ON (dis)ORDERED AGGREGATION OF PROTEINS
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

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

GENERAL RULE BASED ON THE GROWTH RATE
M d R

M
 v
1
,  , v
M
; p
1
,  p
N
; t
 d t
- mechanism – dependent continuous function v i
system’s main variables p i
- control parameters t
- time d R
 d t

ONE-NUCLEUS BASED SCENARIO
GENERAL SCHEME FOR THE MASS CONSERVATION LAW
 t
1
 t
1
 t
 t
C r C r
V t
V t
1 V t
V t
1
V

- volume
- surface t
- time c r C r c r c r
C r c

- internal concentration (density)
- external concentration
- position vector

 t
1
 t
C r
V t c r
V t
1 c r m
 
1
 
C
 
V dV
 m
 t
 t
1
1
 t
V
 
C r

 t
1
 t
C r
V t
V t
1
C r c r m
 

C
 
V
 dV

V t
1


V c r
  dV
 m
 t


 j
 d S dV d d t
V
 
C
 d V



 
1 j
 
 d S

EMPHASIS PUT ON A CLUSTER – CLUSTER MECHANISM: d R d t

D c external
 c boundary
,
R steady
D


M
0
 t ch
 t
 

1 D f D
 f d f t

1

D
- time- and sizedependent diffusion coefficient
M
0
- initial cluster mass t
- characteristic time constant ch geometrical parameter
(fractal dimension) interaction (solution) parameter of Flory-Huggins type

PIVOTAL ROLE OF THE DOUBLE LAYER (DL):
Na + ion water dipole
Lysozyme protein random walk
DOUBLE
LAYER
Cl ion surface of the growing crystal

Growth rates for the DL-controlled on-one-nucleus-based aggregation model deterministic: d R
~

V ion
, d t supersaturation parameter t

1
Frenkel-like macroion velocity stochastic (an example): d R
~


V ion d t
 
 an ( un ) correlated noise

MANY-NUCLEI BASED SCENARIO
GRAIN (CLUSTER)-MERGING MECHANISM
3
3
1 2
1 2 t
1 t
1
3
3
2
2 t
2 t
2
A spheruliti c : V total

Const .
B aggregatio nal : V total

Const .

RESULTING 2D-MICROSTRUCTURE: VORONOI-like MOSAIC
FOR AGGREGATION
INITIAL STRUCTURE FINAL STRUCTURE

RESULTING FORMULA FOR VOLUME-PRESERVING d-DIMENSIONAL MATTER AGGREGATION d R
 k
 
R

 d

1
 v
 spec
  d t adjusting timedependent kinetic prefactor responsible for spherulitic growth hypersurface inverse term time derivative of the specific volume
(inverse of the polycrystal density)

ADDITIONAL FORMULA EXPLAINING THE MECHANISM M
(to be inserted in continuity equation) j
 
 
σ
D
0
0 b
 f
 x
(!) drift term diffusion term
D b x - hypervolume of a single crystallite
σ
0
, D
0
- independent parameters


D
0 x
α
,
D
0
 x
 x


1
R
  d

1 d surface - to - volume characteristic exponent



 f
AFTER SOLVING THE STATISTICAL PROBLEM
 
 t
 div


Correspond j
  ing f
  is obtained


0
Initial and Boundary Conditions
USEFUL PHYSICAL QUANTITIES: x n where
:

V
 fin
0 x n f
  dx
V fin
 
TAKEN USUALLY FOR THE d-DEPENDENT MODELING

CONCLUSION
 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)

LITERATURE:
- A.Danch, A.Gadomski.
a ; A.Gadomski, J.Łuczka b a Journal 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).