL + üST ut €!-ç Jo srsêr{+uÁso1oq¿ 'L? 8UnÐI.{
Qll (w+rl'qþr) LzocfæÞæ-w¿ool
/loI
t¡
/
v,/
/I
IoI
o
/
/;s(ùc\
IS. oöh
t(^\sÈ-
ooz Ðt^twì
IÈe_
ÞÈ
t,(.^
.!
-.J I
v./aat'
v'v
tr---tro
I
I
I
I
I
Io
T3SNI
(AI
a
09
o
o
/
aIoæ
-Ûð
oÞ az
o
@,
o
oI'/)ñ
þd̂
ö€s$ +o"ÐÈ*ìsè\
)
/o$os
/
À/
o
/ l
A
^
^
/
/o
I
I
L '/ tr
i" l^ / /
E- ^
A
-otr
-¿Â
- -À
5o
Ã-
.^
foô 15(o^
Ecozf GalcuJatú) flL)
_- o)- -O - o
/¡
Ivl
60INSET
Tq_
Ø
TIGIIRE 42. ?hotos¡rnthesis of u.rigitta in ASW + 10 nM.[ES, pE I.:QO - 7.72, at various concentrations of co2 for: (o) piece
or thalrus in B(o),.çffi"iri,"tål'i"*nti;i'"',ã qÃj-i"'Äi;i'. ü""t - photosynthesis or srices' ( surracearea not detemined). T - 25"C.
ooa
o ./
ln -orlI
Ë
so\
.È
.eoò\.sÈ too\.o'{-r
€ô+Jq)s
,//
/
/tr
Á
/
zl" tr
I tr/
,/
/o
/
"lo
//'
/zz 6 6 lo
fcqf (calurlaral (fi,f"
FTGURE 43. photos¡mthesis of u.ri-gicla in as
perspex holders not present (i.e. if the piece of
thallus were "bare"). In Table 5 the initial slopes of
the curves in Figs. 39 43 are corrected for surface
area effects; when surface area is taken into account,
there is little difference between photosynthesis of a
piece of UIva and the same piece in holder B(I) (cf-
Figs 40 and 42).
complicated geometry the system,Because of the
initial slopes of
S\^/armerS are IeSS
meaning. However,
concentration
meaningful, or
the apparent
have a less
slices or
obvious
can be ( see
curves
of
for
TABLE 5
rnitial slopes (m s-l xJ-O-5¡ of the concentrationcurves shown in Figs . 39-43, with photosynthesisexpressed on the basis of the surface areaavailable for diffusion in the boundary layer.
pH of ASW
HoIder 5.45-5 .82
6. s3- 7 .60-7.r0 7.7;
8.40-8.65
B(0)(both sides exposed)
r.13
B(0)(one side exposed)
r.14 3.20 22.L 50.7
B(r) 1.33 22 .0
B(Ia) 0.49r L.97 10.7 28 .4
B(2) 0.222 L.03 4 .62 20 .9
95
KM's f or CO2
below) an indication of diffusional limitations and
these are shown in TabIe 6, together with the xftPP ¡ot
pieces of Ulva thallus. (The XftPP values were obtained
by using the V predicted by the computer programme
FVKUP, and assuming the rate to be Iinear with
concentration for tco2l < Kftpp.) Here again, surface
area must be taken into account for pieces of thallus'
since the holders cover a portion thereof (fig. 98).
The Kt?pp "aIues shown in TabIe 6 are derived f or the
initial, linear part of the concentration curve
corrected for surface area, i.e. with photosynthesis
expressed on the basis of the surface area available for
diffusion in the boundary Iayer.
TABLE 6
Apparent KM values ( ¡,tM CO2) for photosynthetic O
evolution 6y UIva in'buf fefed ASW at various pH's
pH of ASW
2
5.45-5 .82
6. s3-6.60
1.60-1 .15
8.40-8.60
swarmers 3.6
few slices 6.9 0.35
slices 9.7 5.3 0.98 0 .42
piece ( bothsides exposed ) t5
piece ( oneside onlyexposed )
15 tots* 1.5 0.24
*Assumin700 nmo
g V cpulÇ be anywhere between 3 80 andImos
96
In a number of experiments, I used holder B(Ia) with
theholes filled with a2% agar solution of ASW + 10 mM
MES; the external diffusion path length is 1.33 mm.
Here (nig. 44) rates of photosynthesis are expressed on
the basis of the surface area available for diffusion,
i.e. the combined surface area of the holes (9Z.Omm2).
If photosynthesis \^Ias completely dif f usion Iimited by
the unstirred layer, the slope of the line in Fig. 44
would be the unstirred Iayer permeability (i.e. kr).
Using a diffusion coefficient fcr Co2 Ln sea water of
t.BI x 10-9 m2 s-l (itterpolated f rom the data of
Ratcliff and Holdcroft, 1963), the predicted kf is
1.36 x l0-6 m =-1, which is in good agreement with the
observed slope of the line of best fit (1.30 x I0-6
* =-1).
Fig. 45 shows some results from the stirring
gradient tower using 14C-Iabelled inofganic carbon in ASW
at low pH. The predictions of FVKUP for this data are
shown in Table 7.
Rates of photosynthesis at high pH in the presence of
the carbonic anhydrase inhibitor sulphanilamide (r'ig.
46) are markedly decreased at Iow CO2 concentrations
(cf. Fig. 43). The apparent KM for COZ has increased
f rom 0.24 ¡.rM to about 3 ,0{M. This large eff ect is not/timmediate; for the experiments shown in Fig. 46 the
tissue had already been exposed to 0.5 mM sulphanilamide
for two days. The response is similar with or without
97
l'-oulf{(
sN.ì
N
3=s\tSe}\o.L¡o
Ë_{)q)
E
E
E
EE
E5@
I a
E E
2oo 1ao 6æ
a
fco=) 6c^Ìculard.) ?,1,1)
FIGIIRE {{. Photosynthesis of U.rigicla ln ASÏI + 10 mM MES, pH 6.55 - 5.84,Z5oC, at various concentrations of C0, fo.r a pfac_e of thallusin B(la). fhe holes were filled ríth â" $ egar solution of thebathins nediun, giving a diffusion path length of 1.33 mlr. The
solitl iine -(caÍ"utatc¿ by linear regression) nas a slope of1.30 x 1f6'n s-1 (r = 0.899).
+II
+
{
+
t
îrt)
"lsoìto(
Itt
Ao
A
l+
I
50l@Ã (d.uwafo c¡"tvt>
t50 bô
FIGIIRE 45. Photosynthetic 14C02-fi*atj.on in Âff + 10 nX [8S,,ueasllrêd in sti¡ripgl e3adient torer at levels 1 (
pE 5.19 - 5.r), 2 (o), ¡
!O, at varions concent¡atlons of C02.(r), + (¡) and 5 (o). r - 25oc.
5æ
Ã
I
Iúl
.{'È
dEÈ
E
to
2þ
o
Áa
o
oo
o
I0
ÌfÉ¡
L)a)I
oI t
1o[corT ("ol,i"tá)
ç^l/t)
FIGUÎE 46. photos¡mthesis of Llier(la, 1n .å.SiW + 1O nl{ TåPS + 0.5 EM sulp}ranilanide, pE 8.32 - 8.66, at va¡iorrs concentrationsof c02. piece or tnariüFin ¡(õ) ritrr botb sittes of the tbalrus exposecl to the solution; (t) * 5 ¡u PIBA. (o) a¡a(f) "¡" tro ctifferent er¡reriments. T = 25'C.
PABA which suggests that the effect of sulphanilamide is
not associated with one-carbon transfer reactions
(Scheffrahn, L966). The rate of penetration of
(completely dissociated) PABA' however, maY have been
slower than that of (mainly neutral) sulphanilamide. In
the presence of both sulphanilamide and PABA there was
the rare occurrence of net 02 uptake in the tight-
TABLE 1
Predicted VtIues of V, KM and kt for the data shownin Fig. 45 , f rom the progranìme FVKUP.
Level I Level 2 Level 3 Level 4 Level 5
V 2L90 2r7 0 L230 1rB0 1080
(nmor m-2 =-r)
KM 941 1sB 7.11 0.394 5.50
'r*kt*(m s
co2)
IO
-r )
5 0.631 19.6 2.67 3.03 36I
regres s loncoef f icient
0.997 1.000 1.000 0.997 0.s69
* Rates= L4'7caI cu
of photosynthesis corresponding to ICO2]/{/M were omitted by mistake in theseIations.
9B
(ii) A note on the meaninq of Kn, for photosynthetic---¿--wl
c9_z f ixation
The reduction of COZ in photosynthesis, as catalysed
by ribulose bisphosphate (RuBP) carboxylase, is a two
substrate (excluding water) reaction:
coz HZO + RuBP -+ 2PGA (phosphoglyceric acid)+
The KM of RuBP carboxylase for CO2, therefore, will
be affected by lRuep], as wilt the maximum rate (cf. the
reduction of OZ). Farquhar and co-workers (Farquhar'
I979; Farquhar, von Caemmerer and Berry, I9B0; Farquhar
and von Caemmerer, I9B2) have suggested that the binding
of RuBP is so rapid that it is the rate of the supply of
RuBp rather than its concentration which is important
and that the former process governs the maximum rate of
photosynthesis at saturating ICO2) while at Iow (sub-
saturating) [Co2], the enzyme can be regarded as
saturated with respect to RuBP.
On a surface area basis, the maximum rate of
photosynthesis of U. rigida is very variable (Fi9s. 39
45¡ cf. Beer and Eshel, 1983); if the variations are due
to changes in lnuep] or the rate of supply of RuBP then
there wil 1 be large changes in the apparent K¡1 of
ribulose bisphosphate carboxylase for COZ. Thus, a low
V may lead to a low estimate of K¡4t however' since the
maximum (i.e. COZ- and RuBP-saturated) rate of RuBP
carboxylase sets an upper limit on the rate of CO2
99
fixation, â high V will not Iead to a high estimate of
KM, COZ transport Iimitations aside. For the Kt values
in Table 6, all of the maximum rates at high pH (8.40
8.60; Fig. 43) were particularly low. This does not seem
to be a direct effect of pH (nig. 3B). It may be that
the observed K* values are underestimates in this case.
(iii) Photosynthesis at Iow PH
The most extensive data are for photosynthesis at low
pH (Figs. 39, 40 and 45) where diffusional limitations
are very significant for the photosynthesis of U.
rigida. This is true even for a bare piece of thallus'
(Fig. 39) with the result that the apparent K¡4 for CO2
decreases f rom about tS yVt to Iess than 7 ft
tn slices'
to less than 4 uM in swarmers (Tab1e 6). Maximum rates/
of photosynthesis were reasonably similar, so this
comparison is probably meaningful. Indeed, on a
chlorophyll basis, the rate of photosynthesis of the
swarmers was higher than f or sl- ices or a more intact
piece of tissue.
For the Iatter, the internal diffusion pathway might
constitute an important part of the diffusional path
length for COZ transport. Internal diffusion includes
both diffusion in cell walls, and diffusion across
membranes, cytoplasm and within the chloroplast.
Because of their thinness and typical composition, CO2
transport across membranes is likely to be fast
100
(Forster, I969; Gutnecht, Bisson and Tosteson, L977¡
Gros and Bartag, I979); the cytoplasmic diffusion
pathway, too, is relatively short because the
chloroplasts tend to cup the inside of the cell close to
the outside surface. The single chloroplast in each
cell is large, however, and it might be possibl-e for CO2
transport within the chloroplast to become a limiting
factor. At U f*
Co2r the rate of photosynthesis of the
svvarmers is about 50 ¡mo1
(g chI )-1 s-I {nig.:o ) which
is about 2.7 x 10-11 ,^oI s-l fot each s\^Iarmer (based/
on 60 mg chl- m-2, I.4 x 10r0 cerls m 2 and, assuming
(Haxo and Clendenning, I953) eight sh/armers per "body"
cel1 of the thaIIus. It is difficult to estimate the
volume of the chloroplast because it is highly
convoluted. Assuming it is spherical with a diameter
half that of the swarmer cell (,^, l0 f^), the volume is
about 6.5 x I0-17 *3; photosynthesis per unit volume of
chloroplast would then be 420nmol m-3 s-l at 5 ¡,rM Coz.l"
By this samé analysis, the maximum rate of
photosynthesis of the swarmers would be 750nmol m-3 "-I.This compares reasonably well with maximum rates of
photosynthesis in chloroplasts isolated from other
plants; an average value would be 150 ¡mol
(mg cht¡-1
h-l or about r600mmor m-3 s-r (HatI, Lg76). Based on
the above estimate, the modulus 0 (equation (29)) for
swarmer chloropl4sts is about 0.32 at c5 = t f*
COZ.
This assumes that transport limitations external to the
10r
chloroplast envelope are negligible and that the COZ
diffusion coefficient is reduced 10-foId within the
chloroplast - this is probably a low estimate based on
some of the diffusion coefficients which have been
measured in cytoplasm (caillé and Hinke, L974)
rf 0 = 0.32, the chloroplast wiII be 82 1002
"effective", depending upon the value chosen for K¡1r so
it is likely that 3.6 /lM is a good estimate of the true
KM for COZ fixation i.e. COZ diffusion within the
chloroplast is not severely rate limiting. For a piece
of Ulva, internal diffusion Iimitations will be somewhat
Iarger because the CO2 supply to the chloroplasts is only
from the two outer surfaces of the thallus (cf.
swarmers, which are surrounded by the bathing medium)
and because of the tortuosity and solid bulk of the cell
walls. There are also the additional, external ("in-
series") resistancesof the outer celI walI ( witn its
thin, mucilaginous cuticle) and a thicker unstirred
Iayer due to the increase in the size of the body - pp.
16,20). These effects may combine to increase the
observed K¡1 for CO2-fixation to about 15 ¡rM (table 6,
Fis 39).
The external unstirred layers imposed by the perspex
holders also have a marked effect at this pH (fig. 40).
With due allowance for surface area affects, TabIe 5
shows that initial velocities are decreased more than 2
and more than 5 fold by holders B(Ia) and B(2)
L02
respectively.
The results shown in Fig. 45 can be compared with
results from the zinc disk experiment. The stirrer was
on "7" (420 r.p.m.), so kr's must be multiplied by 0.878
(see p 57). using ¡coz (in seawater) equal to 1.BI x
IO-9 m2 s-1 (Ratctiff and Holdcroft, 1963), kT wilI be
futher reduced by { (r.Bl x r0-9 )/(3.01 x r0-9¡¡7s, or
0.703 (cf. p 57). With predicted kr's of 0-573, I-33,
2.67t 3.80 and 5.25 x 10-5 m s I (levels I to 5
respectively)' KM (cOr) = 3.U /^*and V = lI50 nmol m-2
s-I, the set of curves shown in Fig. 41 is generated by
the Briggs-MaskeIl equation.
Taking internal diffusion into account, bY using R =
5 ¡m ( roughly the thickness of the chloroplast "Iayer"
in U. rigicla) and D"¡¡ for CO2 within the chloroplast
Iayer of 0.603 x l0-9 m2 s-I (i.e. one third of ocoz in
sea water), produces the curves shown in Fig. 48. The
fit of the latter set of curves is only a slight
improvement on that of the former at low [CorJ.
Table 7 shows, and it can be seen by inspection,
that the very best fit to the data of Fig. 45 is
obtained with Kg, on the whoIe, increasing going from
level 5 to l. Such an event would be inexplicable in
terms of a simple first-order enzymic reaction, coupled
with diffusional limitations. A possible recourse is
to the oxygen inhibition of photosynthesis; this will be
103
II
+
þo
¡rl)
"l{,J
ìË(
+
tAI
Ða)s.¡
+
A
Lc-o;l (*t uwatio fJvL)t50
+I
+aI
I+124,0
¡ú)
"ld
ù
ìtoÈ
+
I
+
A
A
fco;l (d.u&atio c¡'U>f50
FTGURE {g. photos¡mthesis-fco21 cu:lrres based on the Yanané quation (equation (v.zt¡¡, with the data of Fig. {1. see text' for values of the Parameters.
discussed in the next section. Whatever the predictions
of FVKUP might mean, it is interesting to note that the
predicted, true KU's for the three most well-stirred
Ievels of the tower are still very low (5.33, 0.394 and
7.If ¡,rM) compared with the apparent KM's (-lB, 24 and
29 þLM), i.e. diffusional limitations are signif icant
even at high rates of stirring.
(iv) oXI inhibition of photosynthesis9e n
In most marine Chlorophyceae, including Ulva
IactucarL., the major enzyme catalyzl-ng carbon fixation
is ribulose-1,S-bisphosphate carboxylase (Kremer and
Küppers, 19'7'7; Colman, L9B4). This enzyme is also an
oxygenase' catalysing the reaction of ribulose
bisphosphate with oxygen to form phosphogl-ycerate and
phosphoglycollate. The overall kinetics of carbon
fixation can be regarded as Michaelis-Menten with
competitive inhibition by 02, the overall Kt being given
by
KM
nQz"a
^fli^fio"
( r + (35 )
(Laing, Ogren and Hageman' I974)-
KM's of RuBP carboxylase/oxygenase
respectively and "$ is the oxygen
active sites of the enzyme. With
limitations' the rate equation for
K are the
for CO2 and 02
concentration at the
COZ transport
COZ fixation will
^fio" and O7
M
104
then be
tcoa å{ txfioz ¡r
xfio" ¡ r lof;o. + cfioa¡f;oz-* acfiozk$o'u)
+ + .to'tfio' + V)
2c ]-
+ v)
(36 )
by substituting equation (35) into the Briggs-MaskelI
equation. If the rate of 02 transport away from its
place of generation is slow compared with rates of
photosynthesis, then clz wiII also be a "steady-state
concentration", as we have assumed is the case for .1O?
The rate of 02 transport can be written
,o, k32 ( c?¿ oo
2,) (37 )
based on the simple Nernst description (i.e. ignoring
simultaneous diffusion and reaction, or at Ieast
partially incorporating it into k1). In the steady
state, and with a photosynthetic quotient of one, the
rate of photosynthesis, v' is the rate of COZ fixation
(vCO¿, equation (36)) which will equal the rate of 02
evolution. (voz ,equation ( 37 ) ). Substituting the
expression for "9t from equation (37) into equation (36)
generates a quadratic equation in v; it can be shown
that the solution is
c
r05
ofiork$o, t r .to" o3o'*-l
V + + V
"k 2T
^fio'afio. ¡r
^Oz-b_-t^fiJ
"to"kço"+ v)2- 4t1+ + lc o2 uJ.J
Cb
rf.8^=0andreduces to the
^fi'k3"(38 )
k3" tends to infinity, this equation
Briggs-Maskell equation. rf "8^ = 0 but
f<fris f inite, v wilI still be Iess than predicted with
no oxygen
zero ( see
.9t wilI be greater than
To get an idea of the importance of oxygen inhibition
of photosynthesis, a knowledge or xfi¿ and the relative
magnitude of t$2 ana f$Oz is required. If it is assumed
that the two transport coefficients are roughly the same
and that Kfiz is about ZOO ¡tv,
(Tolbert, l9B0), then
oxygen inhibition can be quite significant at high
values of cflz and Iow values of kt (Fig. 49b). wh"tt "fl2is 1ow, the effect is minor because of the Iow affinity
of the enzyme for 02 compared with co2 (i.e. the factor
xfio'7xff2 i" smarr ) .
In my experiments using the 02 electrode, "fl- tt=
never greater than 120fM and estimates of kt range
from 0.2 Lo 1'3 x l0-5 * "-I (Tabte 5)- For the thicker
unstirred layers, therefore' photosynthesis may have
been slightly Iowered by oxygen inhibition' compared
with the rates which would have been expected were 02
transport infinitely fast. Based on a fit to the
inhibition, because
Fig. 49).
r06
looo (ar
(b)
cþ -- soo¡lvt
o^cú^
k?"
-i"
=O
6oo
+æ
o
5@
o
lo
1I\if)
r{t{^nlSso
dì'
EË
.gvì
0/*s
tho*)o
-c\
\.,
Íds
+æ
=Q
o.5Eco,-f
Ko^=2*r
t6e ¡n s-r
(c)
Kl = zx ld ,ts'l
ét=r*¡.,.ut
oo ¡t/t-o.t-le
-- /x 1166
tn- s-l,ozK¡
5
5oo
o 1.oo1rll{L)
FIGIIRE 49. OZ inhibition of photos¡mthesis, i.e. equation (38)
ît3', lo'0"i,,31r':r5 Æ'"iu lïrl ? =roïoïEi
1 0-6 m s-1 .
Briggs-Maskell equation, val-ues for kf;O' would be
slightly underestimated, i.e. the predicted unstirred
layer thicknesses would be too Iarge. The effect might
be more important for the experiment shown in Fi9. 45,
in wfricfr cf;z was not measured but may have been close to
air-equilibrium (¡ry 2OO ,uM)t then "KM" would indeed
have been increasing with increasing 6 , íf "KM" is
given by equation (35), as FVKUP predicted-
It should be added that the diffusion of 02, compared
to co2, is likely to be faster for the same diffusion
path length because of the ratio of the diffusion
coefficients (DoL/Dco" = L.29 at 25oC in pure water;
Himmelblau, L964). 02 transport, therefore, may impose
smaller Iimitations than CO2 transport on
photosynthesis. However, there can be facilitation of
COZ transport by HCO3 ions (see next sub-section), just
as there can be facilitation of OZ transport by
macromolecular carriers (Raven, L9l7). In the diffusion
boundary layer, k, has weaker dependence on the
diffusion coefficient than in a strictly unstirred
sol-ution (i.e. compared with transport within the
thallus ) .
(v) Photosynthesis at higher pH's
In ASW of higher pH (6.5 or more), it is immediately
apparent that, for the same lCo2), rates of
photosynthesis are considerably increased (Figs- 4L, 42,
r07
43; Table 5). This is the typical response of a
user" and can be taken as evidence of HCOã "use"
f970). What is the nature of this HCOã use in
U. riqida?
increase relative
"HCO' -(Raven,
The simplest explanation would be that HCO3 ions
enhance CO2 transport across the boundary layer in the
same way as HeOfr- ions can enhance H2PO4 transport. A
direct prediction of this mechanism is that the
constraints on photosynthesis due to diffusion of CO2
across the boundary Iayer should become progressively
Iess significant as tHcoJl and tco3-l
to lcorl ti.e. as the pH increases).
On the whole, this prediction is not fulfilled- At
pH 6.55 - 6.84 (nig. 44) the predicted kr f or CO2 is
very close to that observed, i.e. there is little if any
enhancement of the CO2 flux even though IHCOt] is more
than five times Ico2J at the pH of the experiment and
the unstirred layer is very thick. Similarly, at pH 7-6
- 7.75 (Fi9. 42), the initial slopes of the
photosynthesis - [CO2 ] curves for various thicknesses of
the unstirred Iayer are in roughly the same proportions
as they are at pH 5.45 5.82 (Table 5). This is
despite the fact that [CO2] is less than 2% of the total
inorganic carbon at pH 7.6 (cf . /!70? at pH 5.6).
AtpHB
0.25% of
4 8.65 (Fi9. 43), where lcoZl is only 0.L2
total concentration of inorganic carbon,the
IOB
there is some evidence for HCOJ enhancement since the
CO2-response curve for tissue in holder B(1a) is close
to that in B(2). Table 5 shows that the initial slope
of the "B(la)" curve is still about half that of a piece
of thallus in B(0) when this initial slope is based on
the surface area directly exposed to the solution. If
the flux of COZ to those portions of the thallus beneath
the vinyl of holder B(0) was greatly enhanced' then a
better basis for the calculation of the initial slope
would be the total surface area of one face. The slope
would then be 31.8 x l0-5 m s I in B(0), i.e. much the
same as the initial slope of photosynthesis vs. lCorl in
holder e(Ia), and close to that in holder B(2). There
is, therefore, a case for HCO3 enhancenent of the CO2
flux at pH 8.4 8.65. Weakening the argument, however,
is the fact that, if there is strong enhancement, AII
the initial slopes, whatever the type of holder' should
be calculated on the basis of the surface area of one
face (even both faces?) of the tissue since the
diffusion of CO, within tissue not directly beneath a
hole wilI be fast. The initial slopes would then have
just the proportions shown in Fig. 43. Moreover, the
maximum rates of photosynthesis are quite löw"(I60 260
nmol m-2 =-1, cf. 300 - 1500 nmol m-2 s-l fron Figs.3B,
39,40,42 and 45) which would lead to a Iessening of
diffusion restraints not necessarily associated with
HCOã enhancement. Indeed, at pH 6.53 - 7.10 (fig- 42),
109
Reactions involving CO2 , and some values of rateconstants, equilibrium-constants and diffusioncoefficients in sea water of the chlorinity of ASW (20e")at 25oC (from Buch, I960; ndsøll7 1969; Johnson, I9B2).
Reactions
TABLE B
Relevant rateconstants
kl +k2_13.62xL0'
ku.co": k-l
Equil ibriumconstants
lrz co3lLco
2J2.51xI}
(H+)[HCo3]
I . HCO3 kcoz KA
-1 -3S
+ HzO
+k-zKi,rcoo IH CO I- 2 3'
4.02x10-4^H.c03 -
M_lL4.2 s '
2.CO2 + OH HCOã O"or=_A -1 -ll.3xl0=M's'K
1-ottcol -_A -11.17x10 = s !
HCot (-K1'lKoo )
t Hcot lwrxloB ttl-1
(H+)tco3-l_ ')_3. HCo3
- coã
I+ H' K2
4.Hzo q+ H+ +oH
Diffusion coefficients (*2 s-1 x 10-9)
KW
I HCoã l
_oI.l4xl0 ' NI
(H+)toH-l
1.03x10-14 M2
From RatcÌiff and Holdcroft' I963.From D for CO2 using Stokes' Iaw.From Kigoshi and Hashitani' 1963.Corrected for ionic strength usingthe activity coefficients ofPytkowicz, I975.Estimate by analogy with the çffectof ionic strength on D for SOîRobinson and Stokes , L959.
Coz :
H2CO3
HCO3
I.81 A
z L.52
0 .822
b
abC
dcrd
23
cCO 0.523 c r e
photosynthesis in holders B(1a) and B(2) is also low and
the initial slopes are again similar.
The data over the pH range 5.5 - 8-5 do not allow a
firm conclusion as to the importance of HCO3 enhancement
for the COZ flux through the boundary layer and cell
walIs. It is worthwhile to consider whether any
theoretical predictions can be made. The uncatalysed
rates of the reactions betvreen CO2 and HCO3 are
relatively slow (cf. the dissociation of H2Po[) which
means that the diffusion of CO2 also needs to be slow if
the reactions are to keep pace with diffusion and
influence it in any way. Friedlander and Keller (I965),
showed that chemical reactions within the diffusion
boundary layer have a significant effect on the flux of
a particular solute only if the thickness of the Nernst
Iayer, 6 , is much greater than a "characteristic
length" for the reactions involving the solute. The
relevant reactions of Co2r together with equilibrium
and/or rate constants, are shown in Table B. An initial
estimate of the "characteristic length" is simply
VWr"*, which is 22n t^. This woutd be the varue
if the rate of the hydroxylation reaction
COZ + OH- -+ HCOJ was negligible. At pH 8.4, the rate
of this reaction is significant. The "characteristic
Iength", based on both the hydration and hydroxylation
reactions is given by Me1don, Stroeve and Gregoire
(1982 ) :
110
It- +ocoz
ç
tkHcoS Kl (A tHCo3 I )
-lz
2K2lHcoã l
which IHcoa] is a mean tucorl throughout the Nernst
mean diffusion coefficient for HCOã and
A is egual to tHCoJl + 2Íco3-1. Assuming
approximately IHcor] in the bulk
l_n
Iayer, D is a
23CO ions and
that Hco3 I issolution (this gives a maximum value of the
"characteristic length" ) and that il is midway between
DHcoã and gCoz. , the "characteristic length" is 159 ¡mat pH 8.4. By comparison, the thickness of the
unstirred layer is about BI5 ¡m for holder B(2) and
about 370 üm for holder B(la). (These values areI
obtained from the initial slopes at low pH (rable 5).
They are good estimates in that the apparent K¡,1 f or CO2
is so far removed from the "true" KM, and the effect of
oxygen inhibition (Fig. 49) is probably sma11.)
For the thick unstirred Iayer then (holder B(2)),
some enhancement of COZ transport through it is expected
since the "characteristic length" for the uncatalysed
reactions of COZ is less than one fifth of the unstirred
layer thickness. Calculation of the degree of
enhancement is greatly simplified if it is assumed that
the pH gradient across the diffusion boundary layer is
negligible, although given the relatively mild buffering
used (I0 mM TAPS) tfris is probably not a good
lTI
assumption. In any case, the constant pH assumption
gives the maximum enhancement possible based on the
uncatalysed hydration and hydroxylation of CoZ. Using
the solution of Hoover and Berkshire (L969), the
predicted enhancement is 5.45 times the COZ flux in the
absence of chemical reactions in the diffusion boundary
layer. In the calculation, I have used a mean diffusion
coefficient for CO2/HCOa; this is more realistic than
assuming (as Hoover and Berkshire did) that the
diffusion coefficients of the inorganic carbon species
are equal (see Quinn and Otto, I91L). If there were no
buffering at aII in the unstirred layer, apart from the_ ')_
CO2-HCOã-CO|- system, the predicted enhancement
would be something more than 20e" lower (Quinn and Otto,
L97r).
In Fig. 43, then, the similarity in the two CO2
response curves for U. rialda in holders B(Ia) and
B(2) may weIl be due to HCO3 enhancement of the rate of
COZ diffusion through B(2)' and there may also be a
smaIl enhancement of diffusion through B(1a). However,
such a mechanism stil I cannot account for the "use" of
HCO? by U. riqida since' even if the transport of CO2 to
the plasmalemma were infinitely fast, its equilibrium
concentration at the plasmafemma would not be sufficient
to support the observed rates of 02 evolution- From the
experiments at low PH, the half saturation concentration
of Co2 with negligible transport Iimitations is about
LT2
3.6 r' At pH 8.4 8.65, however, (t'ig. 43, Table 6 ),
Klrl f or CO2 is one tenth of this value -the calculated
0.24 - 0.42 üM./
It is possible that the pH of the cel1 wall is lower
than that of the bulk medium, given a Donnan
distribution of H+ ions. Then, at equilibriuml tco2l
would be higher in the celI wall than in the bulk
solution or the unstirred layer. However, equilibrium
would never be approached in the celI wall: the
thickness of the outer cell wall is only about 2 ¡n
aL
the most, so that, tinless diffusion coefficients were
reduced B0 fold or more, the diffusion of CO2 across
the wal1 would be rapid compared with the rate of
transformation of HCOJ into CO2 in the wall. The
effective ICO2] at the plasmalemma,therefore, would not
be affected by the watl pH, unless the wa11 pH was very
Iow (cf. Walker, Smith and Cathers' 1980). This may not
be the case if there vùas a powerful catalyst inthe ceII
walÌ. There are a number of substances that are known
to be CO2-hydration catalysts ( e-g. oxyanions of weak
acids, such as arsenite ion, hypochlorite ion). The
most powerful catalyst is the enzyme carbonic anhydrase.
If the celI wall was catalytic, then equilibrium between
CO2 and HCOt might be approached even for short
diffusion times. The experiments with sulphanilamide
(rig. 46) suggest that carbonic anhydrase is involved in
photosynthesis, but the lengthy pre-treatment that is
equil-ibrium between CO2 and HCOj/Cos in thethe equilibrium COz(waIl ) =
CO2 (butk) .
t< That is,cell wall,
thenot
115
required for a significant effect implies that the
enzyme is not located in the cell waII, which should be
readily accessible to the inhibitor.
Even if equilibrium was approached in the cell wall'
for there to be a ten fold decrease in the apparent Kt
for CO2 at pH 8.4, the pH in the celI wall would need to
be about 7.5. This seems unlikeIy, particularly since
there is a net influx of H+ ions during the
photosynthesis of at Ieast two UIva species which
increases the pH of the boundary layer above that of the
bulk solution (Cummins, Strand and Vaughan, I969)-
(vi) Mechanisms of HCO 3 u SC
Given that an enhanced CO2 flux across an unstirred
layer, due to the presence of HCOã, cannot sole1y
account for the observed use of HCOã ions by U. rlgida'
the next (in order of decreasing economy) additional
postulate is that the celI membrane is sufficiently
permeable to HCOt ions to aIlow a significant flux of
the ion across the plasmalemma. If there were no
catalyst for converting HCO3 to CO2, the cell would be
no better off in this situation. With a catalyst' at a
pH of 7 tn the cytoplasm and using an estimated px! for
HCOã of 5.98 (i.e. the same as in sea water), HCOt would
be in equilibrium with about B? as much COZ- The
required steady-state IHCOJ] needed to account for the
rates of 02 evol-ution that are observed can be
r14
calculated, therefore. For instance, ât pH -8.4' 0.5
COZ in the bulk phase gave a rate of 02 evol-ution of
about 80 nmol m-2 =-L (piece of thallus in B(0), Fig.
f*
43). At pH - 5.5 (fig. 39) the comparable rate is only
5.8 nmol m-2 s-I at 0.5 ¡rM CO2. This low rate, however,I
is partly because of the Iimitations imposed by the
transport of CO2 upto the plasmalemma. The results with
swarmers (Fig. 39, Tabte 6) imply that the rate can be
increased more than four times at low lCOrl if the rate
of transport of COZ up to the plasmalemma is increased.
Per unit area of thallus, the rate of 02 evolution would
be 24 nmol *-2 "-1 at 0.5 ¡rM CO 2 tf CO2 Lransport upto
Ithe plasmalemma were as fast as it is in a well-stirred
suspension of swarmers. The rate could be even higher
if transport limitations up to the chloroplast envelope
were entirely removed. (This would approximate the
situation for ceIls with an 'inside'(i.e. cytoplasmic)
source of CO2 from equilibration with HCO3 in the
cytoplasm. ) so cytoplasmic Hco3 woufd need to supply a
sufficient [CO2 ] to give a rate of photosynthesis of
(80 - 24) = 56 nmol 02 *-2 =-l i.e. about I.2 fM
Co2 aL
most by anatogy with swarmers. IHCOã], therefore, would
need to i¡e l5 pM, or possibly less, in the cytoplasm.
rn the bulk phase, [HcO¡] is about 200 fM
at pH 8.4,
0.5 ¡M CO2r so the required tuCOrl could be achieved
within the cytopl-asm provided the electrical potential
difference (P.D.) across the plasmafemma were not too
much more negative than -66 mV. This value of the
1r5
membrane P.D. is plausible; West and Pitman (I961 ) quote
a value of -60 mV for U. lactuca while BIack and Weeks
(1972) found only -42 mV in a species of the related
genus Enteromorpha.
Thus, by this equilibrium argument, the scheme
outlined above could work. The problem is that tHCO3l
is not an equilibrium concentration but a steady-state
one. The CO2 formed from HCOJ is continually removed
not just by the diffusion of CoZ back into the bulk
medium but also in photosynthesis itself. Ignoring the
Iatter for the moment, it is clear that the influx of
HCO3 must far exceed the efflux of CO2, since the
tendency is for ICOZI to equall_ze on either side of the
plasmalemma. It could be argued that diffusion
limitations, both internal and external, could lead to a
decrease in the efflux of CO2 formed from HCO, in the
cytoplasm, i.e. CO2 "trapping". The results shown in
Fig. 39 (Iow pH) imply that diffusion resistances for
COZ between the plasmalemma and the bulk medium are
significantr so that transport between these two could
be rate limiting for COZ efflux. This may not be the
case for HCOJ influx, where the potential concentration
gradients (and therefore rates of transport) are much
larger (at high pH). In this situation, therefore, a
thick Nernst Iayer may be an advantage provided the
influx of HCOJ is not too greatly affected. The results
at hiqh pH (rig. 43) provide some support, since in this
116
case slicing is detrimental from the point of view of
of COZ needed forthe half-saturationphotosynthesis; the
(Table 6).
concentration
apparent KM rncreases upon slicing
The extent of diffusion Iimitations to the membrane
transport of HCO: *iI I depend on the kinetics of the
influx. Assuming they are Michaelis-Menten, an influx
as large as that required for CO2 fixation could occur
even if the KM of the porter was quite high' because of
the high concentration of HCOã. Transport (bulk phase
to plasmalemma) restrictions would then be much less
severe than those that exist for CO2 fixaLion at low
pH. (It is also possible that the maximum HCo3 influx
determines the maximum rate of phtosynthesis. This
would explain why the maximum rates at high pH are low
and would also lead to an underestimate of the "true K*"
for COZ of the carboxylase. No firm concfusions can be
drawn from the measured values of the maximum rates,
however, given their variabifity. )
It is possible, then, to obtain a higher steady-state
concentration of COZ in the cytoplasm than in the bulk
medium provided the inffux of HCOJ is rapid compared
with the efflux of CO2. The work done to maintain the
dis-equilibrium would be the work required to maintain
the pH gradient across the plasmalemma (i.e. to keep the
cytoplasmic pH low in comparison with the bulk medium in
1r7
the face of a rapid
constant electrical
plasmalemma.
base), and to maintain a
difference across the
influx of
potentia I
It is also necessary that CO2 is formed rapidly
enough from HCO] to keep up with the rate of its
assimilation in photosynthesis. The presence of the
enzyme carbonic anhydrase in moderate amounts would
ensure that HCOt and CO2 ure always virtually in
equilibrium. The effects of sulphanilamide (rig. 46l,
are not immediately consistent with the inhibition of
carbonic anhydrase. The prolonged pretreatment that is
required is pûzzling, because even intra-cellular
carbonic anhydrase should be reasonably accessible to
sulphanilamide; Geers and Gros (1984), for example, give
a value of 30 minutes for the penetration of another
sulphonamide (acetazolamide, l0 ¡lI
r) into skeletal muscle
cells and the work of Holder and Hayes (I965) suggests
that sulphanilamide would penetrate considerably faster.
It is possible that U. rigi4a contains an iso-enzyme of
carbonic anhydrase which is more insensitive to
sulphanilamide than most animal enzymes (Everson, I910¡
Graham, ReedrPatterson and Hockley' I9B4). In that
caser rêIativeIy large amounts of sulphanilamide would
be needed within the cell for significant inhibition of
the enzyme. A concentration higher than that in the
bulk medium could be achieved by an accumulation
mechanism analogous to the classical model for ammonia
II8
INH3
accumulation by plant ce11s, i.e. diffusion across the
membrane as the neutral base and trapping in the more
acid cyt-oplasm (and even more acid vacuole) as
SO2NH2; however, there is then some
uncertainty as to whether electron transport in the
chloroplasts remains unaffected (Stern, L963¡ Swader and
Jacobson, I912). It would be worthwhile to study the
effects of more powerful inhibitors of the plant enzyme
at high pH.
The uptake of 02 in the light (no CO2) that was
observed following treatment with sulphanilamide plus
PABA is interesting from the point of view of the
trapping and subsequent (re) fixation of respiratory and
photorespiratory COZ. With carbonic anhydrase present'
the cytoplasm would form an effective CO2 trap since
most of the COZ would be immediately converted to HCO3
ions; this could explain why, in general, O2 uptake (CO2
evolution) was not observed in the light. Inhibiting
the carbonic anhydrase with sulphanilamide in the
presence of PABA has the predicted effect, but not in
its absence. One or other of sulphanilamide or PABA,
therefore, might have other effects on mitochondrial
respiration or photorespiration; it was observed, for
example, that rates of dark respiration were much (up to
five times) higher in the sulphanilamide plus PABA
treated tissue than for tissue treated with
sulphanilamide alone. High rates of dark respiration do
1r9
not necessarily give rise to the uptake of O in the2
tight - U. rigida swarmers had extremely high rates of
dark respiration, but 02 uptake in the light \^/as not
observed.
II. Photosynthesis of ibolis antarctica and
Vallisneria spiralis
( i ) Results
Fig. 50 shows the photosynthesis - COZ curve of A.
antarc'lica leaf pieces at 1o\d pH ¡ CO2 seems to inhibit
photosynthesis at high concentrations. Some data at
high pH, again obtained with the 02 electrode' are shown
in Fig. 51.
using 14c-fixation, the response to tco2l (Iow pH)
of sections of V. spiralis leaf in the stirring gradient
tower does not have any relationship with the degree of
stirring of the bulk solution (r'ig. 52a). rf the rates
of \4c fixation at each concentration are lumped (nig.
52b), the Briggs-Maske1I curve of best fit (FVKUP) is
obtained with a KM (293 þM CO2) virtually the same as
' ,. "-r). Athe apparent KM (V equal to 615 nmol m-z
reasonable fit can also be obtained if kT is small (as
it is for OZ) and KM quite low; the solid line in Fig.
52b, for example, represents the Briggs-MaskelI equation
with K¡1 = 30 f* and k1 = 1.3 x 10-6 * "-1. Again,
CO2 is possibly inhibitory at the highest concentration.
I20
¡
a
ooo
o
60o
п""oþo\s
Ir/)
s{t
{d.
oÈ{
vì
tSL+r{
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{Jq)
s
o
a a
a
oo
aao
o
Io
o
oaa
o
a
o
O
w40ECo;l (calculatd) Çtlvt)
FreuBE 50. Net photos¡mthetic o¿ evorution against [co"'t of A-.Ð1;aretica leayes, in -å's + 50 nM trIES, pH 5'01 - 5'2O, 2OoC'
õI
vì
d(æI
Ë
s.\.s
aa
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a
[æ^] (cak,'.bted.) (N)
FIGTRE 51. Photosynthesis of A,a¡rtarctice leaves ín ASW + 50 nM T¡PS'
a
.9
rrSr\o,o
+J0
\ùq)
o
a
z
pE 8.45 - 8.80, 2æC''
(ct)
ffi
I
200
Itr^
r{I
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r^\)\
+
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o.+Eco--l
1,L l'72
FTGURE 52o.photosynthetic 14coc fixation of v,spiralig (22? I llesh wt. m-
in thã""tir"rts graálent tower (sl,ntors as in tr'ig. 15)i pH 5'76.or z5"c.
b. Dafa at each concentration lumpetl.
,)
lzoo
0
tr
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ilt
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o o^A A
^ A
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FIGURE',.:HJ::il":fr:"'Ïil"l.iËil3Íli",Tlîå#,""*i"T[,(;:?:îìå:"ä:Ë,ïä":ilüî:il}'t?;,Î":Tì.-i:;ii¡.i'-?i:,.
Hil:-;j-rå*:'Jü:kf1ná";i ä¿;mmgt:-;noãtto" iitn v= t4oó o'ãt '-2 "=1' hr = ro¡, kr = 3 x 10-2ttr s '
I
aI
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a¡rd in holcleror 25'C (e,lr,a).
Pr+ "f
¡,IGIIRE !{..photos¡mthesis of sections of V.spiralis leaf (rro):.in the presence of carbonic.anhydrasg (*)t'-' r'(ár-(Å):'-il;;s-tiã car¡on conõãffi2 1.64'rír (t), 1.19 n¡¡ (o,J9, 1.87 nM (a). r = zo'c (r)
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o
The results of a number of experiments on the
response of photosynthetic 02 evolution to the pH of
the bulk medium are shown in Fig. 53 ( A. antarctica) and
Fig.54 (v. spiralis). For the sea grass, the maximum
rate of photosynthesis occurred at about pH 6.8 (total
inorganic carbon concentration = 2.22 mM). The optimum
pH for V. spiral- is (L.2 I.6 mM tota I inorganic carbon )
was about 6. The optimum pH's correspond to ICo2J's of
about 240 ll.Mr and 1 mM respectively- In both plants'/
photosynthesis declined below, as welI as above, this
pH. The addition of an artificial unstirred layer
(B(2))to a section of V. spiralis Ieaf (open triangles'
Fig. 54), with cut ends sealed with dental wax, resulted
in a lower pH optimum, even though the total inorganic
carbon concentration was higher in this case (1.87 mM)
than for the experiments with "bare" leaf sections; the
maximum rate of photosynthesis was also comparatively
Iow. In the presence of carbonic anhydrase (I900
witbur-Andersen units mI-I), there was a very steep
decline in the rate of photosynthesis between pH 6.2 and
6.8 but the rate was more or less constant from pH 6.8 -
8.6, at rB0 nmor m-2 s-l (r'ig. 54,stars)-
The photosynthesis of pieces of epidermis of y=-
spiralis ( with, at the most, 3 underlying layers of
mesophyll) also decreased comparatively sharply'
although this began at the higher pH of 7 (Fig. 55).
Photosynthesis also decreased dramatically at pH less
L2L
than 7 in the presence of L.96 mM inorganic carbon; the
effect was less pronounced at the lower inorganic carbon
concentration of 1.17 mM (cf. Fig. 54).
As mentioned in the "Materials and Methods", 02
evolution in V. spiralis continues all day in the
absence of exogenous inorganic carbon. Table 9 shows
results from experiments in which 02 evolution and I4C
fixation were measured simultaneously in the 02
electrode chamber, ât high pH. The amount of carbon
f ixed was a very smaII proportion of th" O2 evolved,
i.e. 02 evolution was not linked with exogenous C
fixation at this pH.
TabIe I0 shows some ô13c values (relative to the Pee
Dee Belemnite standard) for entire V. spiralis Ieaves
grown under stirred and unstirred conditions. The
values ü/ere kindly obtained by C.B.Osmond: f or methods,
TABLE 9
Net photosynthetic oxygen evolution and 14c
fixation for a section of the leaf of V. spir25oc, total inorganic carbon concentration =I.20 mM.
a1is,
pH net photosynthe si s
nmol OZI m-2 s-l nmol Cf m-2 q_I
9 .36
8.91
8.63
t9B
175
t3B
0. B0
22.7
7 .9L
L22
see Osmond, Valaane, HasIam, Uotila and Roksandic
(I9Bl). The two cultures were identical in every way
except that in one tank the water was vigorously stirred
and pumped using the sLLrrer/impeller sections of two
immersion circulators (as used in constant temperature
baths). The 613c values for the two cultures were not
significantly different; the overall value was L9.79 +
0.I9%
(ii) C supply for photosynthesis of A. antarctica
A. antarctica appears to be a "bicarbonate user" in
that about 200 yVt CO2 is required to saturate
photosynthesis at low pH (rig. 50) whereas less than
twentieth the COZ concentration is needed at high pH
TABLE 10
613c varuesstandard) foror unstirred
(relative toV. spiralis
the Pee Dee BelemniteIeaves gro\^in in stirred
water.
Sample ô13. z. 1t.I)12.tr. B0stirred
-r9.93
t2.rr.80unstirred
-I9.55
5 .2 .8Lunst irred
-20. r0
2.6-8rstirred
-19. r8
2 .6 .8runstirred
-20.2r
L23
one
(fig. 51). FVKUP predicted that photosynthesis is
strongly diffusion limited at low pH, Iow ICOZ ] (kr =
7 x 10-6 m s-I, nfio"= Z.S y,v) and so HCO3- enhancement
of COZ transport, potentially, could explain the
observed ef fect. However, at pH 8.5, the
"characteristic length" of the uncatalysed COZ
hydration,/dehydration and hydroxylation,/dehydroxylation
reactions is about 150¡m. This compares with an
effective diffusion path length of 2a6 yn (from kt
above , Dco, = 1.72 x l0-9 m2 =-r). Arthough the
reaction "length" is shorter than the diffusion length'
the difference between the two is smaII, and so the CO2
reactions in the unstirred I ayer \^7ouId have to be
catalysed to account for the I4-fold increase in the
rate that is observed at high pH cf. Iow pH at the same
(low) [Co2]. Further, it may be that photosynthesis at
low pH is not as diffusion-limited as FVKUP supposed.
The computer chose a mean maximum rate of BB6 nmol m-2
s-I f or the data of Fig. 50; however, if there \^/as
substrate (Co2) inhibition of photosynthesis' the
maximum rate would be considerabty higher and a good fit
to the data would be obtained with a more Michaelis-
Menten response to CO2, together with inhibition at
higher concentrations.
Inhibition by the substrate is also a plausible
explanation for the decline in photosynthesis (fig.53)
when the pH is less than optimal. The optimum pH is
124
about 6.8, i.e. 2A0 yvt Co2r and this lcOrl agrees welI
with the optimum CO2 concentration from Fig. 50. Higher
concentrations of COZ would be present at lower pH's
Ieading to greater narcosis. Inhibition of
photosynthesis at high lcoZl or Iow pH is not uncommon
in water plants. Van Lookeren Campagne (I955) observed
it in his experiments with V. spiralis (cf. Fi 9s.52,54and 54), as did Talling (I976) in phytoplankton, Weber,
Tenhunen, Yocum and Gates (I979) in Elodea, and
MacFarlane and Raven (I985) in Lemanea. High ICO2]
might acidify the cytoplasm. In some of the experiments
shown in Fig. 53, however, Iow pH has little or no
effect on the rate of photosynthesis.
At supra-optimal pH's, photosynthesis again decreases
and this presumably reflects the decrease in ICO2]. The
lines in Fig. 53 represent the way in which the rate
would decline if the response to CO2 were of the Briggs-
MaskelI form. Here kT has been assigned the nominal
value of 3 x 10-5 m s-I (an ef f ective E of 5, f).
rLre
Briggs-Maskell equation successfully predicts the shape
of the initial decline of photosynthesis with pH for
reasonable values of its parameters; however, it fails
miserably in predicting the rates of photosynthesis that
occur at high pH (> S). Indeed, to obtain rates of the
observed magnitude, ofio. would have to be exceedingly
Iow (( I ¡rM) and even then it would be impossible to/
fit the data over the whole pH range, or even from pH 9
r25
- I0. It can be concluded that either (a) the plant is
able to take up HCo3- directly from the solutionr ot (b)
there is external (to the plasmalemma) acidification
with catalysis of the CO2 hydration,/dehydration
reactions.
(iii) C supply for photosynthesis of V. spiralis
The continual
absence of exoge
9, imply that V.
evolution of 02 in the light in the
nous Ct together with the data of Table
spiralis has access to an internal
source of oxidant. The most likely contender is CO2r
particularly in the light of HeIder and van Harmefen's
(1982) findings that V. spiralis accumulates Iarge
quantities of malate (in the vacuoles of mesophyll
cells?) which can be decarboxylated when the external
supply of COZ is limited (cf. Beer and Wetzel, I981)'
In this respect, V. spiralis has some of the
characteristics of cAM even though it is a c3 pIant.
Indeed, the El3c values (table 10) are not unexpected
for fractionation by CAM although the áI3c of the
inorganic carbon suppfj¿ is not known (Raven, Griffiths
and MacFarlane ' 1985 ) .
The dependence of photosynthesis upon exogenous CO2
is seen most clearly for pieces of (mainly) epidermal
tissue (Fig. 55) and, for more intact leaf sections, in
short-term 'nao, fixation studies (nig. 52), rn the
former, there does not seem to be any significant inside
r26
source of COZ (imnediately implying that this source is
in the mesophyll of the leaf) and transport limitations
on the supply of exogenous CO2 wLll be much less
important than for the intact leaf. The response to pH
can be modelled using the Briggs-Maskell equation and
the calculated tCO2l at various PH's; the broken and
solid lines represent fits to the data with KVI fot CO2
equal to 30 yM and k,, for Co2 equal to I x l0-5 m s-1./
V's have been taken as higher than the maximum rates
that were observed because it seems Iike1y that there is
inhibition of photosynthesis at high concentrations of
CO2r i.e. at low pH in this case (Van Lookeren Campagne,
1955). The value of the transport coefficient, kT, has
been assigned arbitrarily; transport from the bulk phase
will be much faster than in the intact leaf but,
nevertheless, not unimportant due to the remaining
underlying mesophyll cel1s. There may even be some HCo3
enhancement o f CO2 transport at pH 9, perhaps accounting
for the underestimate of the rate of photosynthesis
there.
For more intact sections of feaves at low pH (Fig.
52b), the Briggs-Maskell equation is again quite a good
fit to the data but with the much lower kT of I.3 x I0-6
m s-1. The value is about one third of kT for oxygen in
V. spiralis. Such a proportion is unexpected, since
organic polymers generally have a higher permeability to
COZ than to O2 (øarrer, 194L, ch-9). I do not know of
L21
any measurements on the relative rates of penetration of
the two gases through plant cuticles, although if a
solution/ diffusion mechanism is operating then the
oil/water partition coefficient (S for 02, only I.6 for
COZ Forster, L969) might be relevant. If the
penetration of COZ through the plant cuticle is slow, an
important pathway for the diffusion of the gas might be
via the cut edges of the leaf section.
The pH response of 02 evolution of V. spira I i s
54) is complicated because CO2 can be supplied from two
sources, within and outwith the leaf. At high PH, the
outside source wiII be negligible and 02 evolution
probably reflects the fixation of COZ generated
internally. The Briggs-MaskelI equation has been fitted
to the data of Fig. 54 (total inorganic carbon
concentration = 1.19 mM) by taking a contribution of 120
_1 -lnmol m-¿ s-r Lo 02 evolution from endogenous CO2
f ixation. KM (30 ¡,tM) and k, (1.3 x 10-6 * =-I) are the/
same as for the t ine in Fig. 52b and V has been taken as
600 nmol m-2 s-1. The fit to the data is good. The
same cannot be said, however, for photosynthesis in the
presence of carbonic anhydrase nor for the
photosynthesis of a Ieaf section in B(2) even using a
lower value for V.
( Fis.
In
much more
the presence of the enzyme, 02 evolution declines
sharply after pH 6.2 than
with the above values of
the Briggs-Maskel1
equation,
I28
the parameters, would
predict. This is (and was!) a p\rzzling result; the
enzyme causes a decrease in the rate of photosynthesis
over the pH range that it has an effect, whereas
carbonic anhydrase should always enhance the exogenous
COZ supply (if the supply is enhanceable). With the
weight of evidence suggesting that aIl the significant
transport barriers to the supply of nutrients are within
the V. spiralis leaf itself, the result is even more of
a puzzLe since then the added enzyme should have no
effect at aII on the CO2 supply it is unlikely that
the enzyme would penetrate the cuticle, or even the celI
wall (Carpita, Sabularse, Montezianos and DeImer' L979¡
Tepfer and Taylor, 1981). Perhaps the most Iikely site
of action for the added enzyme is in the small volume of
solution which infiltrates the ends of the air l-acunae,
where they are cut. (Although leaf sections showing
obvious infiltration, i.e. almost along the entire
Iength, of the Iacunae were discarded, there was always
some infiltration.) If the permeability of the cuticle
to CO2 is low, then an important pathway for CO2
diffusion would probably be through the Iacunal tunnels
and thence to the epidermal cells (via, in most cases'
the mesophyll). Diffusion along a Iacuna would be very
rapid (t cm of stagnant gas is equivalent to aboua tf*
of stagnant water, diffusion-wise) so the major
diffusion barriers in this pathway are the mesophyll and
the infiltrated water at the cut ends of the lacunae.
L29
With carbonic anhydrase present in the Iatter, it at
least has the potential to have an effect; its effect,
however, still remains opposite to what one might have
reasonably expected. A possible explanation is that the
enzyme enhances the $f lux of endogenous CO2r with a
subseguent loss in the efficiency of fixation of that
coz. The effect must work both ways, i.e. exogenous coz
influx must be enhanced as weIl, but this may be a
quantitatively Iess important source of co2 over the pH
range being considered than the "inside" source-
Further, even if the COZ supplies were initially of
similar importance, the two fluxes may not be enhanced
to the same extent : when carbonic anhydrase occupies
only a portion of a diffusion pathwaY, its effect (as a
COZ-fIux enhancer) is greater when it is located
downstream rather than upstream, with respect to the
direction of the flux (Schulz, I9B0)-
The addition of the artificial unstirred layer (Fig.
54), with cut Iacunae sealed with wax, significantly
Iowers the pH optimum for photosynthesis (i.e. the ICO2l
required for the attainment of the maximum rate is
increased). Thus, for a fit to the data using the
Briggs-Maskell equation as before, the effective k''
needs to be small. (The main reason is probably the
sealing of the lacunae; holder B(2) had no effect on
rates of 02 transport - Fig. 37). However' 02 evolution
is then predicted to decrease quite sharply beyond the
r30
optimum pH, whereas the decline is gradual. As before'
there may be complications due to the internal CO2
source ; the fixation of this COZ is potentially more
efficient with the unstirred layer, and the lacunae
sealed, than with out. Again, therefore, there is the
suggestion that endogenous COZ fixation makes a
significant contribution to 02 evolution even at near-
neutral pH's.
In aII of the foregoing there are numerous
complications which might occur, although I have tended
to side with William of occam in discussing possible
explanations for my results. It has been assumed, for
example, that endogenous COZ is supplied at a constant
rate regardless of the exogenous CoZ supply : it is
possible that at high bulk CO2 concentrations the
internaL CO2 supply is shut off. Similarly, PH changes
within the Ieaf accompanying net malate(?)
decarboxylation or synthesis wiII affect the rate of
transport of CO2 through mesophyll cells (which are
mainly vacuole) and the tissue apoplast. This will be
very important in the intact leaf, but for the leaf
sections which I used the somewhat artificial transport
barrier of infiltrated water in cut lacunae would appear
to be the dominant resistance.
Finally, the intracacies of (symplastic ?) malate(?)
transport from the vacuoles of mesphyll cells into the
chloroplasts of epidermal ceIls, and its subsequent
r3t
decarboxylation, have not been discussed at aII. If
carbonic anhydrase is present in the chloroplasts, most
of the CO2 from the decarboxylation will be immediately
converted to HCOt which raises the possibility of net
HCO3 efflux from epidermal ceIIs. It seems unlikely
that HCOt ions would cross the cuticle at a great rate
(see discussion on phosphate influx, pp 13-74), but they
may carry most of the inorganic carbon flux between
epidermis and Iacunae even if they are not transported
across the plasmalemma of epidermal cells as such.
L32
CONCLUSIONS
The main objective of this work was to quantify the
limitations due to diffusion, particularly diffusion
across the boundary layer, on the uptake of nutrients by
some aquatic macrophytes. This requires a knowledge of
the rate of diffusion, and of the intrinsic rate of the
nutrient-utiIizing "reaction"; it is the relative rate
of diffusion which determines whether or not it imposes
any Iimitation on the rate of the reaction, and modifies
the kinetics.
For the membrane transport of 14c-methylamine in
UIva, the intrinsic kinetics of influx do appear to be
first-order Michaelis-Menten, and so in series with
diffusionrinflux should be given by the Briggs-Maskell
equation. This was found to be the case, with the value
of the transport coefficient, kT, being derived
independently from the dissolution of ztnc in acid.
OccasionaIIy, there is an additional burden on reaction
due to diffusion in parts of the tissue where diffusion
and reaction proceed simultaneously. The Briggs-Maskell
equation is not vatid under these conditions, but the
guasi-empirical equation of Yaman/e can be used. Using
reasonable estimates of the two additional parameters'
the thickness of the zone wherein the reaction is
occurring and the effective diffusion coefficient there,
the Yamané equation describes the data quite well. At
hiEh rates of stirring, and for fast reactions, internal
13e
diffusion can be a major transport Iimitation even in
tissue as thin as the Ulva thallus. For thicker
tissues, with reaction sites distributed evenly
throughout the thickness, the boundary layer may be a
relatively minor diffusion resistance. The diffusion
boundary layer thickness is also almost irrelevant for
nutrient uptake by Vallisneria Ieaf sections, because of
the cuticle. It is dangerous, therefore, to generaL:-ze
about the importance of the unstirred layer for aquatic
plants as a class.
Even general statements about the nutrients
themselves, with regard to their rate of transport
across the unstirred layer, can be dangerous. For the
uptake of solutes such as phosphate and inorganic
carbon, which exist in various forms in solution,
chemical transformations along the length of the
diffusion pathway can, effectively, increase kT (or
Deff). The magnitude of the increase depends upon the
relative rate of the diffusion of the species in
question and the rate of its chemical transformation
into other "carrier" species. Because the
interconversion of dihydrogen phosphate and monohydrogen
phosphate ions is so rapid, the uptake of H2PO4 wiII be
significantty enhanced in the presence of HPoî- even
when the diffusion pathway is quite short. However, in
the case of COZ uptake' very thick unstirred layers are
required for enhancement by HCOã ions to be significant,
t3+
unless the rate of the normally slow interconversron is
increased by catalysis.
The degree of enhancement can be quantified if the
"characteristic length" of the chemical reaction is
known, compared with the diffusion path length. For
analytical calculations, however, it must be assumed
that H+ and OH- transport is very fast, i.e. that there
is no pH gradient along the diffusion pathway. This in
itself, can constitute an important difference among
aquatic plants, because there may be net fluxes of H+
across the plasmalemma. For the same unstirred layer
thickness, variations in the magnitude of the H+ flux
will give rise to different pH gradients, and similarly
for equal H+ fluxes but different thicknesses of the
unstirred tayér. Concerning the COZ supply for
photosynthesis, net H* influx (or OH- efflux) wiII give
rise to a somewhat alkaline boundary layer, which will
tend to increase the rate of the reaction CO2 + OH--+HCO3
and enhance rates of COZ diffusion by HCOã "carriage".
on the other hand, net H+ efflux will tend to acidify
the boundary layer which will not lead to any
enhancement of CO2 diffusion but will tend to increase
its concentration near the plasmalemma.
The Briggs-Maskel-l equation is derived on the
assumption of first-order Michaelis-Menten kinetics for
the nutrient-util-rzLng reaction. In a number of cases,
the assumption is not good, The outstanding example is
r35
O2 uptake in dark respiration. Here, the reaction
involves two substrates (OZ and reduced cytochrome
oxidase) which means that the xffa of cytochrome oxidase
is a variable dependent upon the degree of reduction of
itself. Even the apparent order of the reaction can
change if the rate of electron transfer to (oxidized)
cytochrome oxidase is very slow or very fast compared
with the rate of reaction with 02. It is possible'
therefore, to obtain apparent first-order Michaelis-
Menten kinetics even if there are significant diffusion
limitations (e.g. if the reaction order is less than
one) and conversly, if the reaction order is greater
than one, a strongly oblique hyperbola may be obtained
when diffusion limitations are negligible. Under these
circumstances, making predictions from fits to the
Briggs-Maskell equation with no prior knowledge of the
intrinsic reaction kinetics is extremely hazardous.
These problems, and more, also occur for
photosynthetic COZ fixation. The driving reaction
catalysed by RuBP carboxylase is again a two substrate
reaction but with the added complication of O2
inhibition. The observed K¡4 for COZ of RuBP carboxylase
is a variable depending on the concentrations of RUBP
and 02, and the order of the reaction can also change
depending on the rate of RuBP supply. In addition'
hiqh [CO2 J can be inhibitory. The complexity of the
intrinsic kinetics can be reduced by conducting
l-36
experiments at Iow bulk O2 concentrations and C02
concentrations that are not inhibitory' and in many
cases the response of photosynthesis to ICo2] has the
Briggs-MaskelÌ form. There is always doubt, however, as
to the meaning of'the "true K, for CO2" that is
predicted byr or that one uses in, the equation this
quantity may be less than the K¡,1 f or CO2 of the RuBP-
saturated enzyme if RuBP is Iess than saturating in
vivo. Further, unless rates of Coz transport are known
independentty, the value of kT predicted by the Briggs-
Maskell equation is also in doubt since the intrinsic
kinetics of co2 fixation may themselves describe an
oblique hyperbola with ICO2]. Nevertheless,
photosynthesis of U. rI ida is clearly affected bY
stirring, especialty at low PH, and independent
measurements of k, describe the response quite weIl
(Fis. 4s).
At high (sea water) pH's, both in U- rigida and
A. antarctica, the membrane transport of HCOã ions is
likeIy to contribute to the supply of- CO2 within the
cel1s. Because tttCOrl is high, and because the rate of
uptake is low, diffusion in the bulk medium and within
the tissue is unlikely to be rate-limiting for HCO3
uptake although it may yet be for COZ. Thus, the
transport limitations that exist for CO2 influx can be
turned to advantage at high pH by the plant since they
wilt timit CO2 (derived from HCOJ) efflux- In V.
L37
spiralis, the cuticle will be very effective at
preventing the Ioss of COZ from endogenous sources-
This plant grows in culture in a solution of pH 9 or
more, so Ico2] will probably be negligible. Even
floating leaves will be exposed to only about la ¡At
CO2
in air, which is again a negligibte Ico2 ] for V.
spiral is photosynthesis. It is possible that a major
source of CO2 for the leaves is the sediment' with
transport via the air lacunae. A cuticle with a
permeability to Co, of L.3 x 10-6 ,n =-I is equivalent to
a diffusion path length in air of more than I0 m(!)
which is much longer than a V. spiralis 1eaf. The
sediment may also be an important source of mineral
nutrients (e.9. phosphate) for the Ieaves (oenny, l9B0),
given the relative impermeability of the cuticle.
TabIe lI summarizes the results for the plants and
processes which I studied. The maximum values of the
transport coefficient (kT) are for well-stirred
conditions and smaIl sections of tissue. They are
equivalent to unstirred layers 30 - 60 ¡m thick. For
comparison with kT, I have included an estimate of V/2x;n¡-
which is a measure of the intrinsic rate of the process
(i.e. in the absence of transport Iimitations); unless
this ratio is much smaller than k1, the transport
limitations imposed by the boundary layer should
significantly modify the response of influx or net
influx to concentration.
13 g
TABLE 1I
Summary of plants and Processesof stirring and the val iditY orMaskell equation.
studied and the effectotherwise of the Briggs-
Pl ant Ef fectof
Stir-r ing
Range of kTpredictedfor unstir-red ]ayef(m 5rxlU5 )
v /2KM
(m srxt05 )
Validity Commentsof Briggs-MaskeI 1
equation
CHTNHe
-J-J0.36-3.9
0.53-4.9
I' Tnflux
r.2-r. B7U. riqida Yes
V. spiralis No
U . r iqida Weak
V. spiralis No
U. rigida
Yes
Yes Cuticle !0.7 ?
U. rigida Weak 0. r-3.3
V. spiralis No 0.25-3.5
H^PO; Influx
0.34-3.1 2L
0.40-3.1 5?
O^_Z ttptake (Dark Respiration)
CO.--z
Uptake ( photosynthesis )
0.22-3.2 2.I-2I
0.23-3.3 L.7?
- 3.2 L2?
Yes
No?
No
Yes ?
( Iowlowand
( highpH)
Transportenhancçdby HPo[
CelI WaIls?
Compl exk inet ic s
Cuticle
Yes(low pH)
Yes(Iow pH)
YesPH'oz)No
Yes (butnot for O
evo I ution
MembraneHCO3transport
Endogenouscoz
V. spiralis No
A. antarctica
2)
No (highpH)
MembraneHCO3
transport?Acid zones?
t39
Often, however, this is not the case, because of the
complications just discussed. In Vallisneria , for
example I v/2KM (estimated) for CH3NHj influx and
photosynthetic CO2 uptake, is (somewhat) Iower than the
maximum k, but the reactions are still very diffusion
Iimited by the cuticle. In UIva, V/2KM for H2POn influx
and CO2 fixation can be quite f,tgh compared with kT, and
yet the effect of stirring on H2PO4 influx or CO2 uptake
at high pH is slight.
It can be concluded, then, that besides a knowledge
of the unstirred layer thickness and the intrinsic rate
of nutrient uptake, a considerable amount of information
must be known about the plant (and nutrient) concerned
before anything definite can be said regarding the rôle
of the boundary layer as a (rate) limiting factor.
Nevertheless, when the boundary layer !s an important
transport barrier, its rô1e can be crucial.
140
Appendix I: T,'ICK S LAWS
Fick (1855) was the first to describe
mathematically the process of diffusion under the
influence of c concentration gradient.
If diffusion in only one dimension is considered,
may write Fick's first law of diffusion:
à"D
IJ
we
(I.I)
(r.2)
l l à"
where Jj is the flux of the diffusing species j (amount
of j which crosses a plane of unit area per unit time),
caused by a gradient in the concentration of j in the xàc'
direction, ãOJ. The minus sign indicates that the
direction of the flux (with respect to x) is opposite to
the way in which the concentration changes with respect
to x. Oj is the diffusion coefficient of species j.
Partial derivatives are written because the
concentration gradient itself changes with time.
The time dependence of the concentration gradient is
described by the continglly equation
ètjà"
à"jàt
This equation simply means that if the flux of j isà¡..'
faster at x than at x + dx (i.e. *:is negative), then
the IocaI concentration of j (i.e. in the volume element
r41
enclosed by x and x + dx) must be increasing with time -more j is coming in than going out.
By substituting Fick's first Iaw into the continuity
equation, and assuming Di constant, Fick's second law is
obtained:
Þàt òæ
ò..(-oj àx')
à (r.3)Dj
The equation can be solved fot.j in a number of
specific cases (see Crank, 1957).
It should be noted that Fick's laws do not include a
term for any electrical potential gradient and so may
not hold for the movement of ions. Later equations, due
to Nernst and Planck, include a term for the electrical
potential.
r42
Appendix II: Oriqin of the auadratic Describinq an
Enzyme-Catalysed Reaction in Series with a
Diffusion Resistance
In a letter to the late Professor G.E.Briggs (l2th
Nov.IgBl ), F.A. Smith and N.A. WaIker enquired about the
origin of equation (I2), which they had previously referred
to as the Hill-Whittingham equation (after Hill and
Whittingham, 1955, in which the equation appears). Part
of Professor Briggs's reply is reproduced below:
v./
143
øØ*lf
In fact, in the Preface to their monograph, Hill and
Whittingham acknowledge the teaching of Professor Briggs
in chapters 2 and 4, and state that he allowed them free
access to his unpublished lecture notes. They also
acknowledge the criticism of Professor E. J. MaskeII.
Maskell (I928) did not explicitty solve the quadratic
for v (n in Briggs' terminology), but he published its
limits and its behaviour for various values of kT (D/L).
I shall refer to it as the Briggs-Maskell equation.
r44
Appendix III: The Meaning_of the Apparent K*f or an
Enzyme -Catalvsed Reaction in Series with a
Transport Process
The apparent K¡1, Kftpp, is found by putting v =
the Briggs-Maskell equation
v/2 in
v= i { xrtt + cbkr + V (KMkT+cbkr+V) - 4c5krVÌ
åv = iix*t,and solving for cb. Thus'
+c6kr+V- (Kitlkr+c5kt+v) - 4c5krv Ì
(Kukr + cSkr)2 = (KMkT + cbkr + v)2 4cSkrVrl.e
t<r2 {x,, + "a)2
kr2 (Ki,l + "a)2 + v2 + 2kT(Ku + c5)v -4c.krv,
V + 2kT(r* + c5)V - 4c5ktV = 0,
i.e. v{v + 2kT(KM c5)Ì 0
2or
For Vf 0,V+2kT(KM
and cb=KM+ V2TT
"u)0
Thus KMaPP KM + (rrr.l)
l. when KM >> v/2kT (i.e. when kr )> v/2KM), KMaPP,-v
K, and the kinetics are determined by the enzymic
reaction.
2. Equation(rrr.f) is the equation of a straight line.
If KMapp is determined at various values of k1, and the
results plotted with KMupp on the ordinate and I/kt
r45
given by the intercept on the ordinate (cf.
r9B2) .
(proportional to the unstirred layer thickness) on the
abscissa, the slope should, be V/2 and the intercept on
the ordinate should be Kt (cf. Thomson and Dietschy,
Ig77). Similarly, tf KMapp is plotted against V, the
predicted slope is L/(2kT), with the true Kt again being_1 -,l,IVanSJçy,
L46
Appendix IV: Relaxation of Diffusion to a FIat Plate
Consicler a solution in Iaminar flow (butk velocity,
U) passing over a flat plate of length X and breadth Z
as shown in Fig. IV. 1. A component of the solution
(concentration in the bulk phase = .n) reacts with the
plate.
+ X
Z.
î.
FlCr, lV .1
At any point on the plate, the limiting flux of the
reactant (i.e. the maximum possible flux normal to the
plate's surface) is given by
Jri* B
in which, provided the diffusion coefficient of the
reactant, the kinematic viscosity of the solution and U
are constant, B is a constant (equation (IB)). The
total diffusional flow, IIim, over the entire plate
is then
x
JIim dx dz
3,\Æ'
-lrm
I47
2 z.B "n Æ
and therefore the averagç. flux to the whole plate is
= rri*-1ñ 4r.jl:
-=
¿ts c5Z.X X
Now, upto X = Xo, the plate is coated so that it can
no Ionger react (rig. IV. 2). The ffux at any point
Xo4r¿_ X
,z
I,
ttG.ñ. L(x ) Xo ) is given by (Levich , l-962, pf 06)
cbJ(x,0) = B
1Et 1
r - z.B ".Þ*(
(xo/xl'* l"
and so the total diffusional flow to the uncoated
portion of the plate is
XX
I dxI J (x, 0 )dx dz = Z.B cb
Xo24
xo xo
which has the solution (cf. Petit-Bois, I961' p95)
[1,t
x-4 lv
or r - 2z.B "rrtr [r (Y1':The average flux to the uncoated portion of the plate is
148
therefore
J2
Hence,
z(x - Xo)^lr2B c5
xXoI
t-
j== x l, þY1"11 x xo L \x/ J
(13)
L49
Appendix V: Diffusion and Reaction in ParaIleI
(i) The dif f usion-reaction eguation
Consider a plant ceII or a piece of plant tissue in
the shape of a sIab, cylinder or sphere (rig. V.1). The
first two either have their edges (or ends) sealed' or
the edges or ends are such a small proportion of the
total surface area that they can be ignored. Substrate'
B, surrounds the ce1I or tissue at a bulk concentration
of c5 and at a surface concentration of cs. The
interior of the ceII or piece of tissue will be
considered as an homogeneous enzyme suspension, which is
at a constant temperature throughout and which can be
regarded as being in direct contact with the bathing
medium - i.e. the permeability of any ceI1 membrane is
very l arge.
The basic equation of diffusion and chemical reaction
within the body is given by the Poisson equation
V (V.I)
where Deff is the effective diffusion coefficient inside
the p1ant, V2 th. Lapracian operator ( Y2" = * * ò%
À2;¡ -
in Cartesian coordinates) and v the rate ofò12
consumption of substrate by chemical reaction. This
equation follows directty from Fick's second law (see
Appendix I) applied to the three dimensional case, and
).î=DeffV-côt
150
Jft{ír,;ø ølcø'J I çtat'i'-1JrLÍlrrffi ylÀ*
Cp Cs C¡"c¡o C5 Cø
6
-R
FIGURE V.1. The reacting plate, cylincler a.ncl
of the Nernst laYer is denoted bY
ß-
Spí'o''r"-
T.
Iv
C,.
\
sphere. Ihe thiclaress6.
the law of
e lectri ca I
conservation
interact ions
of matter, provided that
can be ignored.
- r,in which k"is the rate constant
I will assume that within the plant the enzyme-
catalysed reaction of Bâproducts follows f irst-order
Michaelis-Menten (eriggs-Haldane) kinetics, i.e.
equation (I0). (rn form, these kinetics are the same as
those known to physical chemists as Langmuir-Hinshelwood
for adsorption and subsequent chemical reaction on ak'c 5̂
f--R-cS
for the chemical reaction and K is an adsorption
constant. )
wirh
state (
Michaelis-Menten kinetics, and in the steadyàcõt 0), equation (V.1) becomes
surface: v =
d,2 c P+1 dc+ (-) -lrdr
Deff t VC
Kl¿+c (v.2 )2dr
in which the Laplacian has been expressed in terms of r
(see Fig. V.1), and p is -1 for the slab, O for the
cylinder and +l for a sphere (e.9. BIand, 1961).
Equation (V.2) is non-Iinear and the exact solution
can only be obtain by numerical methods. However,
Michaelis-Menten kinetics simplifies to first-order or
zeroth-order kinetics when c is very Iow or very high
and so it is worthwhile exploring equation (V.2) in
these two limiting situations. Equation (V.2) is al-so
simplified for the slab because it represents the one
151
dimensional case and the equation becomes
ð,2 cDeff (
")d.r'
cV (v.3)
Krrl + c
The solutions for
different as wilI
the other shapes are not greatly
be shown l-ater.
(ii) First-order kinetics
For c <<
d2c V(-) c
KM
(v.4)
(e.9.
genera I
or
where
Here V is taken to
mor m-3 =-l ) which
solution is
Deffd,r2
d2c 2
dr 2(K) c
Krtl'Def f
be expressed per unit volume
renders / dimensionless. The
V
þ=R
øc = A sinh t(-) rl + B
R
øcosh t(-) rl
R
(v.s)
B are integration constants. A is quickly
sinceatr=0, r.e
where A and
el iminated 0d.cdr(A fl cosh t(ff) rl + B €sinh ø= 0 and coshø=
s inh r (€) r])r=0 =
Equation
0. Because
0
L52
Lt A (V.5) is now
øc = B cosh t(-) rlR
At R, c = cs, so we can write
cswhence
cosh þ
and substituting back into equation (V.6),
T
(v.6 )
(v.7)
cs=þcosh/
B
Def f Def f k1(c6
øcosh t(-) rl
R
cs (ø/R)sinh I
cosh þ
The concentration of substrate at the surface is usually
unknown, but \,ve can express it in terms of the bulk
concentration using Nernst's theory (Introduction,
section I). Thus, at the surface, the flux is
Deff t$f)r=p which is equal to kr(c5 "") where kT is
the velocity constant for external mass transport. From
equation (V.7),
(dc
c^)ùdr r=R
krcband so cs
Deff (þ/s) srnh glkT
cosh þand substituting into equation (V.7 ) yields
+
153
kr cb cosh l(Ø/P) rl(V.B)
kT cosh p + Deff (ø/R) srnin þ
The flux of B into the plant through each face of the
slab is
kr cb {þ/n) sinh /J Def f ) (v.e)
r=R kT cosh þ
cosln þ
+ Def f (ø/R) sinh gf
1+ -)kT
or
l.e
11
J(v. 10 )
(v.r1)
cb Def f (9/n) sinh þ
In this last equation, the .-arm cosh ótermW IS
simply the inverse of the flux that would occur if there
were no external diffusion boundary layer. The otherlterm,
" {-, is the inverse of the maximum f lux throughbT
the diffusion boundary Iayer. In fact, the two terms
within the brackets have the units of a resistance
(s m-I, say) and the overall resistance is simply the
sum of the resistances internally and externally. This
additivity of resistances is a property of the symmetry
of the system and holds for infinite cylinders and
spheres as weIl (Aris, 1975).
The rate of uptake of substrate into the plant per
unit volume, v, is JrlR
t 1 R2 coshþ R
V+
If there were no external resistance, v would be given
c6 Deff þ sinh þ kT
Deff þ sinh /by cb (
" ) which can be rearranged to give
R' cosh þ
r54
cb
The overall velocity falls below its maximum value
(.b F) by the factor La?rh c. This factor is theM
effectiveness facLor, \ , used by chemical engineers.
(Briggs and Robertson , 1948, also made use of a similar
term in their experiments with carrot disks.) Equation
(V.12) is quite general and holds for any first order
reaction in a s1ab, with the rate constant replacing the\7term ^r.M
Fig. V.2 shows some plots of ry against the Thiele
modulos, Ø, for various shapes. It can be seen that all
the curves are similar, with the sphere being the least
effective shape and the slab the most effective. The
difference is reaIly only significant al ø values of t
or 2, where the slab is I0 - 15% more effective than the
sphere.
Fig. v.2 shows that for small values of ø (less than,
sây, 0.2)L is virtually 1 and the rate of consumption
of substrate is determined by the chemical reaction.
This corresponds to a situation where the reaction rate
is very much slower than the rate of diffusion.
Throughout the solid body, c is practically equal to cs.
For values of ø larger than, say, 2, catalyst
effectiveness is low (in fact 1- tø
"t ø --+ os ) and
V Lanh þ(-)(-)KMø
(v.12 )
t55
O,E
>lorb
0rltün-,òeña
rL s
0'5
0.5
0,1
?- 4 5 G
ø
FIGURE V.2. Effectiveness factorrQ, against the [Teiele modulus, l, fora sphere, a cylinder.aira a slab ( adapted fron Aris, 1957 anð'
A-ri;, i975, Fig. 3.7). ø t" based on a characteristic dlmensionwhich ls the ratio of the volume to the surface area of theborly. For the elab this is sinply the half-thiclmess, R (Fi'g.V.1), while for the cylinder and sphere it is $ tfre radiusend + the radius respectively (Aris, 1957; Roberts antlSatterfieltt, 1965).
0.
1.0
0.J
0.L
0
0
3I
the overall consumption of B is constrained by the rate
of diffusion; for much of the slab, c (( c". Fig.V.3
illustrates the concentration gradients that form withina slab for various values of ø.
When an external mass-transfer resistance is present,
it can be shown from eguation (V.11) tfrat
Løø2
L Lanh þ Bi(v. 13 )+
where Bi is the dimensionless Biot
be seen as a ratio of external (kT)
rates of mass transport.
Rkrnumber, D-.effto internal
Bi can
(Deff,/R)
Fig.V.4 shows the effectiveness factor as a function
of / for various values of the Biot number. For a
particular value of the ThieIe modulus, catalyst-
effectiveness gets progressively lower as Bi decreases
(i.e. as kT decreases). For spheres, it can be shown
that k,, .pproaches a limiting value of O,/R as the
thickness of the Nernst diffusion Iayer increases (see
Langmuir, 1918). The Biot number, therefore, cannot be
Iess than DÐ-. Generally, Deff is less than D and soeff
Bi for spheres will usually be more than l. For
infinite plates and cylinders there is no such
restriction on kT (and Bi) which continues to decrease
as 6 increases (SatterfieId, lr9BL,Þ L]-2).
rs6
1
?,
0.9
0,6
0.4
o.Z
0 o,2 o'+ 0'6 o;8
tR,,
X'IGURE V.õ. Profiles of concentratLon in one half of a reactlng slab(first-orðer klnetics) for varioue values of the [t¡le1enodulue, y'.
o
(=o'l
=l
/.
f= lo
n
to
o.1
o.8
o,1
o,6
o.5
0.1
o.ã
O,2
O,1
Bit"o
Ei= 6
þi= L
Di= I
0 3z0ø
FIGIIRE V.4. Effectiveneee factorrf[r versuø þ fox varlous Biot n¡mbers
lar "c tn).
1o I a;â oo
6
FTGURE V.5.
Cb q^J'l
I
ItVì
frìI
ËÈ)
o.ËÈ'
þ
Bí=Z6
4þi=l
2
zo 30)
Relationshlp between the reaction velocity a¡rd the bulkconcentration of reactarrt,
ents the kinetÍos in theconplete absence of transport llnttations (i.". Deff,k* + oo).
The actual uptake rate, v, wiII be given by
v = cotþ ){ which is equivalent to equation (v.11).M
Fig.V.5 shows the v versus cb curve based on
this equation, compared with the kinetics if there were
(a) no external resistance and (b) if there \^7ere no
resistance at aIl, either external or internal.
( iii ) Zeroth-order k inet ic s
For c >>
which has the solution (BIum and Jenden,
V
c=cb
22This f unction is discontinuous at c = v (b * åTÉ)I effbecause c cannot be negative. This is clearly seen in
Fig.V.6, where concentration profiles in a slab are
sketched for various values of c5 (k1 ? oo ). Due to the
nature of the kinetics, for those regions of the tissue
where c ) 0, the local rate of reaction wilI be V.
overall, the reaction velocity will be V f;ana tne
influx into the slab wilI be V 2R', where R'is the
depth to which substrate penetrates (see Fig.V.6).
There wilI be a critical c5, cb, above which R'is
simply R (Fig.V.6b). cb' occurs when the concentration
of substrate in the middle of the slice just equals zero.
From equation (V.I4)' putting r = 0,2
cmiddte=c5-v(Ê-+2frlT eff
Rv (- +
kT
R2 12)
2 Deff(v.r4)
L57
I
(c)(b)(ø)
jt (+
C6
C6
Cu
C=o
=fç
FIGURE V.6. Concentration profiles within a reacting slab (zeroth-orcler reaction kinetics). For low 9¡, reactant onlypenetrates part way into the slab (a)¡ ?t."¡ = cd:-reactant JuÈt penetrates tq the middle (b);-.for allct > 4, ã 1" ãver¡mhere greater than zero (c).
I
lo
6
2
IIú)
all
Ë
oËË
Þ
I
4
þí-æ
BL= zBî= 1
20lo c5 (fM )
FIGIIRE V.7. Velocity vernrs ooncentration curvos for e zeroth-ortlerreaction within a slab 10Orrn thlck. Darr - 5'x tO-10 n2 c-1.lltre reection rate equation'is v = v'(c]'0) or^v =-o (c - O),
. xeprescntetl by the dashcdt' l-1ne. V - 10 n¡rot t-J q-1. Blotnumberg of 1a¡tt 2 eorrcapondl to klrs of 1x 1O-5 anct
2 x 1C-5 n s-1 respectlvely.
therefore
cb (v. 15 )
and correspondingly solving for R'
2 2 Deff cb Deff(v. r6 )
V kT
Equation (V.I6) agrees with the equation of Kidder
(1970), which was based on Warburg's (1923) equations
but with an additional external resistance. Fig. V.J,
which can be compared to Fig. V.5, shows the uptake
kinetics based on equation ( V.1 6 ). *
The complete solution of the diffusion-reaction equation
(equation (V.3)) will approach the solutions shown in
Fig.V.5 and V-7 at low and high cb respectively. An
analytical expression for the flux over the entire range
of c5 will be necessarily approximate; the accuracy
*In terms of Thiele moduli and effectiveness factors,
R
v (- +kT
+R
with Ø defined as Rno external mass-tr
V Def f .c bans fer res ]- S ta
and YL = R'/R, then f ornce
(v. 17 )
2
1 + 2/BL
(Ø ,< ^F)
(p >/ ^El
while with an external resistance,(F<
Bi.2 ø2 Bi
I
I 212+
q
l5B
(Ø >,
r + 2/BL
(v. 18 )
can be checked by comparing the approximate solution
with the exact, numerically calculated one.
(iv)
The Briggs-MaskelI equation (p B) holds if the
reaction takes place only on the surface of the plant
ceII or tissuer ot if mass transfer in the interior of
the plant is fast due, for instance, to cytoplasmic
streaming. It holds, therefore, for small values of Bi.
Where these conditions are not met, i.e. where there
are significant internal mass transfer resistances, the
diffusion-reaction equation is non-Iinear. BIum and
Jenden (I957) approximated it to Iinearity by a Taylor
series expansion. They derived the equation
cbKMR2R
V (p + 2)(p + 4) Deff (p + 2) kr
which for a plate (p -I ) becomes
Approximate solutions of the eguation
++VV
cb KM 2R R+
(c5 +
+ (v. 1e )
V V
This is a quadratic in v, and solving gives
v 3 D.ff kr
(.n Kl,t + aV Kl,t + av )2 AacbV)+v=
R3D
2a
RrT
Ë- is rarge compared
(v. 20 )
2
Tin which a =
eff+
1s9
When
2with # (e.s.
effequation (V.20 ) reduces to the
if Deff were large), Rk-A ^.)T
Br iggs -Ma ske I I
and
equation.
Murray (f968) used the mathematical technique of
singular perturbation to obtain solutions to the
diffusion-reaction equation, His method, however, isonly valid when the dimensionless parameter
o "l is smallVR2
(certainly less than O.I); this is often not the case
for the plants studied in this project.
Other approximations rely on empirically-derived
equations, obtained by adjusting parameters until the
derived equation produces values close to those obtained
numerically (Atkinson and Daoud, 1968; Kobayashi, Ohmiya
and Shimizu, L916). These expressions (which are in
terms of the effectiveness factor) do not consider an
external mass-transfer resistance. Recent1y, Yamané
(I98I) modified the equations of Kobayashi et aI. (I976)
to rectify this. His equation is
\tu (2.6w0 't )t-1,+
I + (2.6K0.8)(v.2 r )
KMwhere W is the dimensionless Michaelis constant, c-
bThe effectiveness
zeroth-order and
"overalI" ThieIe
f actors .l"[0
f irst-order
modulus
and T[1 are those for
reaction, based on an
a
V
Def f (Kvl"n)
ø R
160
+(v .22)
Using this value ot þ, 4o is given by equation (V.IB)
and 4f byequation(V.13) foraslab. Theoverall \is simply the arithmetic mean of these two, with the
weighting factor 2.61d0'B determined empirically so as
to minimize the difference between the calculated and
the true effectiveness factor. This difference is
generally slight (<4%); over a very limited range of ø
(0.4 < þ < 0.8) and with 0.1 < l+ ( l, the dif f erence
may be 6Z or more at small Biot numbers (Yamané, 19Bl).
Yamané's effectiveness factor is shown as a function of
ø in Fig.V.B for various values of lL, with Bi (and
theref ore kt ) --> æ
Fig.V.9 shows some graphs of v versus c5 based on
equations (V.20) and (v.2L) for various values of kt.
For the values of K¡1, V, R and Deff used in Fig.V.9,
Yamané's equation is in virtual agreement with the
numerical result because the ThieIe modulus is 1arge,
particularly at low c5's (K > 0.I). BIum and Jenden's
equation, which is the simplest , tends to straighten
the curve but still shows reasonable agreement with it;
at a velocity of half V, the agreement is very close
which makes the equation useful for predicting (and
describing) apparent KM's. In fact, KraPP from the
Blum-Jenden equation is
KMapp
( cf . Appendix III ) .
t6r
Ktvt+aY (v. 23 )
I'IGIIRE V.B. Yana¡rers cffectiveneEs factorrTf , against / fot variousva.lues of K (qn/"t) rnd 81+oo.'
.a
6
II
ú\llt
È
oË\
þ
þi-+ooBi= 5
b
tt=Z
4
2
tsi= I
?ßlo þC ( ru)
FIGURE V.!. ConcentratÍon cuxves for the rate of an enzylne-catalysetlreaction ln a s1ab, l00rum t\ick: Deff = 5 x 10-10 t2
"-1,K, - 1 rrM and V = 10 mmol m-J s-1.füict Íitt"" - Ya¡ra¡r'e rs equation (equation V.21).ttrin lines - Blum-Jenden equation (equation V.20)'Dottecr line - ffiffiii:;i"if:.""iill"l;l*,no transPort
Appendix VI: List of Symbols and their Usual Units
A
,A1 ,42 ' 43
B
Bic
D
e
g
J
kkrk+r,k+2,k¡3,k-2K
KM
PrR
tu
U
vol-
Greek letters
surface area; m2
parameters used in equation (32) , p 77;
a number; dimensionlessBiot number (=R kt/Deff ); dimensj-onf ess
concentration ; fM (¡mol dm-S = Inrìol m-3 )
diffusion coefficient; m2 s-1total- concentration of cytochrome oxidasein mitochondrion;
/uMacceleration due to gravitY; m s -2
flux; nmol m-2 s-1velocity constant; nmol s-1rate constant; s-1rate constants for electrontransfer reactions (p 76); M-l s-1 or s-lequilibrium constant; dimensionfess, M or M2
Michaelis constant i /^JVPrandtl- number (= ),/D); dimensionlesscharacteristic dj-mension; m
time; s
loca1 fluid velocity; m s-1fluid vel-ocity in butk mediumi m s-lvelocity in reference to an enzyme-catalyzedreaction; nmol m-2 s-l or mol m-3 s-lmaximum velocity of an enzyme-catalyzed reaction;nmol m-2 s-l or mol m-5 s-1volume; m3
distance in x direction (e.9. along flat platef rom J-eading edge ) ; m
length of flat plate; m
length of coated section of pJ-ate (p L48); m
molar activity coefficient; dimensionless
r*', P /t =-ts,
V
xo
v
x
X
o<
E
å
nK
density coefficient ( = þ #r, m3 mol-lthickness of diffusion boundary layer:' f^thj-ckness of hydrodynamJ-c boundary J-ayer; /*^effectiveness factor; dimensionlessdimensionless Michaelis constant (= t<¡4/c6)
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NOTE:
This publication is included in the print copy of the thesis held
in the University of Adelaide Library.
It is also available online to authorised users at:
http://dx.doi.org/10.1093/jxb/33.2.195