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Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20081
Modeling of Biochemical Reactions
Dr. Carlo CosentinoSchool of Computer and Biomedical Engineering
Department of Experimental and Clinical MedicineUniversità degli Studi Magna Graecia
Catanzaro, [email protected]
http://bioingegneria.unicz.it/~cosentino
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20082
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20083
Biochemical Energy
The equilibrium of a reaction is linked to the variation of biochemical standard free energy, ∆G’0
The velocity, instead, depends on the activation energy, ‡PSG →∆
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20084
Reaction Rate
The reaction rate is determined by
The concentration of the reactants
The kinetic constant, usually denoted by k
The reaction rate for is
From transition-state theory it is possible to derive the relation
PS k⎯→⎯
[ ]SkV =
RTGehTk
‡∆−=k k: Boltzmann constant
h: Planck constant
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20085
Reaction Equilibrium
Let us consider the following basic reversible reaction
It can be described by the system
The equilibrium constant is a function of ∆G’0
PSPS
k
k
⎯⎯←⎯→⎯
−1
1
[ ] [ ] [ ][ ] [ ] [ ]PkSkdtPd
PkSkdtSd
11
11
−
−
−=
+−= [ ][ ] 1
1
−
==′kk
SPKeq
eqKRTG ′−=°′∆ ln R=8.315 J/mol·K (gas constant)
T=temperature [K]
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20086
Enzymatic Reaction Kinetics
Enzymes are a family of proteins specialized in the catalysis of reactions
Catalysts do not react and do not affect the reaction equilibrium, however they increase the reaction rate by decreasing the activation energy
The basic mechanism is the formation of an enzyme-substrate complex, which creates a more favorable condition for the formation of the product
After the substrate has been transformed into product, the complex dissociates and the enzyme is free to catalyze the next reaction
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The Role of Enzymes
Enzymes play a central role in all the biological processes (metabolism, regulation, signaling)
Many diseases are caused by deficiency of some enzyme
Many drugs act by interacting with enzymes
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20088
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20089
Michaelis-Menten Model
The basic model of enzymatic reaction was proposed by Michaelis and Menten in 1913
The law of mass action (LMA) states that the reaction rate is proportional to the product of the concentrations of the reactants
L. Michaelis (1875-1949)
M. Menten (1879-1960)
CBA k⎯→⎯+[ ] [ ][ ]BAkdtCd
=
mA+ nBk−→ C
d[C]
dt= k[A]m[B]n
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200810
Michaelis-Menten Model (cont’d)
Applying the LMA to the model reaction scheme, we obtain
with initial conditions s(0)=s0, e(0)=e0, c(0)=0, p(0)=0
Note that the equation of p(t) is decoupled and yields
( )( ) ckpckkeskc
ckkeskeckesks
2211
21111
,,
=+−=++−=+−=
−
−−
&&
&&
[ ] [ ] [ ]PpEScEeSs ==== :,:],[:,:
( ) ( )∫=t
dcktp0
2 ττ
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200811
Simplified M-M Model
The total amount of enzyme (free+bound) remains constant over time, indeed, summing the equations of e(t) and c(t)
from which we can derive e(t) and substitute into the other eqs, obtaining
( ) ( ) 00 etctece =+⇒=+ &&
( ) ( )( ) ( ) 00,
0,
21101
01101
=++−==++−=
−
−
cckksksekcssckskseks
&
&
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200812
Simplified M–M Model (cont’d)
Typically, the initial formation of the complex ES is much faster than the product formation, hence it can be assumed to be instantaneous
This assumption lets us pose dc/dt ≅ 0, that is
where the positive constant
is the Michaelis–Menten constant
( ) ( )( ) mm Ks
seks
Ktstse
tc+
−=⇒+
= 020 &
1
21
kkkKm
+= −
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Simplified M–M Model (cont’d)
Furthermore, the amount of enzyme is typically much less than that of substrate (at least in metabolic reactions), so one can assume that s(0)=s0also in the simplified model
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Adimensionalization
A change of variables is often used in biochemical models, in order to obtain adimensional normalized parameters
This helps in analyzing the behavior of the system for different values of the parameters
Furthermore, the adimensionalization typically reduces the number of parameters
For the M–M reaction we can use the following change of variables
( ) ( ) ( ) ( )
0
0
001
21
01
2
0001
,,
,,,
se
sK
skkkK
skk
etcv
stsutek
m ==+
==
===
− ελ
τττ
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200815
Adimensional M–M model
By applying the change of variable above, the simplified model reduces to
If the amount of enzyme is much less than that of substrate, then ε¿ 1
( ) ( )( ) ( ) 00,
10,=+−==−++−=
vvKuuvuvKuuu
&
&
ελ
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200816
Analysis of the Time Constants
Assuming that s0 does not undergo a significant variation during the initial transient, it is possible to evaluate the length of such interval by the eq.
The time constant of this first-order system is
An estimate for the substrate transformation time constant can be derived by using the maximum derivative
( )cKsksekc m+−= 01001&
( )mc Ksk
t+
=01
1
02
0
max
0
ekKs
dtdss
t ms
+≈≈
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200817
Analysis of the Time Constants
Finally, we can give analytical conditions for the validity of the simplified model in terms of tc and ts
The amount of s consumed during the initial transient has to be negligible, that means |∆ s/s0|¿ 1
Actually, even in the case when e0/s0=O(1), the latter assumption is satisfied for large values of Km (that is when the reaction is slow)
( )12
01
02 <<+ mKskek
10
0 <<+
=mKs
eε
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200818
Experimental Parameters
In the experimental practice, not all the kinetic parameters of the reaction are measured, but rather
the M–M constant, Km
the maximum reaction rate,
[ ] 02max0 ekRQ ==
mm KsQs
Kssek
R+
=+
=0
0
0
0020
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200819
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200820
Quasi–Steady–State Approximation
The assumptions exploited in the derivation of the simplified M–M model take the name of quasi steady-state approximation (QSSA)
Large differences in the time-scales of the reactions may create huge difficulties both in terms of simulation and of understanding the basic principles of operation
To overcome this limitations, theoreticians (especially in the biochemistry community) often use QSSA to eliminate the fastest (and the slowest) equations in the system of ODE
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Validity of the QSSA
The validity of the QSSA depends on
How large is the difference in the time–scales of the reactions
How large is the difference between the amount of enzyme and that of substrate
In the case of protein interaction networks (PINs) the QSSA assumptions are not satisfied, indeed
Enzymes have multiple substrates
Substrates are acted upon by multiple enzymes
Enzymes and substrates often swap roles (e.g. two kinases can phosphorylate each other)
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Use and Abuse of the QSSA
An enlightening work concerning the validity of the QSSA
The authors performed a compared analysis of the Van Slyke–Cullen mechanism, a special case of the M–M reaction, with and without applying the QSSA
E.H. Flach, S. Schnell, Use and abuse of the quasi-steady-state approximation, IEE Proc.–Syst. Biol. 153(4), 187–191, 2006
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200823
Van Slyke–Cullen Model
By exploiting the conservation of the total amount of enzyme, we obtain
Which can be rescaled, in order to get rid of the parameters k1 and k2, by applying the change of variable
In the next slides we will not make use of the bars to refer to the new variables
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200824
Consideration on the Fluxes
The mass is conserved within the system over time, therefore
If v is constant the system can be reduced to a second-order system
When vi=0, the system is said to be in closed form, since there is neither input nor output
If v1(p) is constant, the equation of p(t) is decoupled (as in the M–M model)
s+ c+ p = v = v1 − v2
s = v1(p)− k1s(e0 − c)c = k1s(e0 − c)− k2c
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200825
Stability Analysis of the Closed System
Having a second-order system, it is possible to visualize the trajectories on a phase-plane
The first step consists of finding the null surfaces, by setting to zero the derivatives, which yields
The intersection of the null surfaces gives the steady state
which depends on k1 and k2, as can be seen by substituting for the original state variables
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200826
Stability Analysis of the Simplified Model
Once the equilibrium point has been computed, analysis of the linearized system provides information about the local stability in the neighborhood of that point
In the case of constant v the eigenvalues of the linearized system are real and negative; this corresponds to a so–called stable–node in the phase plane
c = 0 surface
The trajectories are attracted by a slow invariant manifold, which is confined to the region bounded between the null surfaces (the null surface of s is not shown)
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200827
Stability Analysis of the Full Model
We can repeat the stability analysis for the full third–order model, in order to check if the two models have always the same behavior
In this case, linearization around the equilibrium point and computation of the eigenvalues can lead to different cases:
The system is still locally asymptotically stable
For certain values of v1 and v2, complex eigenvalues arise, leading to a so–called stable focus in the phase plane
c = 0 surface
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200828
QSSA Model of the Open System
We will now assume v1(p) ≠ v2(p), that is the open system
Assume that, after the initial transient, the amount of complex changes very slowly, such that dc/dt≈ 0
It is therefore possible to express c as a function of s
Then, substituting in the other equations, we get
s ≈ v1(p)−e0s
1+ s
p ≈ e0s
1+ s− v2(p)
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200829
Comparison of QSSA and Full Model
Also in this case the systems behavior can be very different
The two phase plane below have been obtained using the same parameters and fluxes, with the full model (a) and QSSA model (b)
The full model exhibit a limit cycle, whereas the trajectories of the reduced one follow a spiral stable mode
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200830
QSSA is not Always Reliable
The QSSA is probably the most frequently used method for reducing the complexity of biochemical pathways models
Nonetheless it has been shown, also in other works, that it can conceal some aspects of the transient dynamics or even alter the long-term dynamics, and thus the qualitative behavior, of the original system
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200831
total Quasi-Steady-State Approximation
A possible way to overcome the limitations of QSSA in enzyme-catalyzed reactions has been proposed in
They simply proposed that, for conditions when ET and S0 have comparable values, the proper intermediate timescale variable is
This yields
JAM Borghans, RJ De Boer, LA Segel, Extending the Quasi-Steady-State Approximation by Changing Variables, Bull. Math. Biol. 58(1), 43–63, 1996
S(t) = S(t) + C(t)
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200832
Validity of tQSSA
Tzafriri and Edelman (J. Theor. Biol., 2004) derived sufficient conditions for the validity of tQSSA, which can be summarized by
that is, the dissociation rate of the enzyme–substrate complex is much faster than the catalytic conversion of substrate into product
Thus, the tQSSA is likely to be an excellent approximation for any ratio of enzyme to substrate and for any ratio of timescales
An interesting reading for applications of tQSSA to several kinds of PINs is
k−1 À k2
A Ciliberto, F Capuani, JJ Tyson, Modeling Networks of Coupled Enzymatic Reactions Using the total Quasi–Steady State Approximation, PLOS Computational Biology 3(3), 463–472, 2007
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200833
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200834
Cooperative Reactions
In the basic model of enzymatic reaction we have assumed that each molecule of enzyme binds only one substrate molecule
It is quite common to have multiple binding sites, e.g. the hemoglobin has four binding sites for oxygen
An enzymatic reaction is said to be cooperative if the binding of one molecule to one site affects the binding affinity at other sites
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Allosteric Effect
The mechanism causing such phenomenon is named allosteric effect
A substrate can be an activator or inhibitor, depending on whether it increases or decreases the binding affinity at other sites
If the substrate and the modulator are the same species, then the interaction is called homotropic, otherwise heterotropic
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Cooperative Reaction – Example
The simplest cooperative mechanism is shown below
By applying the law of mass action and then the QSSA, analogously to what done for the M–M model it is possible to derive the maximum reaction rate
( ) 200
04200
000 ssKKK
skKkse
dtdssR
mmm
m
t +′+′+′
===
3
34
1
21 ,k
kkK
kkkK mm
−− +=′+
=
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200837
Hill Plot
Plotting the reaction rate as a function of s0 it is possible to observe the difference with respect to standard reactions
For the sake of simplicity, we plot the case when k2=0, which yields
2000 0 sRs ∝⇒→
( ) 0,0
000 >
+= n
KsQs
sRm
n
n
In this case the behavior is usually described by a Hill curve
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Hill Coefficient
The quantity n is termed the Hill coefficient
The same term can be found by considering an ideal reaction, with an enzyme binding n substrate molecules at the same time (complete cooperativity)
In real cases, the value of n has not to be an integer, rather it is a real number, because usually the substrates do not bind contemporaneously to different sites, thus
n>1 positive cooperation
n<1 negative cooperation
n=1 no cooperation
Therefore the Hill coefficient is a measure of the cooperativity of the reaction
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200839
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200840
Lineweaver–Burk Plot
The L–B plot (or double reciprocal is a common tool in biochemistry analysis of enzyme kinetics
It is easily derived by inverting the equation of S in the M–M model
[ ][ ]SV
SKV
m
max0
1 +=
[ ] maxmax0
11VSV
KV
m +=
It is very useful because it enables linear regression of experimental data
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200841
Regulation of Enzymatic Reactions
Enzymes can increase the rate of a reaction by several orders of magnitude, but they can also be used for fine regulation of reaction pathways
The production and degradation are often adapted to the current requirements of the cell
Furthermore, the may be targets of effectors, both inhibitors and activators
The effectors are proteins or other molecules that can modify the enzymatic reaction rate
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200842
Enzyme Inhibition
Different types of inhibition mechanism exist, depending on
the state in which the enzyme can bind the effector
the ability of different complexes to release the product
The general pattern of inhibition is shown in the figure below; the several subcases derive by eliminating some of the reactions
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Competitive Inhibition
The inhibitor competes with the substrate for the binding site (or inhibits substrate binding by binding elsewhere to the enzyme)
[ ] [ ][ ]SKSV
dtPd
m +=α
max
[ ] [ ][ ][ ]EI
IEKKI
II
=+= ,1α
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200844
Competitive Inhibition L–B Plot
The competitive inhibition can be characterized experimentally by means of the L–B plot
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200845
Uncompetitive Inhibition
The inhibitor binds only to the ES complex
This may be due to conformational changes of the enzyme caused by the substrate binding (allosteric effect), which makes a new binding site accessible
[ ] [ ][ ]SK
SVdtPd
m α′+= max
[ ] [ ][ ][ ]ESI
IESKKI
II
=′′
+=′ ,1α
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200846
Uncompetitive Inhibition L–B Plot
The L–B significantly differs from the previous case
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200847
Mixed Inhibition
The inhibitor can bind both to the enzyme and to the ES complex
Therefore it binds to a different site than that used by the substrate
[ ] [ ][ ]SK
SVdtPd
m αα ′+= max
[ ] [ ][ ][ ]ESI
IESKKI
II
=′′
+=′ ,1α
[ ] [ ][ ][ ]EI
IEKKI
II
=+= ,1α
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200848
Mixed Inhibition L–B Plot
Also in this case the L – B plot can be used to distinguish from the other types of inhibition mechanism
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Noncompetitive and Partial Inhibition
Noncompetitive inhibition can be viewed as a special case of mixed inhibition, when the parameters α and α’ are equal
In this case the inhibitor has the same affinity to the enzyme with or without bound substrate
It is rarely encountered in experimental practice
If the product can also be formed from the enzyme–substrate–inhibitor complex, the inhibition is said to be partial
If the production rate from ESI complex is high, the effect of this latter mechanism can even yield an activating instead of an inhibiting effect
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200850
Dependence on Environmental Factors
It is worth pointing out that enzymatic reactions are strongly dependent on factors like pH and temperature, as shown in the two examples below
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200851
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200852
Stochastic Models
The models considered so far are deterministic, i.e. given the initial conditions (the state at t0) and the exogenous signals perturbing the model, the response is univocally determined
A stochastic model, on the contrary, involves random variables, therefore its behavior cannot be predicted a priori, although it can be statistically characterized
The figure shows two realization of the same stochastic process, starting from the same initial condition
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200853
Biological Systems are Stochastic
Certainly biological ones fall in the category of stochastic systems, indeed the very basic steps of every molecular reaction can be described only in terms of its probability of occurrence
Moreover, the diffusion of molecules is a realization of a random walk process (Brownian motion)
So, why deterministic model are so widespread?
Usually the phenomena under consideration involve a large number of molecules, therefore the average effect is well described through deterministic equations
When is it necessary to use stochastic models/simulations?
When the mechanism to be described is based on the interaction of few molecules, or we want to simulate the functioning of a little pool of cells
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200854
Gillespie Algorithm
The algorithm has been first presented in
The purpose is to simulate chemical reactions with limited computational resources
The outcome is a realization of the underlying stochastic process
DT Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. of Computational Physics 22, 403–434, 1976
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Basic Principles
The basic working principles of the Gillespie algorithm derive from the description of the collision of particles in a vessel
When two molecules collide, the reaction happens only if they have proper orientation and kinetic energy
A first hypothesis is that the frequency of non–reacting collisions is much larger than that of reacting ones
Secondly, the algorithm assumes that each reaction involves no more than two molecules
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200856
Sketch of the Derivation
The method starts from the observation of the volume occupied by a molecule A that is in relative motion with respect to another molecule B
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Sketch of the Derivation
Assuming a random uniform distribution of the molecules in the volume V, the probability of collision can be computed as
If in such volume we have X1 molecules of the species S1 and X2 of the species S2, the probability of collision is
tvrVVV δπδ 122
121
coll−= Probability of collision of two
molecules in the interval �t
tvrVXX δπ 122
121
21−
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Stochastic Reaction Constant
In this framework, the reactions can be characterized by means of a probability of reaction per unit time, instead of a kinetic rate as in ODE models
For instance, given the reaction
we can define a constant c1, dependent on the chemo–physical properties of the molecules and on the temperature, such that
X1 X2 c1dt = Probability that the reaction happens in the volume Vwithin the interval dt
In general, given a system of N molecules and M reactions, each reaction is characterized by a specific stochastic reaction constant cµ (µ=1,…,M)
1211 2: SSSR →+
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Stochastic and Deterministic Constants
Intuitively, the stochastic reaction constant, cµ, must be linked to the kinetic constant of the corresponding determinist equation, kµ
In the validity ranges of the deterministic models, it is possible to establish the simple relation
The presence of the factor V depends on the fact that deterministic models take as state variables the concentrations, whereas stochastic ones use the number of molecules
11 Vck =
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200860
Stochastic Simulation
There are two main approaches to solve the stochastic system and compute the system evolution
Master Equation Approach
Stochastic Simulation Algorithm (SSA) with Gillespie method
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200861
Master Equation Approach
The key element in this approach is the probability function
The master equation describes the evolution of the function P
The probability P(X1,…,XN;t+dt) can be derived as a combination of the probability of all the possible reactions that can happen within dt
The differential equation that is derived by this argument does not usually have analytical solution, nor it admits a computationally efficient solution
( )tXXXP N ;,,, 21 KProbability that at time t there are X1molecules of species S1, …, XN of species SN
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200862
Outline
Modeling of biochemical reactions
Deterministic models
Michaelis–Menten model
The Quasi–Steady–State Approximation
Allosteric reaction
Regulation of enzymatic reactions
Stochastic models
Stochastic derivation
The Gillespie algorithm
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200863
Stochastic Simulation Algorithm
Disregarding the formalism of the master equation, looking for a more practical approach, we need to answer two questions
Starting at time t, which is the next occurring reaction
When this reaction will occur
Clearly, the answer can be given only in terms of probability, thus let us define the pdf of the reaction, P(τ,µ), such that
( ) τµτ dP , Probability that, given X1,…, XN at time t, the next reaction in V will be Rµ and that it will occur within (t+ τ,t+ τ +dτ)
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200864
Stochastic Simulation Algorithm
In order to find an analytical expression for P(τ,µ) let define
hµ → number of possible distinct combinations of reactants in Rµ in the state (X1,…,XN), µ=1,…,M
If Rµ has two–reactants (of the type S1+S2 → …), then hµ =X1X2
If Rµ has one reactant (of the type 2S1 → …), then hµ =1/2 X1(X1-1)
aµdt = hµcµdt → probability that a reaction Rµ occur in the interval (t,t+dt) starting from the state (X1,…,XN), µ=1,…,M at time t
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200865
Stochastic Simulation Algorithm
From this quantities it is possible to derive the following expression
where
Having a random number generator with uniform distribution, it is possible to generate the exponential distribution above
( ) ( )⎩⎨⎧ =∞≤≤−
=altrimenti
,,100
exp, 0 Mandaa
PKµττ
µτ µ
( )
∑∑==
==
==MM
chaa
Mcha
110
,,1
ννν
νν
µµµ µ K
Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 200866
Steps of the Gillespie Algorithm
Step 0 (Initialization) – Define the number of molecules of each species, the kinetic constants and the random number generator
Step 1 (Monte Carlo) – Generate random numbers, to determine which is the next reaction and the length of the interval dt
Step 2 (Update) – Increase time by dt and update the number of molecules of each species on the basis of the occurred reaction
Step 3 (Iterate) – If the number of reactants is greater than zero and the simulation stop time has not yet been reached, iterate from Step 1
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Drawbacks of the Stochastic
High computational load
Not possible to derive reduced order model (like M–M or Hill terms)
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References
DL Nelson, MM Cox, Lehninger Principles of Biochemistry, WH Freeman, 2004
JD Murray, Mathematical Biology, Springer, 2007
E.H. Flach, S. Schnell, Use and abuse of the quasi-steady-state approximation, IEE Proc.–Syst. Biol. 153(4), 187–191, 2006
A Ciliberto, F Capuani, JJ Tyson, Modeling Networks of Coupled Enzymatic Reactions Using the totalQuasi–Steady State Approximation, PLOS Computational Biology 3(3), 463–472, 2007
DT Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. of Computational Physics 22, 403–434, 1976