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 scenario
A. (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
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
II. Crystal growth – a polynuclear path
C. Protein spherulites’ formation – a competition-cooperation effect between biomolecular
adsorption and “crystallographic registry” effects (towards Muthukumar’s view)
Plan of talk (continued):
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
OBJECTIVE:
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
Routes of modeling – a summary
N\*
* Relevant Variable
* Dynamics
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-timecorrelated fluctuations (StokesLangevin type)
Sm-Ki-St
Cluster mass M
(Flory-Huggins polymer-solution
interaction parameter)
Stochastic (e.g., Poisson)
process N( t ), and its
characteristics
N>1
Bo-Gi-On
Legend to Table:
Bo-Gi-On: Boltzmann-Gibbs-Onsager
Sm-Ki-St : Smoluchowski-Kirkwood-Stokes
Fr-Ste-Po: Frenkel-Stern-Poisson
Effect of chain connectivity on nucleation
[from: M. Muthukumar, Advances in Chemical Physics, vol. 128, 2004]
Matter aggregation models, leading to (poly)crystallization
in complex entropic environments:
(A) aggregation on a
single seed in a
diluted solution,
(B) agglomeration on
many nuclei in a
more condensed
solution
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 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)
 t1
 t1
t
t
C r
C r
V t
V t1
V t
C r
c r
c r

mt    C r  dV 
V t 
dm
1

dt t1  t
c r

mt1    C r  dV

 cr  dV
V t1 V t 


 C r   cr dV
V t1 V t 
V t1
V t1 
t1 t


d
C r   cr dV 

dt V t 

dm
  j[c(r )]  dS
dt  t 

 jcr  dS
 t 
EMPHASIS PUT ON A CLUSTER – CLUSTER MECHANISM:
cexternal  cboundary
dR
D
,
dt
Rsteady
D  M 0 tch  t 
D
1 D f
- time- and sizedependent diffusion
coefficient
M 0 - initial cluster mass
t ch - characteristic time constant
t  1
Df  d f 
geometrical
parameter
(fractal dimension)
interaction (solution)
parameter
of Flory-Huggins type
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: R  R (t )  0 ;
(iii) The feeding field is convective: j[c(~
r ,, )]  c( R)v( R, t )er ;
(iv) The generalized Gibbs-Thomson relation:
c(~
r , ,  )  c( R)  c0 (1  1 K1  22 K 2  a.t.)
1
2
where: K1 
; K2  2
R
R
additional
terms
(curvatures !)
and c0  c( R) when R   (on a flat surface)
i : thermodynamic parameters
Growth Rule (GR)
i=1 capillary (Gibbs-Thomson) length
dR
 A( R)v( R, t )
dt
i=2 Tolman length
DL-INFLUENCED MODEL OF GROWTH (continued, a.t.
neglected): specification of A(R) and v( R, t )
R 2  21 R  22
A( R)   2
R  21 R  22
For 2  0 A(R) from r.h.s. of GR reduces to
R  21
A( R)  
, R  Rc
R  Rc
where
Rc  21 
For nonzero -s: R~t is an asymptotic solution to GR – constant tempo !
c0
 - supersaturation dimensionless parameter;  
C  c0
v( R, t )  velocity of the particles nearby the object
Could v(R,t) express a truly mass-convective nature? What for?
DL-INFLUENCED MODEL OF GROWTH: stochastic part
Assumption about time correlations within the particle velocity field
[see J.Łuczka et al., Phys. Rev. E 65, 051401 (2002)]
v( R, t )  V (t )
where
V (t )  0,
V (t )V (s)  K ( t  s )
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:
dR
 A( R)V (t )
dt
Fokker-Planck representation:


P ( R, t )  
J ( R, t )
t
R
 

2 
with J ( R, t )   D(t ) A( R)
A( R)  P ( R, t )  D(t )[ A( R)]
P ( R, t )
R
 R

t

and D(t )  K ( s)ds (Green-Kubo formula),
0
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
P( R, t )   D( R, t ) 
P( R, t ) 

P( R, t )  D( R, t )


t
R  k BT R
R 
where the energy (called: entropic potential)
and the diffusion function
Most interesting: D(t )  t
  kBT ln A( R)
D( R, t )  D(t )A( R)
2
1
for
t  t0 (dispersive kinetics !)
Especially, for readily small  it indicates a superdiffusive motion !
The matter flux:
D( R, t ) 
P( R, t )
J ( R, t )  
P ( R, t )  D ( R, t )
k BT R
R
DL-INFLUENCED SCENARIO: when a.t. stands for an elastic
contribution to the surface-driven crystal growth (2=0)
R  21R  Ry 
2
A( R)  
R  21R  Ry 
2
y    y (  ) - positive or negative (toward auxetics) elastic term
  1,2,3 - 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
y (  ) - elastic dimensionless displacement
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.
  0
  0
POLYNUCLEAR PATH
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.
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
dR
 k t  R d 1vspec t 
dt
adjusting timedependent kinetic
prefactor responsible
for spherulitic growth:
it involves orderdisorder effect
hypersurface
inverse term
time derivative of the
specific volume
(inverse of the
polycrystal density)
ADDITIONAL FORMULA EXPLAINING THE MECHANISM
(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
 Correspond ing 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   or V fin  finite
TAKEN USUALLY FOR THE d-DEPENDENT MODELING
AGAIN: THE GROWTH MODEL COMES FROM MNET

f x,t
jx,t  bx 
f ( x, t )  D  x 
x
x
drift term
(!)
diffusion term
x - hypervolume of a single cluster (internal variable)
T, D0 - independent parameters
Dx   D0 x α ,  Note: cluster surface is crucial!
d  1 surface - to - volume
bx   D0 k BT x α

d characteristic exponent
scaling: x  R d holds ! f ;   kinetic & thermodyna mic
GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM
OF DERIVED POTENTIALS (FREE ENERGIES) AS

‘STARTING FUNDAMENTALS’ OF CLUSTER-CLUSTER LATETIME AGGREGATION
S  1 T   ( x, t )f dx
   ( x, t ) -internal variable and time dependent chemical potential

-denotes variations of entropy S and f  f ( x, t ) (and f-unnormalized)
(i) Potential for dense micro-aggregation (for spherulites):
   ( x)  ln( x)
(ii) Potential for undense micro-aggregation (for non-spherulitic
flocks):
1d
   ( x)  x
CONCLUSION & PERSPECTIVE

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
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 ...