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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
mt C r dV
V t
dm
1
dt t1 t
c r
mt1 C r dV
cr dV
V t1 V t
Cr cr dV
V t1 V t
V t1
V t1
t1 t
d
C r cr dV
dt V t
dm
j[c(r )] dS
dt t
jcr 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
tch - 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 ;
r ,, )] c( R)v( R, t )er ;
(iii) The feeding field is convective: j[c(~
(iv) The generalized Gibbs-Thomson relation:
2
~
c(r ,, ) c(R) c0 (1 1K1 2 K2 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 21R 22
A( R) 2
2
R 21R 2
For 2 0 A(R) from r.h.s. of GR reduces to
R 21
A( R)
, R Rc
R Rc
where Rc 21
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) k BT ln A( R)
and the diffusion function
D(R, t ) D(t )A(R)
Most interesting: D(t ) t1
2
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 21R Ry
2
A( R)
R 21 R 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 - spherulitic : Vtotal Const.
t2
B - aggregational: 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 1vspec 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
jx,t
Bx f x,t Dx
D0
x
drift term
(!)
diffusion term
x - hypervolume of a single crystallite
σ 0 , D 0 - independent parameters
Dx D0 x α ,
Bx D0 x 1
scaling:
x R d holds !
d 1 surface - to - volume
d
characteristic exponent
AFTER SOLVING THE STATISTICAL PROBLEM
f x, t
divjx, t 0
t
Corresponding Initial and BoundaryConditions
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
jx,t bx
f ( x, t ) D x
x
x
drift term
(!)
diffusion term
x - hypervolume of a single cluster (internal variable)
T, D0 - independent parameters
Dx D0 x α , Note: cluster surface is crucial!
d 1 surface - to - volume
bx 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 ...