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).
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