Download Deterministic Global Parameter Estimation for a Budding

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.
cell division
mitosis
(M phase)
G1
G2
DNA replication
(S phase)
Modeling Techniques
We use ODEs that describe the rate at which each
protein concentration changes

Protein A degrades protein B:
d [ B]
 c[ A]
dt
… with initial condition [A](0) = A0.
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
Modeled cell must execute a series of events, in order
1.
a)
b)
c)
d)
e)
2.
3.
4.
[Clb2] + [Clb5] drops below Kez2.
[ORI] goes over 1 before two divisions of wild cell.
[SPN] increases above 1.
[Esp1] increases above 1.
[Clb2] drops below Kez.
Cell is inviable if [BUD] does not reach 0.8 before (e)
Squared relative differences of masses and G1 phase
lengths in last two cycles is less than 0.5.
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.