some foundations of cellular simulation nathan addy scientific programmer the molecular sciences...
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Some foundations of Cellular Simulation
Nathan AddyScientific Programmer
The Molecular Sciences InstituteNovember 19, 2007
The alpha pathway:The system we study here at the Molecular Sciences Institute
A full model of this system requires one simulation algorithm per chemical reaction.
There are two basic methods for simulating cellular systems.
• Stochastic Algorithms • Differential Equations
Stochastic Simulation algorithms
Use probabilities and the random numbers generation togenerate a statistically accurate time course of the system.
Define Initial Conditions
State
Probability distributionsSample random numbers
Gillespie Algorithm
A
B
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
C
A+
+A
B +
A A
A
B
C
A well-mixed container. 3 reactions.
B
Gillespie Algorithm
B
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C +A AB
Propensity = 39 * pAB
3 * 13 ways this reaction can occur
pAB = ‘intrinsic’ probability of the A + B reaction
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
Propensity = 6 * pAA
3 * 2 ways this reaction can occur
pAA = ‘intrinsic’ probability of the A + A reaction
A A+ A A
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
Propensity = 13 * pBC
13 * 1 ways this reaction can occur
pBC = ‘intrinsic’ probability of the B + C reaction
CB +B
C
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
Total propensity = propensityAB + propensityAA + propensityBC
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
Probability that the next reaction to occur given this state will be the ‘A + A’ reaction is
Propensity of ‘A + A’ reaction Total Propensity
6 * pAA 39 * pAB + 6 * pAA + 13 * pBC
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
P(A + A) =
P(B + C) =
P(A + B) =
6 * pAA 39 * pAB + 6 * pAA + 13 * pBC
13 * pBC 39 * pAB + 6 * pAA + 13 * pBC
39 * pAB 39 * pAB + 6 * pAA + 13 * pBC
Gillespie Algorithm
A
AA
B
BB
B
B
B
B
B
B
B
B
B
C
Sample the uniform distribution to determine which reaction fires next.
Sample the exponential distribution to determine what time the next reaction will occur at.
Gillespie Algorithm
Define Initial Conditions
State
Probability distributionsSample random numbers
Gillespie Algorithm
Population of A starts at 3.Population of B starts at 13.Population of C starts at 1.
State
Probability distributionsSample random numbers
Gillespie Algorithm
A = 3B = 13C = 1
Probability distributionsSample random numbers
Gillespie Algorithm
A = 3B = 13C = 1
Sample random numbers P(rxni) =Propensity(rxni)
Propensity( rxnn )
Gillespie Algorithm
A = 3B = 13C = 1
Random number generated.Suppose A + B->AB occurs
P(rxni) =Propensity(rxni)
Propensity( rxnn )
Gillespie Algorithm
A = 2B = 12C = 1AB = 1
Random number generated.A + B->AB occurs
P(rxn0) =Propensity(rxni)
Propensity( rxnn )
Gillespie Algorithm
A = 2B = 12C = 1AB = 1
Probability distributionsSample random numbers
Gillespie Algorithm(standing in for all stochastic algorithms)
Pros:• Simulates things exactly, at the level of the individual molecules.• Assumptions don’t break down in very small systems. • Model behavior includes noise, which is often important in celluart behavior
Cons:• Very Slow compared with other methods because reactions only happen one at a time.• Generates statistically accurate simulations, so multiple runs may be needed.
Differential Equations
Equations that describe the behavior of objects over time in terms of their derivatives.
Simplest example:
Solving
dy
dt f (t)
y(t) f (x)dx0
tA car a starts from rest at position 0 at time 0 and travels with velocityv(t) = 10m/s * t afterwards. Where is it after 10 seconds?
Differential Equations
A very common style of mathematical modeling:
F md2p
dt 2m1
m2
m3
F Gm1m2
r2N-body problem
Differential Equations
Assumptions that differential equations make:
• All variables have continuous derivatives.
• Change to model state happens deterministically. There is no system noise.
A form of differential equations that model reversible chemical reactions.
Mass Action equations
A
AA A
A
AA A
forward
backward
Mass Action equations
Assumptions:
• Everything is well mixed.• Reaction mechanism is very simple.• Everything is in high enough concentrations so that lack of noise and continuous changes to populations of species are acceptable.
Mass action equations say that the rates of both the forward and backward reactions are proportional to the product of their concentrations raised to the power of their stochiometries.
Mass Action equations
A A
dforward
dtp
forward[A] [B]
The p’s are the proportionality constants that act as rate constants.
dreverse
dtp
reverse[AB]
B
B
The reaction 1A + 1B 1AB comes out to have equations
Mass Action equations
Ste2 Ste2
On and off rates for mass action equations are determined Experimentally, and recorded at places like the alpha modelwiki.
kon ?
koff ?
yeastpheromonemodel.org
Mass Action equations
Ste2 Ste2
kon ?
koff ?
* koff = 9 * 10-4 s-1. Experiment was done using 35S-labeled pheromone at 22°C. TAME was used to prevent pheromone degradation, and cells were treated with NaN3 and KF to prevent growth and other energy-dependent processes. Kd was also measured, but later discounted by the same group. Jenness et al. 1983 PMID 6360378
* From a computational model. kon = 2 * 104 M-1 s-1, koff = 1 * 10-2 s-1 (corresponding to a Kd = 500nM) was used. This these numbers were referenced to Yi et al 2003 PMID 12960402 who used the same koff, but a kon 100x greater. Kofahl and Klipp 2004 PMID 15300679
yeastpheromonemodel.org
Simulating mass action equations
If we have a system composed of multiple mass action equations, we cannot find solutions analytically; we can (and must) solve them numerically.
In general, systems of differential equations must be solved numerically.
Numerical methods for solving DiffEqs
We saw before the simplest types of differential equations involvesimple integrals. If we cannot solve the system directly, we canapproximate a solution by using Riemann sums.
A car a starts from rest at position 0 at time 0 and travels with velocity . What is its position after 10 seconds?
v(t) Sin(Cos(1 t1 t 2
))
Numerical methods for solving DiffEqs
By taking Riemann sums, we can approximate a solution to this differential equation.
Where the error is
Where M2 is the maximum absolute value of f’’(x) on the interval.
Euler’s method as a numerical solution to mass action equations.
Define Initial Conditions
State
Calculate derivativesUpdate state
Euler’s method as a numerical solution to mass action equations.
State
Calculate derivativesUpdate state
dreverse
dtp
reverse[AB]
dforward
dtp
forward[A] [B] A and B start at an initial concentration of 1M.
AB starts at an initial concentration of 0M.
Euler’s method as a numerical solution to mass action equations.
Calculate derivativesUpdate state
t = 0[A] =1M[B] = 1M[AB] = 0M.
Euler’s method as a numerical solution to mass action equations.
t = 0[A] =1M[B] = 1M[AB] = 0M.
Update State
dAB
dtpon[A][B] poff [AB]
dA
dtdBdt
poff [AB] pon[A][B]
Assume pon = .005 s-1 M-1 and poff = .001 s-1
dAB
dt t0
.005s 1M 1 *1M *1M .001s 10M .005Ms 1
dAdt t0
dBdt t0
poff [AB] pon[A][B] .001s 10M .005s 1M 1 *1M *1M .005Ms 1
Euler’s method as a numerical solution to mass action equations.
t = 0[A] =1M[B] = 1M[AB] = 0M.
With a time step of .2 seconds:
dAB
dtpon[A][B] poff [AB]
dA
dtdBdt
poff [AB] pon[A][B]
dAB
dt t0
.005s 1M 1 *1M *1M .001s 10M .005Ms 1
dAdt t0
dBdt t0
poff [AB] pon[A][B] .001s 10M .005s 1M 1 *1M *1M .005Ms 1
ABdAB
dt* t .005Ms 1 * .2s.001M
ABdAdt
* t .005Ms 1 * .2s .001M
Euler’s method as a numerical solution to mass action equations.
t = .2[A] =.999M[B] = .999M[AB] = .001M.
With a time step of .2 seconds:
dAB
dtpon[A][B] poff [AB]
dA
dtdBdt
poff [AB] pon[A][B]
dAB
dt t0
.005s 1M 1 *1M *1M .001s 10M .005Ms 1
dAdt t0
dBdt t0
poff [AB] pon[A][B] .001s 10M .005s 1M 1 *1M *1M .005Ms 1
ABdAB
dt* t .005Ms 1 * .2s.001M
ABdAdt
* t .005Ms 1 * .2s .001M
Euler’s method as a numerical solution to mass action equations.
We have an error condition for solutions producted usingthis method:
If
If we expand y(t0 + h) as a Taylor series around t0 in h We get
Error rises with h2.
Euler’s method as a numerical solution to mass action equations.
We can use this error condition to alter the step size as we go along, in order to minimize global error.
Other numerical methodsThere are other more complicated and more accurate methodsfor simulating differential equations.
They all typically use current state to simulate into the future by small time steps. Some use a history of past state as well.
Fourth order Runge-Kutta:
Cumulative error is on the order of h4
Other numerical methodsDifferent methods can be particularly good for particular systems. (fewer steps, less error)
A good ODE solver can combine different numerical methods and switch between them during simulation in order to maximize performance.
Pros:Usually much faster than stochastic simulation
Cons:Eliminates some biological information.Maybe oversimplifies the system.
Parameter Estimation
We have a system of differential equations. We have measured mostof the reaction constants, but a few are left unknown.
We have done experiments that give us overall system input/outputrelationships.
Our system can be represented as:
OF(I, p,t)
where O is system output,
I is system input,
and p is the value of the unknown parameter.
Parameter Estimation
x
0 Oexperimental(t) F(Iexp, p, t) dt
Parameter estimation here can be thought of as equivalentTo trying to find the value of p that minimizes the function
Sensitivity Analysis
The majority of parameters in pathway models aren’t known.
Some parameters are more important than others.
It would be nice to be able to find the really important ones, sothat we can focus on energy on them. This is sensitivity analysis.
Sensitivity Analysis
If we have our model
OF(I, p,t)
where O is system output,
I is system input,
and p is the value of the unknown parameter.
Sensitivity analysis is trying to find parameters, such that the partial derivative is maximized.
piF(I, p, t)
AnalysisParameter analysis and sensitivity analysis can be used with models of all sorts, not just differential equations.
Because the systems have to be solved numerically, solving theseproblems directly is usually impossible.
Numerical methods for solving these problems must be used.
• Numerical methods for finding zeroes of functions.• Numerical methods for function minimization.
Hybrid modelsWe have seen examples of modeling using stochastic algorithmsand modeling using ODEs.
Both have strengths and weaknesses.
If…Speed is important.We don’t believe noise is relevant.We can get by approximating integer populations as continuous concentrations.
ODES are probably best.If….
Population numbers are low.Noise may be important.
Stochastic algorithms are probably best.
Hybrid models
When we want to make large models, some parts will probably bebest represented with ODEs, others stochastic algorithms.
Hybrid models are models that use both types of algorithms.
Methods have been developed to simulate models that are built from both differential equations and stochastic equations.
They amount to interpreting the process of solving differential equations numerically as a “discrete event process” in the same way that stochastic algorithms produce discrete events.
Hybrid models
Simulating a hybrid model amounts to having a list of events that occur at discrete times. These events can either be stochastic eventsor corrections to differential equations.
These events are executed in order and after each event, any dependent events are rescheduled.
This type of modeling combines the strengths of both types of algorithm.
ConclusionReactions can be modeled either using differential equations or stochastic simulations.
Systems can be modeled using combinations of both types. Eachhas advantages/disadvantages.
Numerical methods for solving systems of differential equationstypically amount to extensions of methods you are familiar with: Riemann sums, the parallelogram method, approximation with Taylor series, etc.
Analyzing model output often requires solving minimizationAnd maximization problems, although these two must usuallybe solved numerically.