ON (dis)ORDERED AGGREGATION OF PROTEINS Adam Gadomski & Jacek Siódmiak Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz,

ON (dis)ORDERED AGGREGATION OF PROTEINS

Adam Gadomski & Jacek Siódmiak

Institute of Mathematics and Physics

University of Technology and Agriculture

Bydgoszcz, Poland

Workshop on Structure and Function of Biomolecules

May 13 - 15, 2004, Będlewo near Poznań, Poland

OBJECTIVE

TO PROPOSE A CONCEPTUAL AND THEORETICAL

STRATEGY, BASED ON THE GROWTH RULE AND

GROWTH MECHANISM, POSSIBLY OF USEFULNESS FOR

QUALITY AND MANUFACTURE TESTS IN PROTEIN-BASED

TECHNOLOGY AND PROTEIN-CLUSTER DESIGN

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

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

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

tppvvMtd

RdNM ;,;,, 11

GENERAL RULE BASED ON THE GROWTH RATE

M

iv

ip

- mechanism – dependent continuous function

- system’s main variables

- control parameters

t - time

consttd

Rd(desirable behavior in time: )

ONE-NUCLEUS BASED SCENARIO

GENERAL SCHEME FOR THE MASS CONSERVATION LAW

Vt

Cc

- volume

- surface

- time

- internal concentration (density)

- external concentration

r

- position vector

tV tV1tV1tV

t t

1t 1t

rc

rc

rc

rC

rCrC

tVtV

dVrcrCttt

m

11

1

1t

dt

mSj

1

SjttV

drcVdrcrCtd

d

dVrCtmtV

1

1

tVtVtV

dVrcdVrCtm1

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

PIVOTAL ROLE OF THE DOUBLE LAYER (DL):

Cl- ion

DOUBLE LAYER

surface of the growing crystal

Na+ ion

water dipole

Lysozyme protein

random walk

deterministic:

1,~ tVtd

Rdion

stochastic (an example):

ionVtd

Rd~

an (un)correlated noise

Frenkel-like macroion velocity

supersaturation parameter

Growth rates for the DL-controlledon-one-nucleus-based aggregation model

MANY-NUCLEI BASED SCENARIO

GRAIN (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

RESULTING 2D-MICROSTRUCTURE: VORONOI-like MOSAIC FOR AGGREGATION

INITIAL STRUCTURE FINAL STRUCTURE

RESULTING FORMULA FOR VOLUME-PRESERVING

d-DIMENSIONAL MATTER AGGREGATION

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

ADDITIONAL FORMULA EXPLAINING THE MECHANISM

(to be inserted in continuity equation)

M

(!)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

fin

V

nn

V

dxtxfxtxfin

0

,:

where

ConditionsBoundary and Initial ingCorrespond

txdivt

txf

0,,

j

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)

CONCLUSION

LITERATURE:

- A.Danch, A.Gadomski.a; A.Gadomski, J.Łuczkab

aJournal of Molecular Liquids, vol.86, no.1-3, June 2000, pp.249-257 b IBIDEM, 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; *Vacuume, vol50, pp.79-83

ACKNOWLEDGEMENT !!!

This work was supported by KBN grant no. 2 P03B 032 25 (2003-2006).