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
GENERAL RULE BASED ON THE GROWTH RATE
dR
 M v1 ,, vM ; p1 ,  p N ; t 
dt
M
vi
- mechanism – dependent continuous function
pi
- control parameters
t
- time
- system’s main variables
dR
 const )
(desirable behavior in time:
dt
ONE-NUCLEUS BASED SCENARIO
GENERAL SCHEME FOR THE MASS CONSERVATION LAW
 t1
 t1
t
t
C r
V t
C r
V t1
V t
C r
c r
c r
V - volume
 - surface
t - time
V t1
c r
C - internal concentration (density)
c - external concentration
r
- position vector
 t1
 t1
t
t
C r
V t
C r
V t1
V t
C r
c r
c r

mt1    C r  dV
V t1 
m
1

 t t1  t
V t1
c r

mt    C r  dV 


 C r   cr dV
V t1 V t 


d
C r   cr d V 

d t V t 
V t 

 cr  dV
V t1 V t 
m
  j  dS
 t  t1 

 jcr  dS
 t1 
EMPHASIS PUT ON A CLUSTER – CLUSTER MECHANISM:
cexternal  cboundary
dR
D
,
dt
Rsteady
D  M 0 tch  t 
1 D f
D
- time- and sizedependent diffusion
coefficient
M 0 - initial cluster mass
t ch - characteristic time constant
t  1
Df  d f 
interaction (solution)
geometrical
parameter
parameter
(fractal dimension) of Flory-Huggins type
PIVOTAL ROLE OF THE DOUBLE LAYER (DL):
Na+ ion
Lysozyme protein
water dipole
random walk
DOUBLE
LAYER
Cl- ion
surface of
the growing
crystal
Growth rates for the DL-controlled
on-one-nucleus-based aggregation model
deterministic:
supersaturation
parameter
dR
~  Vion ,
dt
t  1
Frenkel-like macroion velocity
stochastic (an example):
dR
~   Vion   
dt
an (un)correlated noise
MANY-NUCLEI BASED SCENARIO
GRAIN (CLUSTER)-MERGING MECHANISM
3
3
1
1
2
2
t1
t1
3
3
2
2
t2
A - spheruliti c : Vtotal  Const.
t2
B - aggregatio nal : Vtotal  Const.
RESULTING 2D-MICROSTRUCTURE: VORONOI-like MOSAIC
FOR AGGREGATION
INITIAL STRUCTURE
FINAL STRUCTURE
RESULTING FORMULA FOR VOLUME-PRESERVING
d-DIMENSIONAL MATTER AGGREGATION
dR
 d 1
 k t  R
vspec t 
dt
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)
σ0
f x,t 
jx,t    bx  f x,t   Dx 
D0
x
drift term
(!)
diffusion term
x - hypervolume of a single crystallite
σ 0 , D 0 - independent parameters
Dx   D0 x α ,
bx   D0 x 1
scaling:
x  R d holds !
d  1 surface - to - volume

d characteristic exponent
AFTER SOLVING THE STATISTICAL PROBLEM
 f  x, t 
 divj x, t   0

 t
 Corresponding Initial and Boundary Conditions
f x, t 
is obtained
USEFUL PHYSICAL QUANTITIES:
x t  :
n
V fin
 x f x, t dx
n
0
where
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.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).