SCU Colloquium Series: Extinction and Persistence in Discrete - - PowerPoint PPT Presentation
Background Stochastic Extinction Future Work SCU Colloquium Series: Extinction and Persistence in Discrete Chemical Reaction Networks Matthew Douglas Johnston Assistant Professor San Jose State University One Washington Square San Jose, CA
Background Stochastic Extinction Future Work
Matthew Douglas Johnston Assistant Professor San Jose State University One Washington Square San Jose, CA 95192 (Joint work with D. Anderson, G. Craciun, and R. Brijder)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work
1 Background
Overview Deterministic Model Stochastic Model
2 Stochastic Extinction
Background Network Properties Conditions (technical details!)
3 Future Work
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
1 Background
Overview Deterministic Model Stochastic Model
2 Stochastic Extinction
Background Network Properties Conditions (technical details!)
3 Future Work
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Objective: Determine the dynamical behavior of systems of interacting biochemical species.
1 Long-term behavior:
Steady states/stationary modes/stability? Persistence/boundedness?
2 Transient behavior:
Oscillations/limit cycles? Hysteresis/switching behavior?
3 Parameter/Model Choice:
Bifurcations? ODE/PDE/Stochastic/Discrete/Continuous?
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Figure: Picture courtesy of Wikipedia.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Synthetic/systems biology depends heavily on the study of modularized networks. Example networks include:
Protein Activation A + B − → 2B B − → A Enzymatic Futile Cycle S + E ⇄ SE − → P + E P + F ⇄ PF − → S + F Signaling Network XD ⇄ X ⇄ XT − → Xp Xp + Y ⇄ XpY − → X + Yp XT + Yp ⇄ XTYp − → Y
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Two basic modeling frameworks (well-mixed systems): Deterministic model:
Keep track of reactant concentrations, xi ∈ R≥0 Assume reactions continuously and deterministically Modeled as system of differential equations
Stochastic model:
Keep track of reactant numbers, Xi ∈ Z≥0 Assume reactions occur discretely and stochastically Modeled as a Continuous-time Markov Chain
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic model: Rate of reaction is proportional to product of concentrations, e.g. X + Y = ⇒ [rate] ∝ [X][Y ] Stochastic model: Probability a reaction occurs is proportional to combinatorial ways constituent species could meet, e.g. ∅ = ⇒ [Prob] ∝ 1 X = ⇒ [Prob] ∝ X X + Y = ⇒ [Prob] ∝ X · Y 2X = ⇒ [Prob] ∝ X(X − 1)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic model: Rate of reaction is proportional to product of concentrations, e.g. X + Y = ⇒ [rate] ∝ [X][Y ] Stochastic model: Probability a reaction occurs is proportional to combinatorial ways constituent species could meet, e.g. ∅ = ⇒ [Prob] ∝ 1 X = ⇒ [Prob] ∝ X X + Y = ⇒ [Prob] ∝ X · Y 2X = ⇒ [Prob] ∝ X(X − 1)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Protein activation model: (A inactive, B active) A + B
α
− → 2B (activation) B
β
− → A (de-activation) Ordinary differential equation system: ˙ xA = − αxAxB + βxB ˙ xB = αxAxB − βxB.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Protein activation model: (A inactive, B active) A + B
α
− → 2B (activation) B
β
− → A (de-activation) Ordinary differential equation system: ˙ xA = − αxAxB + βxB ˙ xB = αxAxB − βxB.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Protein activation model: (A inactive, B active) A + B
α
− → 2B (activation) B
β
− → A (de-activation) Ordinary differential equation system: ˙ xA = − αxAxB + βxB ˙ xB = αxAxB − βxB.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General mass action system: dx(t) dt =
r
ki (y′
i − yi) m
xyij
j
where, for each reaction, ki > 0 is the rate constant y′
i − yi ∈ Rm is the reaction vector
m
j=1 xyij j
is the interaction term Polynomial differential equations arise frequently in mathematical biology! (e.g. infectious disease, ecosystems)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General mass action system: dx(t) dt =
r
ki (y′
i − yi) m
xyij
j
where, for each reaction, ki > 0 is the rate constant y′
i − yi ∈ Rm is the reaction vector
m
j=1 xyij j
is the interaction term Polynomial differential equations arise frequently in mathematical biology! (e.g. infectious disease, ecosystems)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General mass action system: dx(t) dt =
r
ki (y′
i − yi) m
xyij
j
where, for each reaction, ki > 0 is the rate constant y′
i − yi ∈ Rm is the reaction vector
m
j=1 xyij j
is the interaction term Polynomial differential equations arise frequently in mathematical biology! (e.g. infectious disease, ecosystems)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General mass action system: dx(t) dt =
r
ki (y′
i − yi) m
xyij
j
where, for each reaction, ki > 0 is the rate constant y′
i − yi ∈ Rm is the reaction vector
m
j=1 xyij j
is the interaction term Polynomial differential equations arise frequently in mathematical biology! (e.g. infectious disease, ecosystems)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General properties: ˙ xA = −αxAxB + βxB = 0 ˙ xB = αxAxB − βxB = 0
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General properties: ˙ xA = −αxAxB + βxB = 0 ˙ xB = αxAxB − βxB = 0 Steady states: αxAxB − βxB = 0 = ⇒ xB = 0 or xA = β α
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General properties: ˙ xA = −αxAxB + βxB = 0 ˙ xB = αxAxB − βxB = 0 Steady states: αxAxB − βxB = 0 = ⇒ xB = 0 or xA = β α Invariant Subspace: ˙ xA ˙ xB
−1 1
−1
Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
General properties: ˙ xA = −αxAxB + βxB = 0 ˙ xB = αxAxB − βxB = 0 Steady states: αxAxB − βxB = 0 = ⇒ xB = 0 or xA = β α Invariant Subspace: ˙ xA ˙ xB
−1 1
−1
−1 1
Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
E S+x0 xA xB
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
E S+x0 xA xB
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
E S+x0 xA xB
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
E S+x0 xA xB
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Protein activation model: (A inactive, B active) ˙ xA = −αxAxB + βxB ˙ xB = αxAxB − βxB. Conservation law: (Total proteins constant) ˙ xA + ˙ xB = 0 = ⇒ xA + xB = xA(0) + xB(0) = C. Reduces 2-D system to 1-D! = ⇒
xA = −αxA(C − xA) + β(C − xA) xB = C − xA
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
X1 X2 X3
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
X0 X1 X2 X3
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
X0 X* X1 X2 X3
Figure: State space is partitioned (invariant spaces)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Stochastic Model: If molecular counts are low (e.g. biochemical/genetic models), more realistic to model systems stochastically. Keep track of reactant numbers: Xi ∈ {0, 1, 2, . . .} Reactions occur discretely and at separate times Modeled as a continuous time Markov chain (CTMC): X(t) = X(0) +
r
Yi t λi(X(s)) ds
i − yi)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Stochastic Model: If molecular counts are low (e.g. biochemical/genetic models), more realistic to model systems stochastically. Keep track of reactant numbers: Xi ∈ {0, 1, 2, . . .} Reactions occur discretely and at separate times Modeled as a continuous time Markov chain (CTMC): X(t) = X(0) +
r
Yi t λi(X(s)) ds
i − yi)
Yi(·) a unit-rate Poisson process
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Protein activation model:
− → 2B B − → A
S+x0
A B
Deterministic
1 2 3 4 1 2 3 4
A B 1 2 3 4 1 2 3 4
Stochastic
Moral of the story: State space is the same but discretized!
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Two approaches for analyzing Continuous Time Markov Chains.
1 Generate sample paths:
Think numerical integration with noise Commonly simulated with Gillespie’s Algorithm (can be computationally-intensive!) (Gillespie, 1976 [7])
2 Evolve probability distribution:
Tracks probability of being in a given state at a given time Evolution given by Chemical Master Equation (linear ODE but very high-dimensional!)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 A B
Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise).
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 A B
Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise).
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 A B
Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise).
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
10 20 30 40 0.2 0.4 0.6 0.8 1
Time = 0
molecules of A Probability 10 20 30 40 0.02 0.04 0.06 0.08 0.1
Time = 0.5
molecules of A Probability 10 20 30 40 0.02 0.04 0.06 0.08 0.1
Time = 1
molecules of A Probability 10 20 30 40 0.05 0.1 0.15 0.2
Time = 5
molecules of A Probability
Probability evolution can be determined through the Chemical Master Equation (Kolmogorov’s forward equations)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]):
a(t) = A(t) V and b(t) = B(t) V .
1 2 3 4 5 1 2 3 4
V = 1
A B
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]):
a(t) = A(t) V and b(t) = B(t) V .
1 2 3 4 5 1 2 3 4
V = 10
A B
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]):
a(t) = A(t) V and b(t) = B(t) V .
1 2 3 4 5 1 2 3 4
V = 100
A B
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Overview Deterministic Model Stochastic Model
Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]):
a(t) = A(t) V and b(t) = B(t) V .
1 2 3 4 5 1 2 3 4
V = 1000
A B
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
1 Background
Overview Deterministic Model Stochastic Model
2 Stochastic Extinction
Background Network Properties Conditions (technical details!)
3 Future Work
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
CAUTION: Discretization can create extinction events not permissible in the deterministic model!
A B 1 2 3 4 1 2 3 4
Stochastic
Protein activation model:
− → 2B B − → A It is possible to irreversibly lose the last activated protein through reaction B − → A.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
CAUTION: Discretization can create extinction events not permissible in the deterministic model!
A B 1 2 3 4 1 2 3 4
Stochastic
Protein activation model:
− → 2B B − → A It is possible to irreversibly lose the last activated protein through reaction B − → A. .
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
2 4 6 8 10 1 2 3 4 5
A B
On the unbounded interval [0, ∞) extinction is inevitable. The deterministic and stochastic models do not agree.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Extinction phenomena in well-known in many stochastic models: Population biology models (e.g. Lotka-Volterra, etc.) Disease spread models (e.g. SIS, SIR, SIRS models etc.) Chemical reaction models (e.g. Keizer’s paradox (1987 [10]), Anderson et al. (2014 [1]), Brijder (2015 [2]), etc.) Alternative statistics required (e.g. expected time to extinction, quasi-stationary distribution, etc.)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
For realistic biochemical networks, it may not be obvious that a stochastic extinction event occurs! Signaling Pathway: (Shinar and Feinberg, 2010 [12]) XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11 Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
For realistic biochemical networks, it may not be obvious that a stochastic extinction event occurs! Signaling Pathway: (Shinar and Feinberg, 2010 [12]) XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Stochastic system inevitably converges to an extinction state (Xp = Xtot, Yp = Ytot, rest = 0) (Anderson et al., 2014 [1])
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Objective: We want to derive network-based conditions for extinction and/or persistence of chemical species. Tools: Use elements from Chemical Reaction Network Theory (CRNT) and Petri Net Theory. CRNT has been used extensively for deterministic models: Deficiency Zero Theorem [3, 8, 9] Deficiency One Theorem/Algorithm [4, 6, 5]
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Technique: Construct conditions for system to exhibit non-extinction (and then violate them!) We are going to need two things:
1 Sequence of reactions corresponding to modes of
stoichiometric balance (a priori known)
2 Sequence of reactions corresponding to admissible recurrent
behavior under the assumption of non-exinction.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Technique: Construct conditions for system to exhibit non-extinction (and then violate them!) We are going to need two things:
1 Sequence of reactions corresponding to modes of
stoichiometric balance (a priori known)
2 Sequence of reactions corresponding to admissible recurrent
behavior under the assumption of non-exinction.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #1: State of system at time t ≥ 0 is given by X(t) = X(0) + Γ N(t) where Γ is stoichiometric matrix and N(t) contains reaction counts. If a state recurs (i.e. X(t) = X(0)) then we have N(t) ∈ ker(Γ) ∩ Rr
≥0.
Generators of cone ker(Γ) ∩ Rr
≥0 are known (stoichiometric flux
modes)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Γ = − 1 1 −1 1 1 1 −1 − 1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Γ = − 1 1 −1 1 1 1 −1 − 1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Γ = − 1 1 −1 1 1 1 −1 − 1 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Stoichiometric Flux Modes: Trivial, e.g. (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) ∈ ker(Γ) Non-trivial, e.g. (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) ∈ ker(Γ)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Stoichiometric Flux Modes: Trivial, e.g. (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) ∈ ker(Γ) Non-trivial, e.g. (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) ∈ ker(Γ)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Stoichiometric Flux Modes: Trivial, e.g. (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) ∈ ker(Γ) Non-trivial, e.g. (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) ∈ ker(Γ)
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Objective: S1 Construct another sequence of balancing reactions by assuming recurrence! S2 Compare resulting vector of counts to known stoichiometric flux modes (i.e. generators of ker(Γ) ∩ Rr
≥0).
Technical assumptions: A1 Assume there is a reaction from a non-terminal complex which is recurrent (i.e. may always fire). A2 Assume system has finite state space.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Can only become “trapped” on terminal components.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! XD X XT Xp Xp + Y XpY X + Yp XD + Yp XDYp XD + Y
1 2 3 4 5 6 7 8 9 10 11
Notice X + Yp “dominates” X and XD + Y “dominates” XD.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Reconstruct the network to capture these new connections! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D Only one terminal component in new graph.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2])
1 Fix a subset of reactions which guarantee full access to
reaction graph.
2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal
complex to terminal component, and then repeat.
4 Use A2 to construct a vector of counts v.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2])
1 Fix a subset of reactions which guarantee full access to
reaction graph.
2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal
complex to terminal component, and then repeat.
4 Use A2 to construct a vector of counts v.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2])
1 Fix a subset of reactions which guarantee full access to
reaction graph.
2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal
complex to terminal component, and then repeat.
4 Use A2 to construct a vector of counts v.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2])
1 Fix a subset of reactions which guarantee full access to
reaction graph.
2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal
complex to terminal component, and then repeat.
4 Use A2 to construct a vector of counts v.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2])
1 Fix a subset of reactions which guarantee full access to
reaction graph.
2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal
complex to terminal component, and then repeat.
4 Use A2 to construct a vector of counts v.
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Observation #4: We can fix the reactions to form a rooted tree! XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D NOTE: Unique path from each non-terminal complex to terminal component!
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Basic Idea (con’t) Following rooted tree until a state recurs! (i.e. X(t) = X(0)) (This is guaranteed by the finite state space assumption.) Resulting vector of counts must satisfy...
1 Has support contained in a given rooted tree. 2 Is weighted toward terminal component. 3 No restrictions within terminal components (if applicable).
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Conditions for extinction: This can give conditions for extinction! Example 1: 2X1 X1 + X2 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (0, 1, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Conditions for extinction: This can give conditions for extinction! Example 1: 2X1 X1 + X2 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (0, 1, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Conditions for extinction: This can give conditions for extinction! Example 1: 2X1 X1 + X2 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (0, 1, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Conditions for extinction: This can give conditions for extinction! Example 1: 2X1 X1 + X2 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (0, 1, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3 v ∈ ker(Γ) = ⇒ Extinction!
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Example 2: X1 + X2 2X1 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (2, 0, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Example 2: X1 + X2 2X1 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (2, 0, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Example 2: X1 + X2 2X1 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (2, 0, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Example 2: X1 + X2 2X1 2X2
1 2 3
ker(Γ) = span {(1, 1, 0), (2, 0, 1)} Rooted Tree: v = (v1, 0, v3), 0 ≤ v1 ≤ v3 v ∈ ker(Γ) = ⇒ Extinction!
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D
ker(Γ) = (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) + cycles Rooted tree: v = (v1, 0, v3, 0, v5, v6, 0, v8, 0, v10, 0), . 0 ≤ v10 ≤ v1 ≤ v3 ≤ v5 . 0 ≤ v6 ≤ v8 ≤ v3 ≤ v5
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D
ker(Γ) = (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) + cycles Rooted tree: v = (v1, 0, v3, 0, v5, v6, 0, v8, 0, v10, 0), . 0 ≤ v10 ≤ v1 ≤ v3 ≤ v5 . 0 ≤ v6 ≤ v8 ≤ v3 ≤ v5
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work Background Network Properties Conditions (technical details!)
Signaling Pathway: XD + Yp XDYp XD + Y XD X XT Xp Xp + Y XpY X + Yp
1 2 3 4 5 6 7 8 9 10 11
D D D
ker(Γ) = (0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1) + cycles Rooted tree: v = (v1, 0, v3, 0, v5, v6, 0, v8, 0, v10, 0), . 0 ≤ v10 ≤ v1 ≤ v3 ≤ v5 . 0 ≤ v6 ≤ v8 ≤ v3 ≤ v5
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work
1 Background
Overview Deterministic Model Stochastic Model
2 Stochastic Extinction
Background Network Properties Conditions (technical details!)
3 Future Work
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work
Summary: Deterministic and stochastic models often have different long-term predictions (e.g. extinction behavior) Can construct sequences of reactions which can contradict assumption of non-extinction (v ∈ ker(Γ) ∩ Rr
≥0)
Future Work: Algorithmize process! Develop connection between existing network conditions and Petri Net Theory
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work
Matthew Douglas Johnston Extinction in Discrete CRNs
Background Stochastic Extinction Future Work
Selected Bibliography
David F. Anderson, German Enciso, and Matthew D. Johnston. Stochastic analysis of chemical reaction networks with absolute concentration robustness. J. R. Soc. Interface, 11(93):20130943, 2014. Robert Brijder. Dominant and T-Invariants for Petri Nets and Chemical Reaction Networks. Lecture Notes in Comput. Sci., 9211:1–15, 2015. Martin Feinberg. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal., 49:187–194, 1972. Martin Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors: I. the deficiency zero and deficiency one theorems. Chem. Eng. Sci., 42(10):2229–2268, 1987. Martin Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors: II. multiple steady states for networks of deficiency one. Chem. Eng. Sci., 43(1):1–25, 1988. Martin Feinberg. Multiple steady states for chemical reaction networks of deficiency one. Arch. Rational
Daniel Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22(4):403–434, 1976. Fritz Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration.
Fritz Horn and Roy Jackson. General mass action kinetics. Arch. Ration. Mech. Anal., 47:81–116, 1972. Joel Keizer. Statistical Thermodynamics of Nonequilibrium Processes. Spring-Verlag, 1987. Tom G. Kurtz. Limit theorems for sequences of jump Markov processes approximating ordinary differential
Guy Shinar and Martin Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389–1391, 2010. Matthew Douglas Johnston Extinction in Discrete CRNs