Biotechnology Letters Vol 12 No I0
Received as revised 16th August
E F F E C T
O F
F P ~ C T A L
N A T U R E
E N Z Y M A T I C
R E A C T I O N S
737-742 (1990)
O N
A r c h a n a S h a r m a , U j w a l P. S h i n d e a n d B.D. K u l k a r n F
Division o f Chemical E n g i n e e r i n g , N a t i o n a l Chemical L a b o r a t o r y , P u n e 411 008
INDIA
S U M M A R Y
M o d e l H n g immobih'zed e n z y m e s as a fractal object in the form of a D L A a n d each
e n z y m e molecule as another fractal object in the form of a percolation cluster,
the present work-simulates the performance for a sequence of elementary reactions
a n d transport on the surface. The results s h o w that non-ideah'ties in the
performance, such as mul~'-stationarity a n d substrate inhibi~bn, can also ar~se
in this simple description~
I N T R O D U C T I O N
Surface representations of proteins have provided a powerful approach for
characterization of the structure, folding, interactions and properties of proteins.
A fundamental feature of proteins that has not been
characterized by
these
representations, however, is the texture (roughness) of protein surfaces and its
role in molecular interactions. T h e
degree of irregularity of a surface m a y be
described b y the fractal dimension D (Lewis and Rees, 1985). The ensuing fracta/
surface dimension controls biologica/ly relevant processes.
Mandelbrot's fractal geometry provides a descriptive and a mathematical w a y
to model m a n y of the seemingly complex forms found in nature (Mandelbrot, 1983).
Statistical self-similarity is the essential quality of fractals in nature. Although
the fractal dimensionality is but a single parameter, it is nevertheless a useful
indicator of protein conformation because it provides a quantitative measure of
the degree to which a structure fills the space in which it resides (Wako~ 1989)o
T h e rate of substrate arrival (by diffusion) at a biological receptor depends
very m u c h on whether the diffusion space is 3-, 2- or 1-dimensional. The m a J d m u m
rate of a reaction will depend on the encounter probability of the components.
T h e diffusion process can be treated in terms of Brownlan motion, or the r a n d o m
walk process. It has been s h o w n in the case of lysozyme that substrate molecules
close to the surface are trapped a n d then migrate along the surface to the active
site (Pfeifer et ad, 1985). S u c h diffusion on complex proteins with fractal structures
m a y have important biological implications (Voss, 1988).
737
In the
present
the substrate
than
that
site, and
can
then
attach
occurs
bulkiness
and
diffuse
to the
dimension of the surface, surface
size and
considered
an enzymatic
reaction
in w h i c h
molecules can attach to certain points on the enzyme surface, other
the active
substrate
w o r k , we h a v e
surface
active site.
greatly
On r e a c h i n g
is c o n v e r t e d
substrate
will
the
depend
The amount of
on
the
fractal
area of enzyme exposed to the medium and the
of the substrate.
the substrate
towards
or the
to product.
the active
Either the
site,
the
reaction
intrinsic rate
conversion
of the
availability of it d u e to s u r f a c e
would then
determine the rate of the reaction. Diffusion on fractal structures
of
diffusion
k n o w n t o b e a n o m a l o u s ( A r g y r a k i s a n d K o p e l m a n , 1990) a n d a p a r t i c u l a r l y
is
simple
i l l u s t r a t i o n o f a r a n d o m f r a c t a l is a p e r c o l a t i o n l a t t i c e ( S t a u f f e r , 1985). I f m o v e m e n t
is allowed from one
only to a nearest
site of the
neighbour
percolation
cluster
occupied
by the
s i t e o f t h e c l u s t e r , t h e m o t i o n is r e s t r i c t e d .
of this
w o r k is t o e l u c i d a t e
the
effect of the
surface
diffusion on them, on rates of enzymatic reactions.
S I M U L A T I O N
fractal
nature
substrate
T h e aim
of enzymes, and
D E T A I L S
Considering attachment of a substrate molecule to the e n z y m e surface, the
reaction mechanism is depicted in figure 1. T h e following assumptions were m a d e
KI
to simplify the simulation.
EoS
,~
EsS
1) A simple method frequently used
for e n z y m e immobilization
is chemical
aggregation using glutaraldehyde as
a
bifunctional reagent (Khare a n d Gupta,
Eo+ S
~~._~ p
Es+ S
1990). W e have modelled such an aggregate
as a D L A cluster which is a r a n d o m fractal
(Witten a n d Sander, 1981)
2) A n u m b e r of protein molecules are
k n o w n to possess a fractal dimension and
a n u m b e r of investigators have reported
such a dimension for different proteins FIGW~ 1 : ache.tic repreuestatioa of the r e u t i , aechui.:
(Lewis a n d Rees, 1985; Wako, 1989). In the h -- same active site; Zs -- esme ssrhce otter thu tie
present w o r k we consider such a protein active site; [d -- cosplex or unbotrate at ensure active site;
with a fractal dimension of -1.89. It is EsS -- usbstrate molecule attacked to emse sorhce; P product; [I l [! are pr6portio.i to the oarhce ares of the
k n o w n that the different conformations of eame other tan the active site nd the eame active site,
proteins m a y give rise to the same value respectively; 13 is proportiotal to the probabilitl ofcollidisg
of fractal dimension. Therefore w e can at the active site nd sttachiug there 15 is proportional to
generally represent
s u c h a p r o t e i n a s a the probability of attactuett to the eHIse .rfice oa collidisg
p e r c o l a t i o n c l u s t e r w i t h t h e s a m e f r a c t a l there; 14 | 16 are proportionl to the dissociatiot cosstnts
of hS ud toO,respectively; 17 is proportional to tie reaction
dimension.
probtbilitj os coaplez for.riot; 18 is proportio.l to the
3) T h e r e is n o i n t e r a c t i o n b e t w e e n . r h c e dill.Sou cod[iciest ud the mber of .rhce bo.d
.botrate .lecsleu
individual substrate molecules.
4) The substrates can attach only to the edges of the percolation cluster
to account for the fact that only certain patches on the e n z y m e surface are
capable of interacting with the substrate.
5) Once the substrate is b o u n d to the surface, it moves towards the active
site through the shortest available distance.
738
6) T h e d i s s o c i a t i o n c o n s t a n t s f o r t h e c o m p l e x e s f o r m e d a r e low.
The above given reaction mechanism
o~ CL~S~,
was simulated u s i n g t h e Monte Carlo
method. The enzyme agglomerabe was
a s s u m e d t o b e in t h e f o r m o f a DLA c l u s t e r
~-"
~
and each individual enzyme molecule as a
percolation
cluster
at
the
threshold
( f i g u r e 2). A 2 - d i m e n s i o n a l DLA c l u s t e r
w i t h ~ 104 l a t t i c e p o i n t s ( f r a c t a l d i m e n s i o n
~ 1.7) a n d a 2 - d i m e n s i o n a l p e r c o l a t i o n
c l u s t e r w i t h ~ 2.5 X 103 ( f r a c t a l d i m e n s i o n
~ 1.89) w e r e c o m p u t e d f o r t h e p r e s e n t
.'::'":I
CLUST[N
study. Certain sites on the percolation
o:..:;- :::
c l u s t e r w e r e l a b e l e d a s t h e a c t i v e s i t e . 10
l a t t i c e p o i n t s o n all f o u r s i d e s o f t h i s
cluster were kept vacant to facilitate the
ill.:: : . . : : : : ' : : ' . . : " ' : - : . . : L : :" ,: .
random walk of the substrate.
Particles were released from outside
":.:'-:::!~-:'~:.~T" / .:L~:::~'~:--: :'1:"
o f t h e r e g i o n o c c u p i e d b y t h e DLA c l u s t e r
and followed a random walk trajectory on
".~;i~" "--c-" " :.!::'~.
the lattice. If the particle reached
the
e n d o f t h e l a t t i c e , it w a s r e j e c t e d . On ~l~t~ ~ : Inobili~a eas~e ~yste~depicted b~~ DLIchafer.
hdi~idnl e~s~e ~olec~les are re~rese~t~t b~ ~ ~ercolxtio~
r e a c h i n g a p o i n t o n t h e DLA c l u s t e r , t h a t cluter at t~e per~lttie~ tkreskoldo
point was magnified as a percolation
c l u s t e r . T h e p a r t i c l e t h e n c o n t i n u e d i t s A o- legiou of acti,e site
random
walk t o w a r d s
the
p e r c o l a t i o n B -- lattice sise .ucc,pied bl percolatiol cluter
C 'De~ ~0ue'...
cluster. If the particle reached the end
-- Perc01ati01 cluter zites re~re,esLi~g t~e xcLive~ite
o f t h i s l a t t i c e , it w a s a g a i n a l l o w e d t o walk
o n t h e l a t t i c e o f t h e DLA c l u s t e r . On r e a c h i n g t h e p e r c o l a t i o n c l u s t e r , t h e
p a r t i c l e was allowed to follow a biased r a n d o m walk t o w a r d s the a c t i v e site b y
t h e s h o r t e s t available d i s t a n c e . Diffusion of s u b s t r a t e p a r t i c l e s b o u n d on the
e n z y m e s u r f a c e was simulated s i m u l t a n e o u s l y . If a p a r t i c l e t r i e d to a t t a c h to a
p o s i t i o n w h e r e a n o t h e r s u b s t r a t e w a s a l r e a d y p r e s e n t , it w a s r e j e c t e d . I f o n e
p a r t i c l e h a d all f o u r n e a r e s t n e i g h b o u r s o c c u p i e d , t h e n it d i d n o t m o v e a t t h a t
instant 0n reaching the active site, the complex was formed and reaction took
place with a certain high probability. Assuming different surface diffusion
c o e f f i c i e n t s (D), in t h e r a n g e o f 0.005 t o 8, p r o d u c t f o r m a t i o n a s a f u n c t i o n o f
time w a s s t u d i e d .
S i m i l a r s i m u l a t i o n s w e r e p e r f o r m e d f o r a p e r f e c t s q u a r e l a t t i c e in p l a c e o f
a p e r c o l a t i o n c l u s t e r to d e l i n e a t e t h e e f f e c t o f f r a c t a l n a t u r e o f t h e s u r f a c e .
Variation of r a t e o f p r o d u c t formation with s u b s t r a t e c o n c e n t r a t i o n was s t u d i e d
for both the cases.
R E S U L T S
Many
chemical
characteristic
and
/kl~D
biochemical
behaviour patterns
m o d e l s to e x p l a i n t h e o b s e r v a t i o n s
D I S C U S S I O N
reactions
in
nature
exhibit
complex
which r e q u i r e t h e use of h i g h l y n o n l i n e a r r a t e
(Sadana eta/,
1981}. I n r e a l i t y , h o w e v e r , t h e
b e h a v i o u r m a y h a v e i t s o r i g i n in o t h e r n o n i d e a l i t i e s s u c h a s f r a c t i o n a l d i m e n s i o n s
of the
system
and
the
rate
and
transport
739
processes
on them.
In the present
study we have tried to analyse this effect by considering a sequence of elementary
hioreactions on the enzyme molecules. The analysis, however, can also be carried
out for any other heterogenous
Profiles for product
system.
formation
,
~
/
coefficients
/
shown
in
(percolation c l u s t e r ) a n d
lattice). The presence
periods
of
time
figure
3b
where
3a
the
~
../ ' ....//..;
f///
0 005 r
"
o..oo/o.o,,~'~ "............
(perfect
of regions
1.0
(a)
as
a function of time for different diffusion
are
,
over
O
:
-/ "/"/"/'/ii
"/''" ......
product
0-01
~
0"001
formation
is
discerned
relatively
in
this
slow
figure.
can
be
This
is
........
especially clear if we examine the profiles
f o r l o w e r d i f f u s i o n c o e f f i c i e n t s . T h i s is,
however,
not
the
case
for
lattice as seen in figure
3b.
This can
following
be
explained
observations
a
perfect
i
I
I
I
fb)
I.(
o
a.
using
during
the
the
0"0
~5/
l
te
i
simulation: At the very beginning of the
r e a c t i o n , v e r y small a m o u n t o f s u b s t r a t e
is
bound
to
diffusion
is
the
enzyme
fast,
these
surface.
/
0
slow
they
could have
in
!
0.2
!
!
0-4
O.E
DIMENSIONLESS TIME
substrate
O0
10
FIGOU 3 : Pr0dect f0r.ti01 u af.cti01 0f Time.
particles can reach the active site faster
than
I
I
If
(~) Ii cue of reactiol ol a percolltioi cluter; (b) ll cue
of resctio~ o, a perfect sq.re lattice
the case of
diffusion.
After lapse of some time, more substrates bind but the hinderance to their
m o v e m e n t is not very high. As time proceeds, the surface is almost saturated
with the substrate a n d resistance to diffusion is high. D u e to the presence of
lattice sites unoccupied b y the percolation cluster (shown in figure 2), achievement
of equal distribution of substrate over the e n z y m e surface t a k e s time. The large
a m o u n t of substrate that binds initially takes time to reach the active site and
further attachment is hindered. Only w h e n most of these have reacted can more
substrate bind and
regions
diffuse slowly towards the active site. This gives rise to
of relatively
slow
product
formation. O n
a
perfect
square
lattice,
achievement of a more or less equal distribution of substrate on the surface is
faster than on a percolation cluster. U n d e r conditions of high diffusion coefficients,
lattice sites unoccupied b y percolation cluster do not cause resistance to m o v e m e n t
a n d the surface also does not get saturated.
740
Figure 4 depicts
surface
as a f u n c t i o n of s u b s t r a t e
~ractal nature
of t h e s u r f a c e ,
percolation cluster
10
the profiles for the
i
1 ~
i
II
I
I
~
on a square
I
1il
concentration.
I
I
to the
effect of the
for reaction on a
lattice.
1"0[ . . . . . . ]
I
I
t it---
n
I
O- 0.00~
Z
Ow
u
~- ~
2: m
attached
To d e l i n e a t e t h e
we h a v e c o m p a r e d t h e p r o f i l e s
with that
~
amount of substrate
0"1
I
~
I
~
~
F"----T
.
m
.
ft._m_._.
n-o-oo~
i ~
I - I t O H E 'l
1i \
,-zo.~ 2
'
~O
i o.,, :._._ --.-...__
@,
"-
~ SQUARE LATTICE
'PERCOLAu
I
0"001
,
,
0;~
,
I
,
04-
CLUSTEF~
I
0,6
i
$OOAR~ L A T T I C E
PEI~COt.ATION CLUST~'R
- i L l _
08-
I'0
I
0.010
-L-
0.;~
I
I
I
0'4
t ~
0 6
t
08
l
DIME~ISIONLF~S SUBSTRATI[ CONCENTRATtON
DIMEEI~;IONLI~S$
IIC6gl 5 : ht e 0t reacti01 u I fucti01 of smb~trtte
FIGUgg 4 : taozzt of sirface bond sibstrate as a fizctiol of
sebstrate coaceiLritioL
At low s u b s t r a t e
concentrations
SUBSTRATF. COHCENT~ATION
co~celtratio~,
( z o n e 1 in f i g u r e 4), t h e a m o u n t o f s u b s t r a t e
t h a t c a n b i n d to t h e s u r f a c e will n o t b e a f f e c t e d b y t h e p e r i m e t e r . T h i s is b e c a u s e
at this concentration
the substrate,
of t h e s u b s t r a t e ,
the ratio of the surface
area occupied by
in c o m p a r i s o n t o t h e o v e r a l l a r e a t o w h i c h t h e s u b s t r a t e
c a n bind~
b o t h in t h e c a s e of a s q u a r e
l a t t i c e a n d a p e r c o l a t i o n c l u s t e r , is small. M o v e m e n t
is f a c i l i t a t e d
lattice and
on the square
on the percolation cluster
At i n t e r m e d i a t e
substrate
surface
that
o f r e a c t i o n is h i g h e r
than
that
( z o n e 1 in f i g u r e 5).
substrate
can bind to the
area available
so r a t e
concentration
surface
( z o n e 2 i n f i g u r e 4), t h e a m o u n t o f
will n o w d e p e n d
for movement towards
the
on the
active site.
perimeter
In the
and
case of a
p e r c o l a t i o n c l u s t e r , t h e p e r i m e t e r is l a r g e r t h a n t h a t o f t h e s q u a r e l a t t i c e . T h u s ,
the a m o u n t of s u b s t r a t e
in i t s m o v e m e n t a n d
t h a t b i n d s to the c l u s t e r
so t h e r a t e s t a r t s
rate of reaction on the square
from less surface
bound
is l a r g e r .
f a l l i n g ( z o n e 2 in f i g u r e 5). T h e fall in
l a t t i c e is f a s t e r a s it u n d e r g o e s
substrate
to s a t u r a t i o n
of the surface
z o n e 3 in f i g u r e 4) a n d h a s a g r e a t e r t e n d e n c y to g e t s a t u r a t e d
At h i g h s u b s t r a t e
It faces restrictions
concentration,
the average
a larger
( f r o m z o n e 2 to
with the substrate.
amount of substrate
b i n d to t h e s u r f a c e o f t h e p e r c o l a t i o n c l u s t e r is l o w e r e v e n t h o u g h
o f t h e s a m e is l a r g e r
than that
is b e c a u s e t h e r e s t r i c t i o n
of the s q u a r e
that can
the perimeter
l a t t i c e ( z o n e 3 in f i g u r e
to movement on the percolation cluster
741
change
4). T h i s
is l a r g e r
due
|'0
to presence
of unoccupied
sites.
and further
attachment
hindered.On
initially,
but
surfaces
of both
saturated
towards
rates
is
move easily over
and
the
figure
5 depict
significant
lattice and a percolation
of reaction
phenomenon
cluster
by
rate
and
cluster.
observed
at
In the
of a percolation
contributes
to the inhibition
inhibition,
different
are defined as regions
concentration
both
in t h e
substrate
cluster,
nature
in the percolation
case
to
get
in m o v e m e n t
The
as shown
cluster,
the same
lattice. The reasons
at high
substrate
in
of a square
concentrations.
arising from the fractal
This
for this
nature
of the
concentration,
and hence the movement of substrate
without it getting
to the double-humped
tend
The
of this excess of substrate.
substrate
the 'dead zones' get completely saturated
to the active site is facilitated
bind
attachment.
lattice
In the case of a percolation
four
substrates
further
square
easily
5.
versus
to the complex kinetics
case
lattice less
the
the binding
substrate
is n o t m a d e v a c a n t
facilitating
is not noticed in the case of a square
can be attributed
surface.
is
square
can be attributed
site caused
of reaction
perimeter
the
coincide as seen in zone 3 of figure
The profiles
rate
the
the surface
percolation
the fall in rate
the active
Thus
stuck
of the curve
cluster
there. This also probably
in f i g u r e
5. ' D e a d z o n e s '
which are connected
to other
regions b y constricted sites a n d do not have a direct access to the active site.
Dead zones are clearly depicted in the figure 2. These are also responsible for
the product formation being less than one in case of reaction on a percolation
cluster.
T h u s the above analysis indicates that substrate inhibition in ce~a2n cases
can be attributed to slow surface diffusion of substrate particles towards the
active site. Rate multiplicity attributed to date to complex reaction kinetics on a
h o m o g e n e o u s surface m a y be attributed to the heterogeneous (fractal) nature of
the surface, as seen here in case of an elementary reaction.
R E F E R E N C E S
A r g y r a k i s , P. a n d K o p e l m a n , P~ (1990) P h y s . Rev. A 41, 2114-2120.
K h a r e , S.K. a n d G u p t a , M.N. (1990) Biotech. B i oengg. 35, 94-98.
Lewis, M. and Rees, D.C. (1985) Science 230, 1163-i165.
Mandelbrot, B.B. (1983) in The Fractal Geometry of Nature W H Freeman and Co.
Pfeifer, P.,Welz, U. and Wippermann, H. (1985) Chem. Phys. Left. 113, 535-540.
Sadana, A., Kulkarni, B.D. and Ramachandran, P.A. (1983) J. Appl. Chem. Biotech.
31, 546-550.
Stauffer,D. (1985) in Introduction to Percolation Theory Taylor and Francis.
Voss, R.F. (1988) in The Science of Fractal Images Peitgen, H. & Saupe, D. ed~
Springer-Verlag.
Wako, H. (1989) J. Phys. Soc. Jpn. 58, 1926-1929.
Witten, T.A. Jr. and Sander, L.M. (1981) Phys. Rev. LetL 47, 1400-1403.
742