MENU [email protected] = *@[email protected] A Delay Chemical Master Equation

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The Chemical MasterEquation

The StochasticSimulation Algorithm

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The Delayed ChemicalMaster Equation

The DelayedStochastic SimulationAlgorithms

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A Delay Chemical Master Equation and aDelay Stochastic Simulation Algorithm

(on request oil, salt and pepper)

G. CaravagnaPh.D. Lunchtime Seminar

“Il partito dominante, non potendo trasformare apertamente le scuole di stato in scuole di

partito, manda in malora le scuole di stato per dare la prevalenza alle scuole private.“ (P.C.,’50)

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StartersThe Chemical Master EquationThe Stochastic Simulation Algorithm

Soup of the dayScenarioThe Delayed Chemical Master EquationThe Delayed Stochastic Simulation Algorithms

DessertExamplesWhat to do

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The CME [’77]

I X(t) = x is the vector state at time t;

I X(t0) = x0 is the initial configuration;

I dt so small that at most one reaction fires;

I P(x, t | x0, t0) is the probability that, given the initialconfiguration, at time t the system is described by thestate vector x.

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The CME [’77]

Two events:

- at time t the system is already in state x and in theinfinitesimal time [t; t + dt[ no reaction fires;

- at time t the system is in state x− νj and reaction Rj

fires.

P(x, t + dt | x0, t0) = P(x, t | x0, t0)

0@1−MX

j=1

aj (x)dt

1A+

MXj=1

P(x− νj , t | x0, t0) · aj (x− νj ).

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The CME [’77]

Two events:

- at time t the system is already in state x and in theinfinitesimal time [t; t + dt[ no reaction fires;

- at time t the system is in state x− νj and reaction Rj

fires.

P(x, t + dt | x0, t0) = P(x, t | x0, t0)

0@1−MX

j=1

aj (x)dt

1A+

MXj=1

P(x− νj , t | x0, t0) · aj (x− νj ).

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The SSA [’77]

1. Initialize the time t = t0 and the system state x = x0.

2. With the system in state x at time t, evaluate all theaj(x) and their sum a0(x) =

∑Mj=1 aj(x).

3. Given two random numbers r1, r2 ∈ U[0; 1], generatevalues for τ and j accordingly to

τ =1

a0(x)ln(

1

r1)

j−1∑i=1

ai (x) < r2 · a0(x) ≤j∑

i=1

ai (x)

then update x = x + νj and t = t + τ , go to step 2.

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Why delays

I application driven in tumor–immune:”An immune system is a collection of mechanisms within an organism that protects against

disease by identifying and killing pathogens and tumor cells [..]. An antigen or immunogen is asubstance that prompts the generation of antibodies and can cause an immune response.“

(Wikipedia)

I more detailed modeling;

I mathematically reasonable (DDEs as extensions ofODEs);

A lot of applications for DDEs and, consequently, for delayedstochastic processes.

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Modifications

I Reactions:I Rj has a delay σj ≥ 0.

I Propensity functions:I aj depends on the state X(t − σj).

I Initial Configuration:I X(t0) = x0 as expected;I function ω, X(t) = ω(t) if t < t0.

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Construction of the DCME

I at time t the system is already in state x and, at delayedtime t −σj , the system was in state xi , no reactions fire;

I at time t the system is in state x− νj and, at delayedtime t − σj , the system was in state xi , reaction Rj fires.

P(x, t + dt) = P(x, t)

0B@1−MX

j=1

Xxi∈I(x)

P(xi , t − σj ) · aj (xi )dt

1CA

+MX

j=1

P(x− νj , t)

0B@ Xxi∈I(x)


1CA

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Construction of the DCME

I at time t the system is already in state x and, at delayedtime t −σj , the system was in state xi , no reactions fire;

I at time t the system is in state x− νj and, at delayedtime t − σj , the system was in state xi , reaction Rj fires.

P(x, t + dt) = P(x, t)

0B@1−MX

j=1

Xxi∈I(x)


1CA

+MX

j=1

P(x− νj , t)

0B@ Xxi∈I(x)


1CA

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The DCME

With some tricks we get:

∂P(x, t | x0, t0;ω)

∂t=

M∑j=1

∑xi∈I(x)

P(x− νj , t; xi , t − σj | x0, t0;ω) · aj(xi )

−M∑

j=1

∑xi∈I(x)

P(x, t; xi , t − σj | x0, t0;ω) · aj(xi )

As expected, if all σj ’s are 0, the DCME is the CME.

∂P(x, t | x0, t0)

∂t=

MXj=1

P(x− νj , t | x0, t0) · aj (x− νj )−MX

j=1

P(x, t | x0, t0) · aj (x).

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The DCME


∂P(x, t | x0, t0;ω)

∂t=

M∑j=1

∑xi∈I(x)


−M∑

j=1

∑xi∈I(x)



∂P(x, t | x0, t0)

∂t=

MXj=1

P(x− νj , t | x0, t0) · aj (x− νj )−MX

j=1

P(x, t | x0, t0) · aj (x).

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The DCME


∂P(x, t | x0, t0;ω)

∂t=

M∑j=1

∑xi∈I(x)


−M∑

j=1

∑xi∈I(x)



∂P(x, t | x0, t0)

∂t=

MXj=1

P(x− νj , t | x0, t0) · aj (x− νj )−MX

j=1

P(x, t | x0, t0) · aj (x).

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Interpreting delays

Two different interpretations for delays:

1. non–scheduled events:I propensities delayed;I reactions as in th SSA.

2. scheduled events:I propensities non–delayed;I reactions scheduled at future time.

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Handling Discontinuities

One common problem:

1. given the system at time t;

2. let τ be the time for next reaction;

The propensities have to be constant in [t; t + τ ].

1. this always holds in the SSA;

2. does not always hold here. :-(

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Handling Discontinuities

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The DSSA (1)


2. With the system in state x at time t, evaluate all theaj(t) and their sum a0(t) =

∑Mj=1 aj(X(t − σj));

3. Given a random number r1 uniformally distributed inthe interval [0; 1], generate values for τ and θtaccordingly to

τ =1

a0(t)ln(

1

r1

) θt = min{θt,j | j = 1, . . . ,M}

3.1 If τ < θt then, if r2 is a random number uniformallydistributed in the interval [0; 1], select reaction Rj suchthat j−1X

i=1

ai (X(t − σi )) < r2 · a0(t) ≤jX

i=1

ai (X(t − σi ))

then update x = x + νj and t = t + τ ;3.2 else τ ≥ θt , then t = t + θt ;

4. go to step 2.

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The DSSA (1)





τ =1

a0(t)ln(

1

r1

) θt = min{θt,j | j = 1, . . . ,M}


i=1

ai (X(t − σi )) < r2 · a0(t) ≤jX

i=1

ai (X(t − σi ))


4. go to step 2.

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The DSSA (1)





τ =1

a0(t)ln(

1

r1

) θt = min{θt,j | j = 1, . . . ,M}


i=1

ai (X(t − σi )) < r2 · a0(t) ≤jX

i=1

ai (X(t − σi ))


4. go to step 2.

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The DSSA (2)



∑Mj=1 aj(X(t));

3. Given two random numbers r1, r2 uniformally distributedin the interval [0; 1], generate values for τ and jaccordingly to

τ =1

a0(t)ln(

1

r1

)

j−1Xi=1

ai (X(t)) < r2 · a0(t) ≤jX

i=1

ai (X(t))

3.1 If delayed reaction Rk is scheduled within [t; t + τ ] thenupdate x = x + νk and t = t + τk ;

3.2 else, schedule Rj at time t + σj + τ , set time to t + σj ;

4. go to step 2.

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The DSSA (2)



∑Mj=1 aj(X(t));

3. Given two random numbers r1, r2 uniformally distributedin the interval [0; 1], generate values for τ and jaccordingly to

τ =1

a0(t)ln(

1

r1

)

j−1Xi=1

ai (X(t)) < r2 · a0(t) ≤jX

i=1

ai (X(t))

3.1 If delayed reaction Rk is scheduled within [t; t + τ ] thenupdate x = x + νk and t = t + τk ;

3.2 else, schedule Rj at time t + σj + τ , set time to t + σj ;

4. go to step 2.

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Predator–prey model with harvesting

dx

dt= r1x(t)

(1− 1

Kx(t)

)− bx(t)y(t)− H

dy

dt= −r2y(t) + cx(t − τ)y(t − τ)

I logistic growth of the preys;

I non–linear predation;

I linear death of the predators;

I constant harvesting;

I delayed effect of predation.

X + Yc7→ X + 2Y with delay τ

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Predator–prey model with harvesting

dx

dt= r1x(t)

(1− 1

Kx(t)

)− bx(t)y(t)− H

dy

dt= −r2y(t) + cx(t − τ)y(t − τ)

I logistic growth of the preys;

I non–linear predation;

I linear death of the predators;

I constant harvesting;

I delayed effect of predation.

X + Yc7→ X + 2Y with delay τ

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Crocodilian population

”In crocodilian populations, the choice of nesting

site determines the sex of hatchlings. In the wet

marsh, cool temperatures primarily produce

female hatchlings. The hot temperatures of the

dry levees result in primarily male hatchlings. The

dry marsh has an intermediate temperature

profile, resulting in hatchlings of both sexes.“

I two populations: juvenile and adults;

I three regions: wet/dry marsh and levees;

I delays: at any time t, the juvenile females born at timet − τ years ago who have survived τ years enter theadult female population.

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Crocodilian population

”In crocodilian populations, the choice of nesting

site determines the sex of hatchlings. In the wet

marsh, cool temperatures primarily produce

female hatchlings. The hot temperatures of the

dry levees result in primarily male hatchlings. The

dry marsh has an intermediate temperature

profile, resulting in hatchlings of both sexes.“

I two populations: juvenile and adults;

I three regions: wet/dry marsh and levees;

I delays: at any time t, the juvenile females born at timet − τ years ago who have survived τ years enter theadult female population.

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Future works

I Extending delays:I time–dependent delays;I stochastic delays;I ...

I Implementation issues delay–dependent;I DSSA and formal methods:

I PT Nets with time;I Timed/History–dependent Automata;I ...

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References

P. Calamandrei. Discorso al III Congresso ADSN. ScuolaDemocratica (2:iv), pp. 1–5, 1950.

R. Schlicht et al., A Delay Stochastic Process withApplications in Molecular Biology, Journal ofMathematical Biology 57(5) (2008), 613–648.

G.Caravagna et al., ??, Journal on My Desk.

A. Martin et al., Predator-prey Models with Delay andPrey Harvesting, Journal of Mathematical Biology 43(3)(2001), 247–267.

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MENU [email protected] = *@[email protected] A Delay Chemical Master Equation

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