Simulation: Software Methods and Biological Processes

1
SimulationSoftware Methods and Biological
Processes

• Marco AntoniottiNYU Courant Bioinformatics Group
  

2
Outline

• Simulation fundamentals
• Modeling
• Modeling Tools
• Mathematical Tools (Differential Equations, Timed
  Systems)
• Languages
• Software infrastructure and implementations
• Models of computations
• Numerics

3
Outline (continued)

• Tools
• General
• Matlab, Mathematica, Maple, Macsyma
• Specialized
• Discrete Event with Extensions
• Hybrid System Simulators
• Biological Simulations
• E-CELL, VirtualCell

4
Modeling

• There are several frameworks that can be used for
  modeling
• Differential Equations (continuous)
• Discrete Systems
• Finite State Automata
• Petri Nets
• Dataflow Diagrams

5
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'

6
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'

7
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)

8
ODE Repressilator System

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

TetR
CI
LacI
9
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)

10
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)

11
Stem Cell Differentiation and Proliferation

• Stem Cell Division/Proliferation
• Asymmetric
• Symmetric
• Environmentally Asymmetric

12
Stem Cell FSA
Asymmetric Division
Ns n. of Stem Cells Np n. of Progenitor Cells
Np Np 1
Ns Ns -1
Symmetric Division 2S
die
quiescent
Ns Ns 1
Np Np 2 Ns Ns -1
SymmetricDivision 2P
13
Stem Cells Population ODE Model
14
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 havelt(100,
  1e5), (101, 1e5), ... (98, 1e5)gt

15
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

16
Simulator Template Architecture
System Specification
Simulation Results (Traces)
Trace Inspection Synchronous GUI Plotter Math
Analysis
Simulation Engine
17
Implementing a Simulator

• The engine of a simulator is essentially a
  loop loop for i from start to finish do
  (evaluate next(i))
• The nature of (evaluate next(i)) determines the
  type of simulator we are using

18
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)

19
Implementing a Discrete Event Simulator

• SpecificationSources, Computing Units and Sinks
• Sources square wave generators
• Units down samplers, logical operation
• Engine
• A queue of pairs lttime, valuegt
• Evaluators for the units
• Each unit can sense if it has inputs

20
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

21
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

22
Implementing an ODE Simulator/Solver (contd)

• Euler Method (unrecommended)
• Runge-Kutta 4

23
Implementing an ODE Simulator (contd.)

• Complete formulation of midpoint computations in
  Runge-Kutta 4

24
Stem Cell Population Octave (Matlab) Code

• The ODE formulation isfunction 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)

25
Running the code
26
A Finite Automata LanguageEsterel

• Consider the following example

?B
?A
?R
?R
?A?B!O
?R
?B!O
?A!O
27
Esterel Rendition

• The previous automata is rendered as module
  ABRO input A, B output O loop await A
  await B emit O each R end
• Each transition happens at each tick

28
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

29
Hybrid Systems

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

k1 gt k2
y gt k1
dy/dt 3
dy/dt -1.5
y lt k2
30
A Hybrid System Implementation

• The main loop of a Hybrid System simulation
  engine becomeslet h ltsome valuegtloop for i
  from start to finish do evaluate the current set
  of ODE's for one step h if (any transition
  becomes enabled) do switch ODE set

31
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

32
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

33
Stem Cells PopulationHybrid System (contd)

• 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)

quiescent
Ns Ns -1
d(time)/dt 1r(die) random()
die
- ln(r(die)) lt alpha Ns time
34
Stem Cells PopulationHybrid System (contd)

• One language that can be used to this end is
  l-Shift (Simsek 2000, UC Berkeley)(defhybrid
  SC-behavior () ... (discrete
  ('quiescent flow ((d/dt time)
  
  (random 1.0)))) (transitions ('quiescent 'die
  when (lt (- (log r) ( alpha n_s time)))
  do ((time 0)
  (n_s
  (decf n_s)))) ))

35
Problems with HS

• There are several Numerical problems
• Guard Crossings

36
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

37
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

38
The ModelCell Differentiation

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

M
V
G2
G1
R
C
S
G0
Apoptosis
39
The ModelCell Differentiation

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

Macrophages
GM-Progenitors
Granulocytes