Multistate Modeling and Simulation for Regulatory Networks

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MultistateModeling and Simulationfor Regulatory
Networks

• Zhen Liu, Clifford A. Shaffer, Umme Juka
  Mobassera, Layne T. Watson, and Yang Cao
• Department of Computer Science
• Program in Genetics, Bioinformatics, and
  Computational Biology
• Virginia Tech

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Goal Modeling the Cell Cycle(John Tyson)
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Regulatory Network Modeling

• Model using a series of chemical reactions.
• The actors are proteins (chemical species)
  whose interaction rates are modeled by rate laws
• Species are created, consumed, combined
• Populations can rise and fall, under the control
  of other species
• Loops and cycles

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Decomposition of Models

• Modelers find it natural to divide into bundles
  of reactions.

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Multistate Phosphorylation Motif

• Blocks relate to naturally occurring motifs
• Example antagonistic interaction between Clb2
  and Cdh1, with Cdc14 as the control variable
  driving phosphorylation of Cdh1
• Forms a bi-stable switch

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Multistate Version

• The reality is more complex, as a protein can
  undergo multiple levels of phosphorylation, which
  can affect the behavior of the larger system

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Multistate Modeling

• Equations on chemical species with multiple
  states, related in some meaningful way
• Expressing as single-state equations would
  require dozens of reactions.

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JigCell Model Builder Support
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Problems

• Complications arise from the potential
  combinatorial explosion of states in complexes
• Example Two multistate species each with 10
  states could form complexes with potentially 100
  states.
• Ai Bj -gt ABi,j
• This presents challenges to simulation.

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Stochastic Simulation

• Reaction models have often been modeled using
  ODEs
• Track concentrations of chemical species
• ODE models cannot account for stochastic effects
• Small numbers for some species (RNA)
• Variations in inputs gt Differing outputs
• Simulation ensemble gt Distribution

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Gillespies SSA (1)

• N molecular species S1, , SN.
• M reaction channels R1, RM.
• For reaction channel Rj
• Propensity function aj
• State change vector vj (v1,j, , vN,j)
• aj(x)dt gives probability that one Rj reaction
  will occur in next infinitesimal time interval
  given state vector x.

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Gillespies SSA (2)

• Select two random numbers r1 and r2
• Let a0(x) be the sum for all the reaction
  propensities on state vector x.
• Time for next reaction to occur is t t
• t 1/a0(x) log (1/r1).

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Gillespies SSA (3)

• Index j for next reaction is given by smallest
  integer satisfying
• S al(x) gt r2a0(x).
• System state updated after each reaction,
  including populations and propensities
• Observations
• A population-based simulation
• SSA calculates propensities for reactions

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Rule-Based Modeling

• A rule defines how a molecular particle reacts
  with other particles
• Aopen,?,? B ---gt AB,?,?
• Subscripts describe the matching configurations
  for binding sites
• Convenient for representation
• Updating propensities of rules faster(?) than
  updating propensities of reactions

krule
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Network-Free Algorithm (1)

• (Sneddon et al. 2008)
• Alternative to turning rules into collections of
  reactions and performing SSA.
• Conceptually similar to SSA, but
• Calculate propensities for rules.
• Particle based (not population based)
• Keep list of particles associated with each rule

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Network-Free Algorithm (2)

• Simulation loop
• Calculate propensity for each rule (cheaper than
  SSA)
• Calculate rule and time of next event
• Select particles from associated list
• Update the particle lists as necessary (major
  expense)

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Population-Based NFA (PNFA)

• (Our first contribution)
• Modification to NFA (go back to) using
  populations for single-state species
• Hybrid particle/population approach
• Attempts to cut down on the size of the lists
  associated with the rules
• Can be viewed as an optimization to NFA
• at worst degrades to NFA

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Full-Scale SSA (FSSSA) (1)

• (Our second contribution)
• Use populations even for multi-state species
• Should work well unless there is a small
  population spread across many states
• Can view as more direct conversion of SSA to
  rules (pure population-based approach)

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Full-Scale SSA (FSSSA) (2)

• For each species, store an array of populations
  (one for each state)
• Might be a sparse array
• Store with each rule the population count for all
  associated reactants

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Full-Scale SSA (FSSSA) (3)

• Simulation loop
• Calculate propensity for each rule (cheaper than
  SSA)
• Calculate rule and time of next event
• Select a state for each reactant from the
  population array
• Update populations of affected species (states)
  and population counts for associated rules (might
  require modifying arrays)

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Comparisons Selection

• SSA does linear search through reactions
• NFA, PNFA do linear search through rules, then
  select qualifying objects from associated
  reactant lists
• FSSSA does linear search through rules, only
  needs to search state lists (populations)

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Comparisons Update

• SSA updates populations of some reactions
  reactants and products
• NFA must create/destroy molecule objects, and
  update associate rule lists
• PNFA same, but does little work on single-state
  species populations
• FSSSA updates sparse matrix info.

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Bi-stable Switch Model

• Reaction-based form
• 12 species
• 44 reactions
• Rule-based form
• 1 single-state species, 1 multi-state
• 7 rules
• Non-zero populations in each state

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Simulation Times Switch
Total CPU Time Propensity Update Reactant Selection System Update Other
SSA 115 72.0 30.6 5.3 7.1
NFA 341 11.1 34.0 286.0 9.9
PNFA 246 9.9 26.2 200.8 9.1
FSSSA 117 9.2 32.4 66.2 9.2
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Cell Cycle Model

• Reaction-based form
• 58 species, 185 reactions
• Rule-based form
• 17 single-state species, 6 multi-state
• 64 rules
• Half the states have zero population
• Observation Affecting one multi-state species
  affects only a smaller fraction of all the rules

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Simulation Times Cell Cycle
Total CPU Time Propensity Update Reactant Selection System Update Other
SSA 171 143.3 23.5 1.4 2.8
NFA 133 36.4 20.4 72.5 3.7
PNFA 113 34.0 17.6 58.6 2.8
FSSSA 64 32.8 18.2 10.5 2.5
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Simulation Quality (1)
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Simulation Quality (2)

• This graph shows distribution of population for
  Clb2, one of the species in the cell cycle model.
• The significance is that it indicates that each
  simulation algorithm gives approximately the same
  ensemble of outputs.

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Complexity Analysis