Towards Systems Biology, October 2007, Grenoble Adam Halasz Ádám Halász joint work with Agung Julius, Vijay Kumar, George Pappas Mesoscopic and Stochastic Phenomena and the Lac operon

Towards Systems Biology, October 2007, Grenoble Adam Halasz Outline Lactose induction in E. coli, an example of bistability Stochastic phenomena in reaction networks Mesoscopic effects in the lac operon Outlook

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: Biological phenomenology lac Zlac Ylac A lac I repressor mRNA permease external lactose internal lactose allolactose  galactosidase

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: ODE model External lactose Lactose L Allolactose A β-galactosidase B Permease P mRNA M Network of 5 substances Example of positive feedback in a genetic network discovered in the 50’s Described by differential equations which are built from chemical rate laws Some time delays and time scale separations ignored and/or idealized

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: ODE model Network of 5 substances Example of positive feedback in a genetic network discovered in the 50’s Described by differential equations which are built from chemical rate laws Some time delays and time scale separations ignored and/or idealized External TMG T e TMG T β-galactosidase B Permease P mRNA M

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: ODE model Because of the positive feedback, the system has an S-shaped steady state structure That is to say, for some values of the external inducer concentration (T e ), there are two possible stable steady states P in P out T external B equilibrium TeTe B P

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: Bistability Switching and memory –Need to clear T 2 in order to switch up TeTe B T1T1 T2T2 B TeTe t t

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: Bistability Switching property is robust –Model parameters perturbed by 5%

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system – first lesson Network of <5 species involved in reactions Reactions decomposed * into mass action laws Can be implemented using simple stochastic transition rules eg: “upon colliding with a B, A becomes C with probability x” Recipe for synthesis of switch with hysteresis Compose motifs to build logical functions,… –Design the network and transition rules –Map the rules to individual stochastic programs

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: ODE is not enough

Towards Systems Biology, October 2007, Grenoble Adam Halasz Stochastic versus deterministic Substances are represented by finite numbers of molecules Rate laws reflect the probability of individual molecular transitions The abstraction/limit process is not always trivial t1t1 N1N1 t2t2 N 2 =N 1 -2 Next transition time distribution: :

Towards Systems Biology, October 2007, Grenoble Adam Halasz Chemical reactions are random events A B A + BAB A B

Towards Systems Biology, October 2007, Grenoble Adam Halasz Poisson process Poisson process is used to model the occurrences of random events. Interarrival times are independent random variables, with exponential distribution. Memoryless property. event time

Towards Systems Biology, October 2007, Grenoble Adam Halasz Stochastic reaction kinetics Quantities are measured as #molecules instead of concentration. Reaction rates are seen as rates of Poisson processes. A + B  AB k Rate of Poisson process

Towards Systems Biology, October 2007, Grenoble Adam Halasz Stochastic reaction kinetics reaction time A AB reaction

Towards Systems Biology, October 2007, Grenoble Adam Halasz Multiple reactions Multiple reactions are seen as concurrent Poisson processes. Gillespie simulation algorithm: determine which reaction happens first. A + B  AB k 1 k 2 Rate 1Rate 2

Towards Systems Biology, October 2007, Grenoble Adam Halasz Multiple reactions reaction 1 time A AB reaction 2 reaction 1

Towards Systems Biology, October 2007, Grenoble Adam Halasz Gillespie simulations When the number of molecules per cell is small*, the respective substance has to be treated as an integer variable The probabilistic transition rules can be implemented in standard ways Gillespie method: instead of calculating time derivatives, we calculate the time of the next transition Many other sophisticated methods exist. As an empirical rule, the higher the number of molecules, the closer the simulation is to the continuous version

Towards Systems Biology, October 2007, Grenoble Adam Halasz Gillespie automata The state of the system is given by the number of copies of each molecular species Transitions consist of copy number changes corresponding to elementary reactions The distribution of the next transition time is Poisson, e -kt where k is the propensity A Gillespie automaton is a mathematical concept [a continuous time Markov chain] Plays the same role differential equations have in the continuum description

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system: spontaneous transitions both ways External TMG concentration mRNA concentration Time (min) # mRNA molecules Increase E

Towards Systems Biology, October 2007, Grenoble Adam Halasz Mixed Gillespie/ODE 1000 cells Aggregate simulations

Towards Systems Biology, October 2007, Grenoble Adam Halasz We create a simplified model, a continuous time Markov chain with two discrete states, high state and low state The transition rates depend on the external concentration of TMG Two state Markov chain model Lo Hi (Julius, Halasz, Kumar, Pappas, CDC06, ACC07)

Towards Systems Biology, October 2007, Grenoble Adam Halasz Transition rates Identified transition rates (Monte Carlo)

Towards Systems Biology, October 2007, Grenoble Adam Halasz Two state Markov chain model Average of a colony with 100 cells Time (min) # mRNA molecules E[M ] t

Towards Systems Biology, October 2007, Grenoble Adam Halasz Lac system – second lesson Bad news: underlying stochasticity can drastically modify the ODE prediction –The price paid for an ‘uncontrolled’ approximation Good news: the ODE abstraction can also be performed rigorously –Gillespie automaton is an exact abstraction –For Gillespie  ODE, all ‘molecule’ numbers must be large Stochastic effects retained at the macro-discrete level –Effects are reproducible and quantifiable –Further abstractions of stochastic effects are possible Lac example: can quantify the spontaneous transitions: –Choose an implementation where they are kept at a low rate –Implement control strategies that use the two-state model

Towards Systems Biology, October 2007, Grenoble Adam Halasz Beyond Gillespie Gillespie method is ‘exact’ – produces exact realisations of the stochastic process Main problem is computational cost –Larger molecule numbers –Rare transitions Several approaches to circumvent ‘exact’ simulations

Towards Systems Biology, October 2007, Grenoble Adam Halasz Towards the continuum limit  – leaping: lump together several transitions update molecule numbers at fixed times: r1 time AB r2 r1 r2 A 

Towards Systems Biology, October 2007, Grenoble Adam Halasz Towards the continuum limit Error introduced by  – leaping is due to variation of the propensities over the time interval (may lead to negative particle numbers!) Acceptable if the expected relative change of each particle number over Δ is small (e.g. if the number of particles is large) If the number of transitions per interval is also large, the variation can be described as a continuous random number  stochastic differential equations Finally, is the variance of the change per interval can also be neglected, the simulation is equivalent to an Euler scheme for an ODE.

Towards Systems Biology, October 2007, Grenoble Adam Halasz Other limiting cases If the number of all possible configurations is relatively small, probabilities for each state can be calculated directly, by calculating all possible transition rates, (finite state projection) or using the master equation (Hespanha, Khammash,..) In some situations (eg. signaling cascades) there is a combinatorial explosion of species, where agent- based simulations are useful (Los Alamos group, Kholodenko)

Towards Systems Biology, October 2007, Grenoble Adam Halasz Transitions in the lac system We used tau-leaping for our simulations The high state can be simulated using SDEs or ‘semiclassical’ methods The lower state can be studied using finite state projection Time (min) # mRNA molecules

Towards Systems Biology, October 2007, Grenoble Adam Halasz Summary Mesoscopic effects in biological reaction networks are due to small numbers of molecules in individual cells They may affect the system dramatically, somewhat, or not at all These effects can be described mathematically and incorporated in our modeling efforts Several sophisticated methods exist; it is important to use an approximation that is appropriate, both in terms of correctness and in terms of efficiency

Towards Systems Biology, October 2007, Grenoble Adam Halasz Thanks: Agung Julius,, George Pappas, Vijay Kumar, Harvey Rubin DARPA, NIH, NSF, Penn Genomics Institute

Towards Systems Biology, October 2007, Grenoble Adam Halasz Control objective Regulate colony behavior, c.q. fraction of induction

Towards Systems Biology, October 2007, Grenoble Adam Halasz Control system design fraction of induction CONTROLLER ext. TMG reference input

Towards Systems Biology, October 2007, Grenoble Adam Halasz Control system design

Towards Systems Biology, October 2007, Grenoble Adam Halasz Beyond modeling: control Initial condition 100% induction Initial condition 0% induction

Towards Systems Biology, October 2007, Grenoble Adam Halasz Beyond modeling: control