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Model reduction for chemical reaction networks
• Formulating Markov models
• Reaction networks
• Scaling limit
• Multiscale models
• General approaches to averaging
• Michaelis-Menten equation
• Forcing onto a lower dimensional manifold
• Appendix
• References
• Abstract
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Intensities for continuous-time Markov chains
Assume X is a continuous time Markov chain in E ⊂ Zd. The Q-matrix, Q = {qkl},for the chain gives
P{X(t+ ∆t) = l|X(t) = k} ≈ qkl∆t, k 6= l ∈ E,
and hence
E[f(X(t+ ∆t))− f(X(t))|FXt ] ≈
∑l
qX(t),l(f(l)− f(X(t))∆t ≡ Af(X(t))∆t
Alternative notation: Define βl(k) = qk,k+l. Then
Af(k) =∑
l
βl(k)(f(k + l)− f(k))
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Martingale problems
≈ is made precise by the requirement that
f(X(t))− f(X(0))−∫ t
0
Af(X(s))ds
be a {FXt }-martingale for f in an appropriate domain D(A).
X is called a solution of the martingale problem for A.
Note that a change of time scale corresponds to multiplying the generator by theappropriate constant.
f(X(ρt))− f(X(0))−∫ ρt
0
Af(X(s))ds
= f(X(ρt))− f(X(0))−∫ t
0
ρAf(X(ρs))ds
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Time change equation
X(t) = X(0) +∑
l
lNl(t)
where Nl(t) is the number of jumps of l at or before time t. Nl is a counting processwith intensity (propensity in the chemical literature) βl(X(t)), that is,
Nl(t)−∫ t
0
βl(X(s))ds
is a martingale. Consequently, we can write
Nl(t) = Yl(
∫ t
0
βl(X(s))ds),
where the Yl are independent, unit Poisson processes, and
X(t) = X(0) +∑
l
lYl(
∫ t
0
βl(X(s))ds).
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Reaction networks
Standard notation for chemical reactions
A+B ⇀ C
is interpreted as “a molecule of A combines with a molecule of B to give a moleculeof C.”
A+B C
means that the reaction can go in either direction, that is, a molecule of C candissociate into a molecule of A and a molecule of B
We consider a network of reactions involving m chemical species, A1, . . . , Am.
m∑i=1
νikAi ⇀m∑
i=1
ν ′ikAi
where the νik and ν ′ik are nonnegative integers
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Markov chain models
X(t) number of molecules of each species in the system at time t.
νk number of molecules of each chemical species consumed in the kth reaction.
ν ′k number of molecules of each species created by the kth reaction.
λk(x) rate at which the kth reaction occurs. (The propensity/intensity.)
If the kth reaction occurs at time t, the new state becomes
X(t) = X(t−) + ν ′k − νk.
The number of times that the kth reaction occurs by time t is given by the countingprocess satisfying
Rk(t) = Yk(
∫ t
0
λk(X(s))ds),
where the Yk are independent unit Poisson processes.
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Equations for the system state
The state of the system satisfies
X(t) = X(0) +∑
k
Rk(t)(ν′k − νk)
= X(0) +∑
k
Yk(
∫ t
0
λk(X(s))ds)(ν ′k − νk) = (ν ′ − ν)R(t)
ν ′ is the matrix with columns given by the ν ′k.
ν is the matrix with columns given by the νk.
R(t) is the vector with components Rk(t).
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Rates for the law of mass action
For a binary reaction A1 + A2 ⇀ A3 or A1 + A2 ⇀ A3 + A4
λk(x) = κkx1x2
For A1 ⇀ A2 A1 ⇀ A2 + A3,λk(x) = κkx1
For 2A1 ⇀ A2,λk(x) = κkx1(x1 − 1)
For a binary reaction A1 + A2 ⇀ A3, the rate should vary inversely with volume, soit would be better to write
λNk (x) = κkN
−1x1x2 = Nκkz1z2,
where classically, N is a scaling parameter taken to be the volume of the system timesAvogadro’s number and zi = N−1xi is the concentration in moles per unit volume.Note that unary reaction rates also satisfy
λk(x) = κkxi = Nκkzi.
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Classical scaling limit
Setting CN(t) = N−1X(t)
CN(t) = CN(0) +∑
k
N−1Yk(
∫ t
0
λNk (X(s))ds)(ν ′k − νk)
≈ CN(0) +∑
k
N−1Yk(N
∫ t
0
λk(CN(s))ds)(ν ′k − νk)
The law of large numbers for the Poisson process implies N−1Y (Nu) ≈ u,
CN(t) ≈ CN(0) +∑
k
∫ t
0
κk
∏i
CNi (s)νik(ν ′k − νk)ds,
which in the large volume limit gives the classical deterministic law of mass action
C(t) =∑
k
κk
∏i
Ci(t)νik(ν ′k − νk) ≡ F (C(t)).
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A multiscale model [1]
Take N to be of the order of magnitude of the abundance of the most abundantspecies in the system.
For each species i, 0 ≤ αi ≤ 1 and
Zi(t) = N−αiXi(t).
αi should be selected so that Zi = O(1).
The rate constants may also scale κ′k = κkNγk , so for a binary reaction
κ′kxixj = Nγk+αi+αjκkzizj
Select βk so that λ′k(x) = Nβkλk(z), where λk(z) = O(1) for all (most?) relevantvalues of z.
The model becomes
Zi(t) = Zi(0) +∑
k
N−αiYk(
∫ t
0
Nβkλk(Z(s))ds)(ν ′ik − νik).
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Identifying the “fast” process
Let ΛN = diag(N−α1 , . . . , N−αm) and ζk = ν ′k − νk. The generator for Z is
BNf(z) =∑
k
Nβkλk(z)(f(z + ΛNζk)− f(z)).
Select the smallest r1 (possibly negative) such that
C0f(z) = limN→∞
N−r1BNf(z)
exists for each f ∈ C2c (Rm), z ∈ Rm, and let D1 be the collection of f ∈ C2(Rm) such
that C0f(z) = 0.
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Identifying r1
Critical exponents: Let
r10 = max{βk : ∃i, αi = 0, ζik 6= 0}r11 = max{βk − αi :
∑βl=βk
λl(z)ζil 6= 0}
r12 = max{βk − 2αi :∑
βk=βi
λk(z)ζ2ik 6= 0}.
Then r1 = max{r10, r11, r12}.
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Second time scale
Select the smallest r2 such that
C1f(z) = limN→∞
N−r2BNf(z)
exists for all f ∈ D1, so in some sense, for general f
N−r2BNf(z) ≈ C1f(z) +N r1−r2C0f(z)
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General approaches to averaging [7]
Models with two time scales: (X, Y ), Y is “fast”
Occupation measure: ΓY (C × [0, t]) =∫ t
01C(Y (s))ds
Replace integrals involving Y by integrals against ΓY∫ t
0
f(X(s), Y (s))ds =
∫EY ×[0,t]
f(X(s), y)ΓY (dy × ds)
≈∫ t
0
∫EY
f(X(s), y)ηs(dy)ds
How do we identify ηs?
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Generator approach
Suppose Brf(x, y) = rC0f(x, y) + C1f(x, y) where C operates on f as a function ofy alone.
f(Xr(t), Yr(t))− r
∫EY ×[0,t]
C0f(Xr(s), y)ΓYr (dy × ds)
−∫
EY ×[0,t]
C1f(Xr(s), y)ΓYr (dy × ds)
Assuming (Xr,ΓYr ) ⇒ (X,ΓY ), dividing by r, we should∫
EY ×[0,t]
C0f(X(s), y)ΓY (dy × ds) =
∫EY ×[0,t]
C0f(X(s), y)ηs(dy)ds = 0
Suppose that for each x, the solution of∫
EY C0f(x, y)µx(dy) = 0, f ∈ D. Thenηs(dy) = µX(s)(dy)
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Michaelis-Menten kinetics
Consider the reaction system A+ E AE ⇀ B + E
modeled as a continuous time Markov chain satisfying
XA(t) = XA(0)− Y1(
∫ t
0
κ1XA(s)XE(s)ds) + Y2(
∫ t
0
κ2XAE(s)ds)
XE(t) = XE(0)− Y1(
∫ t
0
κ1XA(s)XE(s)ds) + Y2(
∫ t
0
κ2XAE(s)ds)
+Y3(
∫ t
0
κ3XAE(s)ds)
XB(t) = Y3(
∫ t
0
κ3XAE(s)ds)
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Scaling
Note that M = XAE(t) +XE(t) is constant. Let N = O(XA) define
VE(t) =
∫ t
0
M−1XE(s)ds, ZA(t) = N−1XA(t)
κ1 = M−1γ1, κ2 = NM−1γ2, κ3 = NM−1γ3
ZA(t) = ZA(0)−N−1Y1(N
∫ t
0
γ1ZA(s)dVE(s)) +N−1Y2(Nγ2(t− VE(t)))
XE(t) = XE(0)− Y1(N
∫ t
0
γ1ZA(s)dVE(s)) + Y2(Nγ2(t− VE(t)))
+Y3(Nγ3(t− VE(t)))
ZB(t) = N−1Y3(Nγ3(t− VE(t)))
Along a subsequence (ZA, ZB, VE) ⇒ (xA, xB, vE) and∫ t
0
ZA(s)dVE(s) ⇒∫ t
0
xA(s)dvE(s) =
∫ t
0
xA(s)vE(s)ds
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Theorem 1 (Darden [3, 4]) Assume that N →∞, M/N → 0, Mκ1 → γ1, Mκ2/N →γ2, Mκ3/N → γ3, and XA(0)/N → xA(0), and
Then (N−1XA, VE) converges to (xA(t), vE(t)) satisfying
xA(t) = xA(0)−∫ t
0
γ1xA(s)vE(s)ds+
∫ t
0
γ2(1− vE(s))ds (1)
0 = −∫ t
0
γ1xA(s)vE(s)ds+
∫ t
0
(γ2 + γ3)(1− vE(s))ds,
and hence vE(s) = γ2+γ3
γ2+γ3+γ1xA(s)and
xA(t) = − γ1γ3xA(t)
γ2 + γ3 + γ1xA(s).
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Quasi-steady state
Assume M is constant κ2 = γ2N/M , κ3 = γ3N/M . Then
f(XE(t))− f(XE(0))−∫ t
0
Nγ1ZA(s)M−1XE(s)(f(XE(s)− 1)− f(XE(s)))ds
−∫ t
0
N(γ2 + γ3)(1−M−1XE(s))(f(XE(s) + 1)− f(XE(s)))ds
Since ZA(s) → xA(s),∫
Cf(xA(s), k)ηs(dk) = 0 becomes
M∑k=0
ηs(k)[(γ1xA(s)M−1k(f(k − 1)− f(k)
+(γ2 + γ3)(1−M−1k)(f(k + 1)− f(k))]
= 0
so ηs is binomial(M, ps), where ps = γ2+γ3
γ2+γ3+γ1xA(s).
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Forcing onto a lower dimensional manifold
Assume
F : Rd → Rd
Ξ ⊂ {x ∈ Rd : F (x) = ∞} a submanifold of dimension m < d
∂F (x), x ∈ Ξ, has rank d−m and the nonzero eigenvalues have negative real parts
ψ(t, x) = x+
∫ t
0
F (ψ(s, x))ds,
Υ = {x : Φ(x) ≡ limt→∞ ψ(t, x) exists}
Then, under additional regularity conditions, Φ is C2 and ∂Φ(x)F (x) = 0 in a neigh-borhood of Ξ (Falconer (1983) [5]).
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Convergence theorem
Katzenberger (1991) [6]
Theorem 2 Let {VN} be a sequence of semimartingales satisfying a uniformity con-dition and VN ⇒ V , {AN} continuous nondecreasing processes such that
inft≥0
AN(t+ ε)− AN(t) →∞
for each ε > 0. Suppose
ZN(t) = VN(t) +
∫ t
0
F (ZN(s))dAN(s)
and VN(0) ⇒ Z(0) ∈ Ξ. Then ZN ⇒ Z satisfying
Z(t) = Z(0) +
∫ t
0
∂Φ(Z(s))dV (s) +1
2
∑k,l
∫ t
0
∂2k,lΦ(Z(s))d[Vk, Vl]s
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Slow time scale limit
Z(t) = Z(0) +∑
k
Yk(
∫ t
0
Nβkλk(Z(s))ds)ΛNζk
= Z(0) +∑
k
Yk(
∫ t
0
Nβkλk(Z(s))ds)(ΛN − Λ0N)ζk
+∑
k
Yk(
∫ t
0
Nβkλk(Z(s))ds)Λ0Nζk
+
∫ t
0
∑k
Nβkλk(Z(s))Λ0Nζkds
Define UN(t) = Z(N−r2t) and assume ζik 6= 0 implies βk − r2 ≤ 2αi and
UN(t) = UN(0) +∑
k
Yk(
∫ t
0
Nβk−r2λk(UN(s))ds)(ΛN − Λ0
N)ζk
+∑
k
Yk(
∫ t
0
Nβk−r2λk(UN(s))ds)Λ0
Nζk
+
∫ t
0
G(UN(s))ds+N r1−r2
∫ t
0
F (UN(s))ds
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Negative feedback with dimerization Bratsun, Volfson, Tsimring, and Hasty [2]
A+ A ⇀ A2 A2 ⇀ A+ A
D0 + A2 ⇀ D1 D1 ⇀ D0 + A2
D0 ⇒ D0 + A A ⇀ ∅
The model becomes
θ(t) = θ(0) + Y1(
∫ t
0
κ−1(1− θ(s))ds)− Y2(
∫ t
0
κ1θ(s)X2(s)ds)
X1(t) = X1(0) + Y3(
∫ t−τ
−τ
κ3θ(s)ds)− Y4(
∫ t
0
κ4X1(s)ds)
−2Y5(
∫ t
0
κ2X1(s)(X1(s)− 1)ds) + 2Y6(
∫ t
0
κ−2X2(s)ds)
X2(t) = X2(0) + Y5(
∫ t
0
κ2X1(s)(X1(s)− 1)ds)− Y6(
∫ t
0
κ−2X2(s)ds)
+Y1(
∫ t
0
κ−1(1− θ(s))ds)− Y2(
∫ t
0
κ1θ(s)X2(s)ds).
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Scaling
Replace κ−1 byNκ−1, κ−2 byNκ−2, κ3 byNκ3, and (X1(0), X2(0)) by (NZ1(0), NZ2(0)),and divide the three equations by N . Define ZN
1 = N−1XN1 , ZN
2 = N−1XN2
θ(t) = θ(0) + Y1(N
∫ t
0
κ−1(1− θ(s))ds− Y2(
∫ t
0
κ1θ(s)X2(s)ds)
X1(t) = X1(0) + Y3(N
∫ t−τ
−τ
κ3θ(s)ds)− Y4(
∫ t
0
κ4X1(s)ds)
−2Y5(
∫ t
0
κ2X1(s)(X1(s)− 1)ds) + 2Y6(N
∫ t
0
κ−2X2(s)ds)
X2(t) = X2(0) + Y5(
∫ t
0
κ2X1(s)(X1(s)− 1)ds)− Y6(N
∫ t
0
κ−2X2(s)ds)
+Y1(N
∫ t
0
κ−1(1− θ(s))ds)− Y2(
∫ t
0
κ1θ(s)X2(s)ds).
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0 ≈∫ t
0
κ−1(1− θN(s))ds)−∫ t
0
κ1θN(s)ZN
2 (s)ds
ZN1 (t) = ZN
1 (0) +N−1Y3(N
∫ t−τ
−τ
κ3θN(s)ds)−N−1Y4(N
∫ t
0
κ4ZN1 (s)ds)
−2N−1Y5(N2
∫ t
0
κ2ZN1 (s)(ZN
1 (s)−N−1)ds) + 2N−1Y6(N2
∫ t
0
κ−2ZN2 (s)ds)
+
∫ t−τ
−τ
κ3θN(s)ds+
∫ t
0
(2κ2 − κ4)ZN1 (s)ds
+N
∫ t
0
2(κ−2ZN2 (s)− κ2Z
N1 (s)2)ds
ZN2 (t) = ZN
2 (0) +N−1Y5(N2
∫ t
0
κ2ZN1 (s)(ZN
1 (s)−N−1)ds)−N−1Y6(N2
∫ t
0
κ−2ZN2 (s)ds)
+N−1Y1(N
∫ t
0
κ−1(1− θN(s))ds)−N−1Y2(N
∫ t
0
κ1θN(s)ZN
2 (s)ds)
−∫ t
0
κ2ZN1 (s)ds+N
∫ t
0
(κ2ZN1 (s)2 − κ−2Z
N2 (s))ds,
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Strong drift
F (z1, z2) = (κ−2z2 − κ2z21)
(2−1
)Ξ = {(z1, z2) : z2 =
κ2
κ−2
z21}
Noting that
ZN1 (t) + 2ZN
2 (t) = ZN1 (0) + 2ZN
2 (0)
+N−1Y3(N
∫ t−τ
−τ
κ3θN(s)ds)−N−1Y4(N
∫ t
0
κ4ZN1 (s)ds)
+
∫ t−τ
−τ
κ3θN(s)ds−
∫ t
0
κ4ZN1 (s)ds
+N−1(θ(t)− θ(0))
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Limiting equation
Let ε = κ1
κ−1and δ = κ2
κ−2. Then
θ(t) =1
1 + εZ2(t)=
1
1 + εδZ1(t)2
and
Z1(t) + 2δZ1(t)2 = Z1(0) + 2δZ1(0)2 +
∫ t−τ
−τ
κ3
1 + εδZ1(s)2ds−
∫ t
0
κ4Z1(s)ds.
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Uniformity condition
VN = MN +RN , a semimartingale adapted to {FNt }
Tt(RN), the total variation of RN on [0, t]
[MN ]t, the quadratic variation of MN on [0, t]
Condition 3 a)
{Tt(RN), N = 1, 2, . . .}
is stochastically bounded.
b) There exist stopping times {τ cN} such that
limc→∞
supNP{τ c
N ≤ c} = 0
andsupNE[[MN ]t∧τc
N] <∞
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References
[1] Karen Ball, Thomas G. Kurtz, Lea Popovic, and Grzegorz A. Rempala. As-ymptotic analysis of multiscale approximations to reaction networks. Ann. Appl.Probab., 2006. to appear.
[2] Dmitri Bratsun, Dmitri Volfson, Lev S. Tsimring, and Jeff Hasty. Delay-inducedstochastic oscillations in gene regulation. PNAS, 102:14593 – 14598, 2005.
[3] Thomas Darden. A pseudo-steady state approximation for stochastic chemicalkinetics. Rocky Mountain J. Math., 9(1):51–71, 1979. Conference on DeterministicDifferential Equations and Stochastic Processes Models for Biological Systems(San Cristobal, N.M., 1977).
[4] Thomas A. Darden. Enzyme kinetics: stochastic vs. deterministic models. In In-stabilities, bifurcations, and fluctuations in chemical systems (Austin, Tex., 1980),pages 248–272. Univ. Texas Press, Austin, TX, 1982.
[5] K. J. Falconer. Differentiation of the limit mapping in a dynamical system. J.London Math. Soc. (2), 27(2):356–372, 1983.
[6] G. S. Katzenberger. Solutions of a stochastic differential equation forced onto amanifold by a large drift. Ann. Probab., 19(4):1587–1628, 1991.
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[7] Thomas G. Kurtz. Averaging for martingale problems and stochastic approxima-tion. In Applied stochastic analysis (New Brunswick, NJ, 1991), volume 177 ofLecture Notes in Control and Inform. Sci., pages 186–209. Springer, Berlin, 1992.
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Abstract
Model reduction for chemical reaction networks
Stochastic models of cellular chemical reaction networks typically involve chemicalspecies numbers and reaction rates varying over several orders of magnitude. Anumber of researchers have proposed exploiting the multiscale nature of these modelsto reduce the complexity of the model to be analyzed or simulated. Systematicapproaches to model reduction will be discussed.