March 19, 2002Marco Antoniotti NYU Bioinformatics Group1 Simulation: Software Methods and Biological...

March 19, 2002 Marco Antoniotti NYU Bioinformatics Group 1

Simulation:Simulation:Software Methods and Software Methods and Biological ProcessesBiological Processes

Marco AntoniottiNYU Courant Bioinformatics Group


Outline

Simulation fundamentals Modeling

Modeling Tools Mathematical Tools (Differential Equations, Timed Systems) Languages

Software infrastructure and implementations Models of computations Numerics


Outline (continued)

Tools General

Matlab, Mathematica, Maple, Macsyma Specialized

Discrete Event with Extensions Hybrid System Simulators

Biological Simulations E-CELL, VirtualCell


Modeling

There are several frameworks that can be used for modeling Differential Equations (continuous) Discrete Systems

Finite State Automata Petri Nets Dataflow Diagrams


Time

The main issue behind all of these modeling frameworks is Time Consequences

What constitues progress How do we implement simulators for systems that

assume a different notion of `time'


Differential Equations

Ordinary Differential Equations (ODE's) are a standard tool used to model physical, biological and engineering systems

y'(t) = F(y(t), t)y(0) = k

Time t is `continuous'


Discrete Models

Finite State Machines Petri Nets Discrete Event Systems

A queue of `events' generated by some `sources' and processed by `components'

Time is a usually a derived notion (e.g. by observing the sequence of events or a Wall Clock)


ODE Repressilator System

The repressilator system (Elowizt, Leibler 2000) comprises three proteins, LacI, CI, and TetR controlling each other.

TetR

CILacI


ODE Repressilator System

The repressilator system (Elowizt, Leibler 2000) is simply described as an ODE system (deterministic version):(i,j) = (lacI,cI) or (tetR,lacI) or (cI,tetR)

dm i

dt= miƒ

·

1ƒ p jn ƒ · 0

dpidt

= ¸ p i mi


Stem Cell Population Model

There are different kinds of (Adult) Stem Cells Hematopoietic, Neuronal, Muscle, Epithelial

A Stem Cell differentiates into Committed Progenitor Cells, which in turn differentiate into Regular Cells (T-cells, Red Blood Cells, Skin Cells)


Stem Cell Differentiation and Proliferation

Stem Cell Division/Proliferation

Asymmetric

Symmetric

Environmentally Asymmetric


Stem Cell FSA

Ns = n. of Stem CellsNp = n. of Progenitor Cells

SymmetricDivision 2S

AsymmetricDivision

die

SymmetricDivision 2P

quiescent

Ns = Ns + 1

Np = Np + 1

Np = Np + 2Ns = Ns -1

Ns = Ns -1


Stem Cells Population ODE Model

dN p

dt= 2¾ƒ ¹ N sƒ ÀN p

K p N p

K p

· p N p

dN s

dt= ¸

K s N s

K s

¾K p N p

K p

· s N s


Simulation Traces

Running a simulation (numerically solving a set of differential equations) produces a trace

A trace is a sequence of vectors of values, possibly symbolic

E.g. for the FSA Stem Cell Model we have<(100, 1e5), (101, 1e5), ... (98, 1e5)>


Discrete Event Systems

The notion of trace gives rise to a more low level notion of system.

Discrete Event Simulators are defined in terms of the underlying timed traces and of their generators


Simulator Template Architecture

System Specification

Simulation Engine

Simulation Results (Traces)

Trace Inspection Synchronous GUI Plotter Math Analysis


Implementing a Simulator

The engine of a simulator is essentially a loop

loop for i from start to finishdo (evaluate next(i))

The nature of (evaluate next(i)) determines the type of simulator we are using


Simulator Runtime

The simulator runtime can be stopped

when stopped, one or more individual components can be instantiated (or retrieved) and activated

restarted when restarted, the simulation will proceed with the

individual components behaving as if part of the overall simulation (this may be considered as a lens effect)


Implementing a Discrete Event Simulator

SpecificationSources, Computing Units and Sinks Sources: square wave generators Units: down samplers, logical operation

Engine A queue of pairs <time, value> Evaluators for the units

Each unit can sense if it has inputs


Implementing a FSA Simulator

Specification A set of Finite Automata Two possibilities

Construct a product automata and then simulate Keep Automata separate and make simulation slightly

more complex

Engine check the set of all enabled transitions from the

current state and produce the next state


Implementing an ODE Simulator/Solver

Specification A set of ODE's

Engine Evaluate at discrete time steps the value of the

function and of its derivative Several different methods are available

Euler Runke-Kutta


Implementing an ODE Simulator/Solver (contd)

Euler Method (unrecommended)

Runge-Kutta 4

ynƒ 1= ynƒ h F x n , yn

y n

y nƒ 1m1

m2

xn xnƒh2

xnƒ h


Implementing an ODE Simulator (contd.)

Complete formulation of midpoint computations in Runge-Kutta 4k 1=hF x n , yn

k 2=hF x nƒh2, y nƒ

k 12


k 22


k 32

y nƒ 1= y nƒk 16

ƒk 23

ƒk 33

ƒk 46

ƒ O h 5


Stem Cell Population Octave (Matlab) Code

The ODE formulation is

function dn = f(N, t)alphap = 0.01; alphas = 0.01;beta = 5; gamma = 0.2; theta = 0.2; kappa = 0.5;Ks = 1000; Kp = 10000;dn(1) = (beta * (Ks - N(1))/Ks

- theta * (Kp - N(2))/Kp - alphas) * N(1);dn(2) = (Kp - N(2))/Kp * ((gamma + 2 * theta) * N(1)

+ kappa * N(2)) - alphap * N(2);endfunctionN0 = [50; 1000]; t = linspace(0, 15, 100)';N = lsode("f", N0, t);


Running the code


A Finite Automata Language:Esterel

Consider the following example

?A

?B!O

?R

?R

?R

?B

?A!O

?A?B!O


Esterel Rendition

The previous automata is rendered as

module ABRO:input A, B; output O;loop

[await A; await B];emit O

each Rend

Each transition happens at each tick


Putting it all together

How do we reconcile a high level modeling view with a low level continuous time based simulation?

Hybrid Systems are one of the latest developments mixing Finite State Automata and Continuous Systems


Hybrid Systems

A Hybrid System is a computational tool that switches among different sets of ODE's accoding to a some conditions

y < k2

y > k1k1 > k2

dy/dt = 3 dy/dt = -1.5


A Hybrid System Implementation

The main loop of a Hybrid System simulation engine becomes

let h = <some value>loop for i from start to finish

do evaluate the current set of ODE's for

one step hif (any transition becomes enabled)

do switch ODE set


Stem Cells PopulationHybrid System Formulation

The Stem Cell ODE formulation is a deterministic and continuous approximation of the real (?!?) model

An Hybrid System formulation can be made closer to the discrete and stochastic counting nature of the model


Stem Cells PopulationHybrid System (contd)

We model the Stem Cell Population as a Hybrid System with only one differential (continuous) variable: time (time)

Our system will therefore integrate

d time / dt = 1



We associate to each transition an exponential decay rate

The exponential decay rate represents the `rate' of an event in the population model (asymmetric division, apoptosis)

Ns = Ns -1quiescent

die d(time)/dt = 1r(die) = random()

- ln(r(die)) < alpha * Ns * time



One language that can be used to this end is -Shift (Simsek 2000, UC Berkeley)

(defhybrid SC-behavior ()...

(:discrete ('quiescent :flow ((d/dt time) (random 1.0))))

(:transitions ('quiescent 'die :when (<= (- (log r) (* alpha n_s

time))) :do ((time 0)

(n_s (decf n_s))))

))


Problems with HS

There are several Numerical problems

Guard Crossings

y(t)

t

time step

(0, 0)

guard: y(t) < 0


Conclusions

Simulations are a good tool for Modeling several systems, hence Biological Systems as well

We saw different kinds of simulation paradigms continous discrete hybrid


Conclusions (2)

A simulation tool must be fitted to the problem at hand

Proper modeling is always a prerequisite to produce data that can be analyzed by matching it with laboratory experimental results


The ModelCell Differentiation

Cell Differention (e.g. multipotent HSCs, Metcalfe et al. 1997)

C

V

G1

R

S

G2

M

G0

Apoptosis

G-CSFGMC-SF

IL-6IL-11

...

M-CSFGM-CSFO-CSF

IL3...

Otherfactors


The ModelCell Differentiation

Cell Differention (e.g. multipotent HSCs, Metcalfe et al. 1997)

G-CSFGMC-SF

IL-6IL-11

...

M-CSFGM-CSFO-CSF

IL3...

GM-ProgenitorsMacrophages

Granulocytes

March 19, 2002Marco Antoniotti NYU Bioinformatics Group1 Simulation: Software Methods and Biological...

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