Aggregation Effects  - Spoilers or

Benefactors

of Protein Crystallization ?

Adam Gadomski

Institute of Mathematics and Physics

University of Technology and Agriculture

Bydgoszcz, Poland

Berlin – September 2004

Plan of talk:

1. CAST OF CHARACTERS – a microscopic view:

I. Crystal growth - a single-nucleus based scenarioA. (Protein) Cluster-Cluster Aggregation – a short overview in terms of its microscopic picture

B. Microscopic scenario associated with (diffusive) Double Layer formation, surrounding the protein crystal

II. Crystal growth – a polynuclear path C. Smectic-pearl and entropy connector model (by Muthukumar) applied to protein spherulites

2. CAST OF CHARACTERS – a mesoscopic view:

I. Crystal growth - a single-nucleus based scenario

II. Crystal growth – a polynuclear path

A. (Protein) Cluster-Cluster Aggregation – a cluster-mass dependent construction of the (cooperative) diffusion coefficient

B. Fluctuational scenario associated with (diffusive) Double Layer formation – fluctuations within the protein (protein cluster) velocity field nearby crystal surface

C. Protein spherulites’ formation – a competition-cooperation effect between biomolecular adsorption and “crystallographic registry” effects (towards Muthukumar’s view)

3. An attempt on answering the QUESTION:

"Protein Aggregation - Spoiler or Benefactor in Protein Crystallization?"

A. What do we mean by ‘Benefactor’: Towards constant speed of the crystal growth

B. When ‘Spoiler’ comes? Always, if … it is not a ‘Benefactor’

4. Conclusion and perspective

Plan of talk (continued):

TO DRAW A (PROTEIN) CLUSTER-CLUSTER AGGREGATION* LIMITED VIEW OF PROTEIN

CRYSTAL GROWTH

__________*Usually, an undesirable aggregation of (bio)molecules is proved experimentally to be a spoiling side effect for crystallization conditions

OBJECTIVE:

N\* * Relevant Variable * Dynamics

N>1

Bo-Gi-On

Protein crystallite’s individual volume – a stochastic variable v

Thermodynamic potentials, and ‘forces’, a presence of entropic barriers

N=1

  Fr-Ste-Po

Crystal radius R Fluctuating protein velocity field – (algebraic) in-time-correlated fluctuations (Stokes-Langevin type)

Sm-Ki-St Cluster mass M

(Flory-Huggins polymer-solution interaction parameter)

Stochastic (e.g., Poisson) process N( t ), and its characteristics

Legend to Table:

 Bo-Gi-On: Boltzmann-Gibbs-Onsager Sm-Ki-St : Smoluchowski-Kirkwood-Stokes Fr-Ste-Po: Frenkel-Stern-Poisson

Routes of modeling – a summary

Effect of chain connectivity on nucleation

[from: M. Muthukumar, Advances in Chemical Physics, vol. 128, 2004]

(A)aggregation on a single seed in a diluted solution,

(B) agglomeration on many nuclei in a more condensed solution

Matter aggregation models, leading to (poly)crystallization in complex entropic environments:

PIVOTAL ROLE OF THE DOUBLE LAYER (DL):

Cl- ion

DOUBLE LAYER

surface of the growing crystal

Na+ ion

water dipole

Lysozyme protein

random walk

Growth of smectic pearls by reeling in the connector (N = 2000).[from: M. Muthukumar, Advances in Chemical Physics, vol. 128, 2004]

GROWTH OF A SPHERE: mass conservation law (MCL)

tt

tVtV

dVrcrCttdt

dm

1

11

1

t

drcdt

dmSj )]([

ttV

drcdVrcrCdt

dSj

tVtVtV

dVrcdVrCtm1

dVrCtmtV

1

1

tV tV1tV1tV

t t

1t 1t

rc

rc

rc

rC

rCrC

EMPHASIS PUT ON A CLUSTER – CLUSTER MECHANISM:

1,

tR

ccD

td

Rd

steady

boundaryexternal

ff dD

geometricalparameter

(fractal dimension)

interaction (solution)parameter

of Flory-Huggins type

fD10

ttMD ch

0M

D

- initial cluster mass

- time- and size-

dependent diffusion

coefficient

cht - characteristic time constant

MODEL OF GROWTH: emphasis put on DL effect

Under assumptions [A.G., J.Siódmiak, Cryst. Res. Technol. 37, 281 (2002)]:

(i) C=const

(ii) The growing object is a sphere of radius: ;

(iii) The feeding field is convective: ;

(iv) The generalized Gibbs-Thomson relation:

where: ; (curvatures !)

and when (on a flat surface)

: thermodynamic parameters

i=1 capillary (Gibbs-Thomson) length

i=2 Tolman length

0)( tRR

rtRvRcrc ej ),()()],,~([

.).1()(),,~( 222110 taKKcRcrc

RK

21 22

1

RK

)(0 Rcc R

i

),()( tRvRAdt

dR

Growth Rule (GR)

additional terms

DL-INFLUENCED MODEL OF GROWTH (continued, a.t. neglected): specification of and)(RA ),( tRv

11 2where,

2)(

ccc

RRRRR

RRA

221

2

221

2

2

2)(

RR

RRRA

For A(R) from r.h.s. of GR reduces to02

For nonzero -s: R~t is an asymptotic solution to GR – constant tempo !

),( tRv velocity of the particles nearby the object

Could v(R,t) express a truly mass-convective nature? What for?

- supersaturation dimensionless parameter;0

0

cC

c

DL-INFLUENCED MODEL OF GROWTH: stochastic part

)(),( tVtRv

where

)()()(,0)( stKsVtVtV

Assumption about time correlations within the particle velocity field [see J.Łuczka et al., Phys. Rev. E 65, 051401 (2002)]

K – a correlation function to be proposed; space correlations would be a challenge ...

Question: Which is a mathematical form of K that suits optimally to a growth with constant tempo?

DL-INFLUENCED MODEL OF GROWTH: stochastic part (continued)

Langevin-type equation with multiplicative noise:

)()( tVRAdt

dR

Fokker-Planck representation:

),(),( tRJR

tRPt

with ),()]()[(),()()()(),( 2 tRPR

RAtDtRPRAR

RAtDtRJ

and dssKtDt

0

)()( (Green-Kubo formula),

with corresponding IBC-s

THE GROWTH MODEL COMES FROM MNET (Mesoscopic

Nonequilibrium Thermodynamics, Vilar & Rubi, PNAS 98, 11091 (2001)): a flux of matter specified in the space of cluster sizes

R

tRPtRDtRP

RTk

tRD

Rt

tRP

B

),(),(),(

),(),(

where the energy (called: entropic potential) )(ln RATkBand the diffusion function 2)()(),( RAtDtRD

R

tRPtRDtRP

RTk

tRDtRJ

B

),(

),(),(),(

),(

The matter flux:

Most interesting: 01 for)( ttttD (dispersive kinetics !)

Especially, for readily small it indicates a superdiffusive motion !

DL-INFLUENCED SCENARIO: when a.t. stands for an elastic contribution to the surface-driven crystal growth (2=0)

RyRR

RyRRRA

1

2

12

2

2)(

Example: =1 (1D case): cs(R)=c0(1 + 1K1 + y1), where y1=1Leff ; here Leff=y(1)=(L-L0)/L0, L and L0 are the circumferences of the nucleus at time t and t0 respectively. In the case of (ideal) spherical symmetry

we can write that y1 = 1 (R-R0)/R0.

)( yy - positive or negative (toward auxetics) elastic term

3,2,1 - specify different elastic-contribution influenced mechanisms linear ( =1), surfacional ( =2) or volumetric ( =3)

- positive or negative dimensionless and system-dependent elastic parameter, involving e.g. Poisson ratio

- elastic dimensionless displacement)(y

0 0

POLYNUCLEAR PATHGRAIN (CLUSTER)-MERGING MECHANISM

.V:cspheruliti-A total Const .V:nalaggregatio-B total Const

1

1 1

22

12

3

3 3

3

2 2

2

t t

tt

TYPICAL 2D MICROSTRUCTURE: VORONOI-like MOSAIC FOR A TYPICAL POLYNUCLEAR PATH

INITIAL STRUCTURE FINAL STRUCTURE

RESULTING FORMULA FOR VOLUME-PRESERVING

d-DIMENSIONAL MATTER AGGREGATION – case A

tvRtktd

Rdspec 1d

time derivative of the specific volume (inverse of the

polycrystal density)

hypersurface inverse term

adjusting time-dependent kinetic

prefactor responsible for spherulitic growth:

it involves order- disorder effect

ADDITIONAL FORMULA EXPLAINING THE MECHANISM

(to be inserted in continuity equation)

(!)x

x,tfxDx,tfxB

D

σx,t

0

0j

00 D,σ

x - hypervolume of a single crystallite

- independent parameters

drift term diffusion term

1

0

0 ,

xDxB

xDxD α

surface - to - volume characteristic exponentd

d 1

scaling: holds !dRx

AFTER SOLVING THE STATISTICAL PROBLEM

txf , is obtained

USEFUL PHYSICAL QUANTITIES:

TAKEN USUALLY FOR THE d-DEPENDENT MODELING

finiteVorV

dxtxfxtx

finfin

V

nnfin

0

,:

where

ConditionsBoundary and Initial ingCorrespond

txdivt

txf

0,,

j

AGAIN: THE GROWTH MODEL COMES FROM MNET

(!)x

x,tfxDtxf

xxbx,t

),(

j

0D T,

x - hypervolume of a single cluster (internal variable)

Note: cluster surface is crucial!

drift term diffusion term

α

B

α

xTkDxb

xDxD

0

0 ,

surface - to - volume

characteristic exponentd

d 1

scaling: holds ! dRx micthermodyna&kinetic; f

- independent parameters

dxftxTS ),(1

GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM OF DERIVED POTENTIALS (FREE ENERGIES) AS

‘STARTING FUNDAMENTALS’ OF CLUSTER-CLUSTER LATE-TIME AGGREGATION

),( tx

-internal variable and time dependent chemical potential

-denotes variations of entropy S and (and f-unnormalized)

(i) Potential for dense micro-aggregation (for spherulites):

(ii) Potential for undense micro-aggregation (for non-spherulitic flocks):

dxx 1)(

)ln()( xx

),( txff

THERE ARE PARAMETER RANGES WHICH SUPPORT THE AGGREGATION AS A RATE-LIMITING STEP, MAKING THE PROCESS KINETICALLY SMOOTH, THUS ENABLING THE CONSTANT CRYSTALLIZATION SPEED TO BE EFFECTIVE (AGGREGATION AS A BENEFACTOR)

OUTSIDE THE RANGES MENTIONED ABOVE AGGREGATION SPOILS THE CRYSTALLIZATION OF INTEREST (see lecture by A.Gadomski)

ESPECIALLY, MNET MECHANISM SEEMS TO ENABLE TO MODEL A WIDE CLASS OF GROWING PROCESSES, TAKING PLACE IN ENTROPIC MILIEUS, IN WHICH MEMORY EFFECTS AS WELL AS NON-EXTENSIVE ‘LIMITS’ ARE THEIR MAIN LANDMARKS

CONCLUSION & PERSPECTIVE

LITERATURE:

-D.Reguera, J.M.Rubì; J. Chem.Phys. 115, 7100 (2001)

- A.Gadomski, J.Łuczka; Journal of Molecular Liquids, vol. 86, no. 1-3, June 2000, pp. 237-247

- J.Łuczka, M.Niemiec, R.Rudnicki; Physical Review E, vol. 65, no. 5, May 2002, pp.051401/1-9

- J.Łuczka, P.Hanggi, A.Gadomski; Physical Review E, vol. 51, no. 6, pt. A, June 1995, pp.5762-5769

- A.Gadomski, J.Siódmiak; *Crystal Research & Technology, vol. 37, no. 2-3, 2002, pp.281-291; *Croatica Chemica Acta, vol. 76 (2) 2003, pp.129–136

- A.Gadomski; *Chemical Physics Letters, vol. 258, no. 1-2, 9 Aug. 1996, pp.6-12; *Vacuum, vol 50. pp.79-83

- M. Muthukumar; Advances in Chemical Physics, vol. 128, 2004

ACKNOWLEDGEMENT !!!

Thanks go to Lutz Schimansky-Geier for inviting me to present ideas rather than firm and well-established

results ...