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) = A 0. Parameter c determines the rate of degradation.

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 order a) [Clb2] + [Clb5] drops below K ez2. 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 K ez. 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 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 phase Real m > 0 is steady state mass at division a (arrest stage) is {unlicensed, licensed, fired, aligned, separated} Integer t > 0 is the arrest type Integer 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 continuous At each iteration, subdivide boxes considered to be “potentially optimal” into three along their longest dimensions The 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 algorithms Alternate 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 Search MADS 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.