Lecture12:PatternsinLivingSystems

DrEileenNugent

Turing’sonlypaperonBiology

THE CHEMICAL BASIS OF MOKPHOGENESIS

BY A. M. TURING, F.R.S. University qf Manchester

(Received 9 November 195 1-Revised 15 March 1952)

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biolo:~irall, unusual system. The investigation is chiefly concerned with the onset of instability. It is faund that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also con- sidered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis.

The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathe- matics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

I n this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. I t is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.

The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points. In the other the matter of the organism is imagined as continuously distributed. The cells are not, however, completely ignored, for various physical and physico-chemical characteristics of the matter as a whole are assumed to have values appropriate to the cellular matter.

With either of the models one proceeds as with a physical theory and defines an entity called 'the state of the system'. One then describes how that state is to be determined from the state at a moment very shortly before. With either model the description of the state consists of two parts, the mechanical and the chemical. The mechanical part of the state describes the positions, masses, velocities and elastic properties of the cells, and the forces between them. I n the continuous form of the theory essentially the same information is given in the form of the stress, velocity, density and elasticity of the matter. The chemical part of the state is given (in the cell form of theory) as the chemical composition of each separate cell; the diffusibility of each substance between each two adjacent cells rnust also

VOL.237. B. 641. (Price 8s.) 5 14August I 952[P~~btished

TuringPatterns• Chemicalspecies

(morphogens)XandYdiffusingextracellularlyandundergoingreactionscharacterisedbyfandg

• Lookforinstabilityandseewhatconditionsthisimposesonfandg

• DX=DY=0systemstablespatiallyhomogenous

• Forsomecombinationsoff,gandDX,DYsamefixedpointunstableandpatternformationpossible

ActivatorX(red),InhibitorY(blue)

ReactionDiffusionEquationsReaction Di↵usion Equations

Consider the following system of 2-component system ordinary

di↵erential equations

@X

@t= DX

@2

X

@x2+ f (X ,Y )

@Y

@t= DY

@2

Y

@x2+ g(X ,Y )

with initial conditions

X (x , 0) = X

0

(x)

Y (x , 0) = Y

0

(x)

We are generally dealing with patterns formed in finite domains

with no-flux boundary conditions and we can choose this or fixed

values X = Xb, Y = Yb.

Perturbations from Steady StateRewrite in vector notation

~X =

✓X

Y

◆F (

~X ) =

✓f (X ,Y )

g(X ,Y )

◆D =

DX 0

0 DY

�

˙~X = F (

~X ) + D

@2 ~X

@x2

In the absence of di↵usion

~X =

✓X

Y

◆˙~X =

✓f (X ,Y )

g(X ,Y )

◆

which has steady states (X

⇤,Y ⇤). We can examine the stability of

these steady states by looking at response of the system to

perturbations of the steady state

~Z =

✓X � X

⇤

Y � Y

⇤

◆

PerturbationsfromSteadyState

Linearized Equations

The time derivative of the response is given by

˙~Z =

✓˙

X � X

⇤

˙

Y � Y

⇤

◆=

✓f (X � X

⇤,Y � Y

⇤)

g(X � X

⇤,Y � Y

⇤)

◆

Finding the Jacobian matrix

A =

fX fY

gX gY

�

(X⇤,Y ⇤)

and linearizing the set of equations in the vicinity of the steady

state (X*,Y*) gives

˙~Z = A

~Z

LinearizedEquations

Stability Requirements

Solutions to

˙~Z = A

~Z are of the form

~Z (x , t) =

2X

n=1

wi ~vi exp(�i t)

where

~vi are the eigenvectors of A with eigenvalues

�i =Tr(A)±

pTr(A)

2 � 4Det(A)

2

and the values wi are set by the initial conditions. The steady

state of the system is linearly stable to perturbations if

~Z tends to

zero as at long time scales. For this requirement to be fulfilled

both eigenvalues must be negative which imposes the following

requirments on matrix A

Tr(A) < 0 and Det(A) > 0

StabilityRequirements

Including Di↵usion

The linearized system in the presence of di↵usion is given by

˙~Z = A

~Z + D

@2 ~Z

@x2

and we look for solutions of the form

~Z = Z

0

exp(�t) exp(iqx)

i.e. the perturbations are spatial waves with wave number q whose

amplitude is either growing or shrinking in time depending on the

sign of �. Substituting this into the equation above gives

(A� q

2

D � �I )Z0

= 0

The system is unstable if at least one of the eigenvalues in the

matrix A� q

2

D has a positive real part. This corresponds to the

determinant Det(A� q

2

D) < 0.

WithDiffusion

Di↵usion Driven Instability

For di↵usion-driven instability we have the following condition

Det(A�q

2

D) = h(q

2

) = DXDY q4�q

2

(DY fX+DXgY )+det(A) < 0

Two of these terms are positive :

DXDY q4

[physical considerations], det(A) [previous analysis] and

therefore we require that

DY fX + DXgY > 0.

The critical case for emergence of patterns occurs for

h(q

2

) = DXDY q4 � q

2

(DY fX + DXgY ) + det(A) = 0.

Diffusion-DrivenInstability

Di↵usion Driven Instability IIWe can calculate the value for q for which the function h is

minimal which respresents the wave number which grows at the

onset of instability.

dh(q

2

)

d(k

2

)

= 2DXDY q2 � (DY fX + DXgY ) = 0

d

2

h(q

2

)

d(k

2

)

2

= 2DXDY > 0

q

2

min =

DY fX + DXgY

2DXDY

Substituting this back into the function h and setting this to less

than zero gives

�(DY fX + DXgY )2

+ 4DXDYDet(A) < 0.

The root of this equation defines the critical di↵usion constant

ratio for the onset of instability (pattern formation).

Diffusion-DrivenInstabilityII

Finite System Size and Admissible Modes

Consider again

˙~Z = A

~Z + D

@2 ~Z

@x2

on the interval [0, L] and with zero flux boundary conditions.

Solutions are combinations of waves with discrete wave numbers

qn =

2⇡nL and of the form

~Z (x , t) =

X

n

wn ~vn exp(�nt) exp(iqnx)

Any given mode will be unstable in the presence of di↵usion if

DXDY q4

n � q

2

n(DY fX + DXgY ) + det(A) < 0

FiniteSystemSizeandAdmissibleModes

ModeSelection

Re(λ)

k

GrowingmodesDampedModes

Formationofstablespatialpatterns

Activator-InhibitorSystems

Diffusion

Diffusion(d>1)ActivationInhibition

a

Activatoru

Inhibitorv

-bu

-vdegradation

Autocatalysis

+

--

LocalActivation

LongrangeLateralInhibition

Gierer-Meinhardt(1972)Activator-Inhibitor Systems (Gierer-Meinhardt 1972)

@u

@t= D

1

r2

u + ↵� �u +

�u2

v

@v

@t= D

2

r2

v + �u2 � ⌘v

Consider again

˙~Z = A

~Z + D

@2 ~Z

@x2

on the interval [0, L] and with zero flux boundary conditions.

Solutions are combinations of waves with discrete wave numbers

qn =

2⇡nL and of the form

~Z (x , t) =

X

n

wn ~vn exp(�nt) exp(iqnx)

Any given mode will be unstable in the presence of di↵usion if

DXDY q4

n � q

2

n(DY fX + DXgY ) + det(A) < 0

v

u

A. J. Koch and H. Meinhardt: Biological pattern formation 1483

Self-enhancement is essential for small local inhomo-geneities to be amplified. A substance a is said to beself-enhancing or autocatalytic if a small increase of aover its homogeneous steady-state concentration inducesa further increase of a. The self-enhancement does notneed to be direct: a substance a may promote the pro-duction rate of a substance 6 and vice versa; or, as willbe discussed further below, two chemicals that mutuallyinhibit each other's production may act together like anautocatalytic substance.Self-enhancement alone is not sufFicient to generate

stable patterns. Once a begins to increase at a givenposition, its positive feedback would lead to an overallactivation. Thus the self-enhancement of a has to becomplemented by the action of a fast-difFusing antago-nist. The latter prevents the spread of the self-enhancingreaction into the surrounding tissue without choking theincipient local increase. Two types of antagonistic reac-tions are conceivable. Either an inhibitory substance 6is produced by the activator that, in turn, slows downthe activator production or a substrate 8 is consumedduring autocatalysis. Its depletion slows down the self-enhancing reaction.

A. Activator-inhibitor systems

The following set of difFerential equations describesa possible interaction between an activator a andits rapidly diffusing antagonist h (Gierer and Mein-hardt, 1972):

is large compared to that of a: ph ) p, . Otherwise thesystem oscillates or produces traveling waves.Though not necessary for pattern formation, the satu-

ration constant v has a deep impact on the 6.nal aspectof the pattern. Without saturation, somewhat irregularlyarranged peaks are formed whereby a maximum and min-imum distance between the maxima is maintained [Figs.1(a) and l(b)]. In contrast, if the autocatalysis saturates(r ) 0), the inhibitor production is also limited. Astripelike pattern emerges. In this arrangement activatedcells have activated neighbors; nevertheless nonactivatedareas are close by into which the inhibitor can difFuse[Fig. 1(c)].Embryonic development often makes use of stripe for-

mation. For example, genes essential for the segmenta-tion of insects are activated in narrow stripes that sur-round the embryo in a beltlike manner (Ingham, 1991).In monkeys, the nerves of the right and the left eyeproject onto adjacent stripes in the cortex (Hubel et al. ,1977). The stripes of a zebra are proverbial.By convenient choice of the concentration units for c

and 6, it is always possible to set p = p and phph (Appendix A). Moreover, some constants involved inEq. (1) are not essential for the morphogenetic ability ofthis system (they are useful if one needs "fine tuning"of the regulation properties). In its simplest form, theactivator-inhibitor model is written:

ga Q—= D~ La+ p~ —p~& + 0~,Bt 1+K a2 6Bh = Dh &h + pea —phh+ oh )

where A is the Laplace operator; in a two-dimensionalorthonormal coordinate system, A = B2/Bx + B /By .D, Dp, are the difFusion constants, p, , ph the removalrates, and p, pg the cross-reaction coeKcients; 0, ahare basic production terms; K, is a saturation constant.As discussed above, lateral inhibition of a by h requires

that the antagonist h difFuse faster than the self-enhancedsubstance a: Dp, && D . This is not yet sufBcient togenerate stable patterns. We show in Appendix A that inaddition the inhibitor has to adapt rapidly to any changeof the activator. This is the case if the removal rate of 6

To simplify the notations, we shall use the same symbolto designate a chemical species and its concentration. Thisshould not lead to any confusion.Here are some orders of magnitude for the diffusion con-

stants in cells. Roughly speaking, the diffusion constants incytoplasm range from 10 cm s for small molecules to10 cm s for proteins. Diffusion from cell to cell via gapjunctions lowers these values by a factor of 10 (Crick, 1970;Slack, 1987).

FIG. 1. Patterns produced by the activator-inhibitor model(1): (a) Initial, intermediate, and final activator (top) andinhibitor (bottom) distribution. (b) Result of a similar simu-lation in a larger field. The concentration of the activator issuggested by the dot density. (c) Saturation of autocatalysis(/c ) 0) can lead to a stripelike arrangement of activatedcells.

Rev. Mod. Phys. , Vol. 66, No. 4, October 1994

Activatoruandrapidlydiffusinginhibitorv

Patternsproducedbyactivatorinhibitormodel

KochandMeinhardt,BiologicalPa\ernForma]on:fromBasicMechanismstoComplexStructuresRev.ModernPhysics66,1481-1507(1994)

CoatpatternsversussimulationsbyReactiondiffusiontypeequation

HowdifferencesinDiffusionCoefficientsforXandYCanArise

Stokes-Einstein:(r–radius,η-viscosity)

SizeDifferences

BindingDifferences:

ActivatorAandInhibitorYdiffuseatthesameratebuttheactivatorisInvolvedinabindingreactiontoanimmobilereceptorR.Effectivediffusionconstant.

D=1/(1+k)

D =kBT

6⇡⌘r

HowdoesacellknowwheretoDivide?

➢ FormationofFtsZringdividesthecell

➢ MinCPreventsFormationofFtsZring

➢ MinDbindsthecellmembraneandRecruitsMinC

➢ MinEbindstoMinDandexpelsitfromthemembrane

“MinproteinpatternsemergefromrapidrebindingandmembraneinteractionofMinE”,LooseetalNatureStructural&MolecularBiology18,577–583(2011)

Invitro,proteinwavesemergefromtheself-organizationoftheseMinProteins.Forwavepropagation,theproteinsneedtocyclethroughstatesofcollectivemembranebindingandunbinding(ATPpresent,supportedlipidbilayer).

InVitroExperiments

Dynamics

MinOscillationsinLiveCells

FluorescentlyLabeledMinD