Converting Macromolecular Regulatory Models from Deterministic to Stochastic Formulation

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Converting Macromolecular Regulatory Models from
Deterministic to Stochastic Formulation

• Pengyuan Wang, Ranjit Randhawa, Clifford A.
  Shaffer, Yang Cao, and William T. Baumann
• Virginia Tech, Blacksburg VA

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The Fundamental Goal of Molecular Cell Biology
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The Cell Cycle
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Cell Cycle Control Mechanism
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Modeling Techniques

• One method Use ODEs that describe the rate at
  which each protein concentration changes
• Protein A degrades protein B
• with initial condition A(0) A0.
• Parameter c determines the rate of
  degradation.
• Sometimes modelers use creative rate laws to
  approximate subsystems

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Simulation Budding Yeast Cell Cycle
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Expermental Data
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Putting it Together
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Chen/Tyson Budding Yeast Model

• Contains over 30 ODEs, some nonlinear.
• Events can cause concentrations to be reset.
• About 140 rate constant parameters
• Most are unavailable from experiment and must set
  by the modeler

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Fundamental Activities of the Modeler

• Collect information
• Search literature (databases), Lab notebooks
• Define/modify models
• A user interface problem
• Run simulations
• Equation solvers (ODEs, PDEs, deterministic,
  stochastic)
• Compare simulation results to experimental data
• Analysis

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Modeling Process
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Stochastic Simulation Motivation

• ODE-based (deterministic) models cannot explain
  behaviors introduced by random nature of the
  system.
• Variations in mass of division
• Variations in time of events
• Behavior of small numbers (RNA, DNA)
• Differences in gross outcomes

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Gillespies Stochastic Simulation Algorithm (SSA)

• There is a population for each chemical species
• There is a propensity for each reaction, in
  part determined by population
• Each reaction changes population for associated
  species
• Loop
• Pick next reaction (random, propensity)
• Update populations, propensities
• Slow, there are approximations to speed it up

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Question

• Given an existing deterministic model, how do we
  convert it to a formulation capable of stochastic
  simulation?
• Can this be automated?
• Is there a fundamental difference in
  representation?
• SSA is known to be CPU-intensive. How much
  computation resource is really needed to simulate
  the converted model stochastically?

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Relation between the Two Formulations

• In common both models describe the same reaction
  network.
• Difference the reaction rate equation is
  replaced by a propensity function describing how
  likely that the reaction will fire in next unit
  time.
• Connection although they have different physical
  meanings, propensity function shares the same
  expression as corresponding reaction rate
  equation (written in number of molecules).
• Caveat except for the creative rate laws

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Missing Information

• Usually ODE models are written in terms of
  normalized concentrations.
• Thus they need to be converted to models in terms
  of number of molecules (population).
• Some information is missing
• Characteristic concentration
• Explicit definition of units
• Volume of the container.

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Conversion

• The relation between normalized concentration,
  real concentration and population of a species

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How Units are Used in the Model

• Every parameter and species is assigned the
  correct unit, scaling factors.
• The conversion algorithm follows units to convert
  the model.

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The Challenge

• Assigning correct units to species and parameters
  is difficult because all the species, parameters,
  and reactions are connected by the whole reaction
  network.
• Once the modeler is forced to provide the
  complete specification, the conversion can be
  automated
• Caveats
• Creative rate laws
• Events

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Events Need Extra Care
/deterministic events/ If (Agtthreshold) Then
event is triggered. (Here gt means
rising above a threshold)
/stochastic events/ If (Altminimum) Then
minimumA If (minimumltcertain low value AND
Agtthreshold) Then event is triggered
minimumA. (we ask for A truly rising from a
low value, not happening to rise by oscillation.)

• Except for events, all other parts of the model
  are automatically converted by JigCell.

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Conversion Tool

• Part of the JigCell modeling suite
• Automatically checks unit consistency inside the
  model
• Every two quantities (a parameter, a species, or
  the result of a sub-expression) connected by or
  - in the rate law equation must have same units.
• All species whose values are changed by the same
  reaction must have the same units.
• The unit of the result from the rate law equation
  must be equal to the unit of the reaction rate.

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The Tool Entering the Data
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The Tool Error Checking
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The Tool Error Correction
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The Tool Results Reactions
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The Tool Results Unit Types
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Simulation Experiments Setup

• Model
• A simplified cell cycle model
• A full-sized budding yeast cell cycle model
• Data
• 38 of 45 species in full-sized model use
  realistic characteristic concentration found in
  the literature.
• Cell volume is set to 50fL.
• Simulator
• StochKit, a C stochastic simulator integrated
  into JigCell, running SSA.

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Distribution of Species on Converted Simplified
Model

• Ensemble result of 10,000 simulations at 200
  minutes simulation time.

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Simulations on the Converted Full-sized Model

• The same model (except events) can be simulated
  either deterministically or stochastically
• The interesting cases are where they do not agree

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Mass at Birth, Full-sized Model

• Mean 1.20, CV 2.96. (Compared with 1.21 from
  deterministic simulation)

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Variance of Mass at Birth vs. Simulation Time vs.
Population
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Simulation Times
Stochastic Time Stochastic Time Stochastic Time Deterministic Time
Model Wall Total Avg./run Deterministic Time
Simplified 145 12305 1.23 0.029
Full-sized 3862 382267 38.2 0.311

• Even a single run of the stochastic simulation
  takes much more time than the deterministic
  simulation.
• Parallel computing is needed and feasible.

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Effect of Random Number Generators

• SPRNG

• random()

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Conclusions

• Improved support for the conversion process
• The JigCell conversion tool
• Deterministic and stochastic formulations are not
  fundamentally different
• Deterministic modelers like to take short cuts
• Real experience with stochastic simulations on
  meaningful models
• Events
• Runtimes
• Approximation results

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Future Work

• Initial conditions distribution
• Truly growing volume
• Our previous model had growing mass but fixed
  volume, which is not realistic
• Change to growing volume will change the reaction
  rate (propensity function)
• Simulations on mutants of particular interest