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

The Fundamental Goal of Molecular Cell Biology

The Cell Cycle

Cell Cycle Control Mechanism

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

]A[]B[

cdt

d

Simulation: Budding Yeast Cell Cycle

Table 6. Properties of clb, sic1, and hct1 mutants

mass at birth

mass at

SBF 50%

mass at

DNA repl.

mass at bud ini.

mass at division

TG1

(min)

changed

parameter

Comments

1 wild type

(daughter) 0.71 1.07

(71’) 1.15 (84’)

1.15 (84’)

1.64 (146’)

84 CT 146 min (time of occurrence of event)

2 clb1 clb2

0.71 1.07 1.16 1.16 No mit k's,b2 = 0

k"s,b2 = 0 Surana 1991 Table 1, G2 arrest.

3 clb1 clb2

1X GAL-CLB2 0.65 1.10 1.19 1.19 1.50 105 k's,b2 = 0.1

k"s,b2 = 0 Surana 1993 Fig 4, 1X GAL-CLB2 is OK, 4X GAL-CLB2 (or 1X GAL-CLB2db) causes telophase arrest.

4 clb5 clb6 0.73 1.07

(65’) 1.30 (99’)

1.17 (80’)

1.70 (146’)

99 k's,b5 = 0 k"s,b5 = 0

Schwob 1993 Fig 4, DNA repl begins 30 min after SBF activation.

5 clb5 clb6

GAL-CLB5 0.61 0.93 0.92 0.96 1.41 73 k's,b5 = 0.1

k"s,b5 = 0 Schwob 1993 Fig 6, DNA repl concurrent with SBF activation in both GAL-CLB5 and GAL-CLB5db.

6 sic1 0.66 1.00

(73’) 0.82 (37’)

1.06 (83’)

1.52 (146’)

38 k's,c1 = 0 k"s,c1 = 0

Schneider 1996 Fig 4, sic1 uncouples S phase from budding.

7 sic1 GAL-SIC1 0.80 1.07 1.38 1.17 1.86 94 k's,c1 = 0.1 k"s,c1 = 0

Verma 1997 Fig3B, Nugroho & Mendenhall 1994 Fig 2, most cells are viable.

8 hct1 0.73 1.08 1.17 1.18 1.69 82 k"d,b2 = 0.01 Schwab 1997 Fig 2, viable, size like WT, Clb2 level high

throughout the cycle. 9 sic1 hct1

0.71 No SBF 0.72 No bud No mit k's,c1 = 0

k"d,b2 = 0.01 Visintin 1997, telophase arrest.

10 sic1 GAL-CLB5

first cycle second cycle

0.71 0.52

0.74

0.73

No repl

0.76

1.20

k's,b5 = 0.1 k"s,b5 = 0 k's,c1 = 0

Schwob 1994 Fig 7C, inviable. First cycle OK, DNA repl advanced; but pre-repl complexes cannot form and cell dies after the first cycle.

Expermental Data

Putting it Together

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

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

Modeling Process

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

Gillespie’s 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

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?

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

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.

Conversion The relation between normalized concentration,

real concentration and population of a species:

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.

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

Events Need Extra Care

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

/*deterministic events*/

If (A>threshold)

Then {event is triggered}.

(Here “>” means rising above a threshold)

/*stochastic events*/

If (A<minimum)

Then {minimum=A}

If (minimum<certain low value AND A>threshold)

Then {event is triggered; minimum=A}.

(we ask for A truly rising from a low value, not happening to rise by oscillation.)

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.

The Tool: Entering the Data

The Tool: Error Checking

The Tool: Error Correction

The Tool: Results – Reactions

The Tool: Results – Unit Types

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.

Distribution of Species on Converted Simplified Model Ensemble result of 10,000 simulations at 200

minutes simulation time.

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

Mass at Birth, Full-sized Model Mean = 1.20, CV = 2.96%. (Compared with 1.21

from deterministic simulation)

Variance of Mass at Birth vs. Simulation Time vs. Population

Simulation Times

Stochastic Time Deterministic TimeModel Wall Total Avg./run

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.

Effect of Random Number Generators

SPRNG random()

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

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