Statistical Mechanics of SloppyModels

Bacterial Cell-CellCommunication and More

Josh Waterfall, Jim Sethna,Steve Winans

Quorum Sensing

•Cell-cell signaling network formonitoring population density

Low populationdensity:pheromonemolecule lost toenvironment.

High density:pheromone ispicked up fromneighbors andsignaling pathwayis activated.

Agrobacterium tumefaciens

plant pathogen -transfersoncogenic DNAto plant, co-opting plantmachinery tomake nutrients.

Quorum triggers sharingof tumor inducingplasmid

pheromone =

•TraI synthesizes small molecule (OOHL) at low, basal rate•TraR needs OOHL to fold properly•Active TraR turns on transcription of other genes, including traI•Hierarchically regulated by separate systems (opine and phenoliccompound metabolism)

OOHL

TraI

TraR

Genes

TraR

TraI

26 reactions19 chemicals24 rate constants

How to live with 24free parameters andstill make falsifiablepredictions???

24 Parameter “Fit” to Data• Cost _ Energy

• Also fit to six other sets ofgenetic and biochemicalexperiments (38 datapoints)

• Still misses data

• Hand-tuning, then fancyoptimizations

• How much can the fitparameters vary, and stillfit the data? (Will giveerror bars)

∑=

−=

RN

i i

iyyC1

2

2))((

2

1)(

σθ

θ

rr

TraR protein is stable only withOOHL

Rad

ioac

tivity

minutes 450

Ensemble of ModelsWe want to consider not just minimum cost solutions, but all

solutions consistent with the available data. New level ofabstraction: statistical mechanics in model space.

Generate an ensemble ofstates with Boltzmannweights exp(-C/T) andcompute for an observable:

O is chemical concentration, orrate constant …

222

1

)()(

)(1

θθσ

θ

rr

r

OO

ON

O

O

N

ii

E

E

−=

= ∑=

bare

eigen

•Huge range of scales from stiffto sloppy : 1 inch = 103 miles•Eigendirections not aligned withbare parameters

Wide range of natural scales

Eigenvalues – not allparameters are createdequal. Range of e20!

Sloppiness – fluctuations ineigenparameters in ensemble upto tens of orders of magnitude!

50

-20

Log Eigenvalues

Eigenparameter

Stiffest eigenparameterpredominantly bacterialdoubling time

Second stiffest adds ratiobetween production of TraRand degradation of TraR andOOHL

Biological insight from eigenparameters

Bare Parameter

Stiffest Eigenparameter2nd Stiffest Eigenparameter

Bare Parameter

Predictions for experiments are constrained

Growth factor signaling model

Although rateconstant valuesare wildlyundetermined,predictions fornew experimentsare not (always).

Other Sloppy systems past, present, future

• Growth Factor Signaling

• Receptor Trafficking

• Translation Dynamics

• Transcription Dynamics

• E-Coli Whole Cell Model

• Nitrogen Cycle in Forests

• Radioactive decay

• Classical Potentials: Molybdenum

Acknowledgements

Biology:Stephen WinansCathy White

Modeling:Jim SethnaKevin Brown

Funding:NSF ITRDOE Computational Science GraduateFellowship