Deterministic Global Parameter Estimation for a Budding Yeast Model

T.D Panning*, L.T. Watson*, N.A. Allen*,C.A. Shaffer*, and J.J Tyson+

Departments of Computer Science* and Biology+,Virginia Tech

Blacksburg, VA 24061

Outline

Application: Cell Cycle Modeling

Optimization Techniques: Dividing RECTangles (DIRECT) Mesh Adaptive Direct Search (MADS)

Computational Results

The Fundamental Goal of Molecular Cell Biology

Application:Cell Cycle Modeling

How do cells convert genes into behavior? Create proteins from genes Protein interactions Protein effects on the cell

Our study organism is the cell cycle of the budding yeast Saccharomyces cerevisiae.

mitosis(M phase)

DNA replication(S phase)

cell division

G1

G2

Modeling Techniques

We 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.

]A[]B[

cdt

d

Tyson’s Budding Yeast Model

Tyson’s 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 “Parameter twiddling” Far better is automated parameter estimation

Mutations

Wild type cell

Mutations Typically caused by gene knockout Consider a mutant with no B to degrade A.

Set c = 0 We have about 130 mutations

each requires a separate simulation run

Phenotypes

Each mutant has some observed outcome (“experimental” data). Generally qualitative. Cell lived Cell died in G1 phase

Model should match the experimental data. Model should not be overly sensitive to the rate

constants. Overly sensitive biological systems tend not to

survive

Transforms

The output from ODE solvers are time course data

Need to convert this to match the qualitative experimental data

We call the function that does this conversion a “transform”

Rules of Viability

1. Modeled cell must execute a series of events, in ordera) [Clb2] + [Clb5] drops below Kez2.b) [ORI] goes over 1 before two divisions of wild cell.c) [SPN] increases above 1.d) [Esp1] increases above 1.

e) [Clb2] drops below Kez.

2. Cell is inviable if [BUD] does not reach 0.8 before (e)3. Squared relative differences of masses and G1 phase

lengths in last two cycles is less than 0.5.4. Cell is inviable if mass has higher ratio than 4 to wild

cell.

Organizing the Observations

Budding yeast phenotype for a given mutant is defined by a 6-tuple (v, g, m, a, t, c).v is {viable, inviable}Real g > 0 is steady state length of G1 phaseReal m > 0 is steady state mass at divisiona (arrest stage) is {unlicensed, licensed, fired, aligned, separated}

Integer t > 0 is the arrest typeInteger c >= 0 is number of successful cycles

Define 6-tuples O (observed) and P (predicted).

Objective Function

We require a scoring mechanism to compare O and P for each mutant.

Objective Function (cont)

The constants are tuned such that a rating of around 10 for a given mutant is a critical error (the model effectively fails for that mutant)

Note that this objective function is not continuous

Optimization Techniques

143 parameters to optimize

DIviding RECTangles (DIRECT) Global optimization

Mesh Adaptive Direct Search (MADS) Local optimization

DIviding RECTangles (DIRECT)

Global optimization algorithm (Jones et al, 1993)Does not require gradient, but the convergence criteria does require the objective function to be continuousAt each iteration, subdivide boxes considered to be “potentially optimal” into three along their longest dimensionsThe algorithm can be tuned to favor “exploration” for good regions, or focus on known good regions

Potentially Optimal Boxes

Mesh Adaptive Direct Search (MADS)

A class of algorithmsAlternate SEARCH and POLL steps.All evaluated points are on a mesh, but the mesh can be adjusted each iteration.Each SEARCH step selects some points on the mesh to evaluate. If an improved point is found, MADS may jump directly to resizing the mesh. (GPS)If no better point is found in the SEARCH step, the POLL searches for a better point within a fixed distance (the frame) of the current best point. (Frame – Coope & Price)Resize the mesh up or down depending on success in the last iteration.

Mesh Adaptive Direct SearchMADS in action

Computational Results

All computation took place on Virginia Tech’s System X supercomputer

We used parallel implementations for DIRECT and MADS.

Experiment 1 MADS started from the modeler’s best point DIRECT used a box normalized around the

modeler’s best point

Experiment 1

MADS evaluated 135,000 points (813 iterations, 128 processors). Final objective function value was 299.

DIRECT evaluated 1.5M points (473 iterations, 1024 processors). Final objective value was 212.

Experiment 2

Looking at the results of Experiment 1, DIRECT does not progress much after 200,000 points.What if we start MADS from this point?What about other points on the plateau?Mixed results: The MADS runs starting at the beginning and end of

the plateau were worse than DIRECT’s best point. The MADS run starting in the middle of the plateau

was better than DIRECT’s best point. MADS made effectively no progress when starting

from DIRECT’s best point.

Contributions

Demonstrated that it is computationally feasible for search algorithms to improve on the modelers’ best point.The best points found by DIRECT were more stable than the modelers’.MADS can (sometimes!) improve on DIRECT when starting from DIRECT’s good points. The relationship is unclear.Demonstrated the relationship in DIRECT’s tuning parameter of tradeoff between splitting large boxes and refining small boxes.