Hybrid systems methods for biochemical networks. Adam Halasz. Outline. Hybrid systems, reachability Piecewise affine approximations of biochemical systems Example I: Glucose-lactose Example II: Tetracyclin resistance. Biomolecular networks as hybrid systems.

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Hybrid systems methods for biochemical networks

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Outline • Hybrid systems, reachability • Piecewise affine approximations of biochemical systems • Example I: Glucose-lactose • Example II: Tetracyclin resistance

Biomolecular networks as hybrid systems • Networks of chemical and molecular processes • State = {values of all concentrations} • Rates of each process are continuous functions of the state • Several layers of processes, different timescales • State space can be huge (O(103) variables for one cell) • Lots of truly discrete behavior: • Genes on/off • Discrete variables • Lots of apparent discrete behavior • Nontrivial continuous dynamics produces multistability, bifurcations • Abstractions – commonly used and/or required for simplification

Biomolecular dynamical systems • Central dogma of molecular biology • DNA encodes genes; it replicates • Genes are transcribed into mRNA • mRNA is translated into proteins • Proteins may: • act as enzymes that catalyze metabolic reactions • act as transcription factors • Metabolic reactions • big network that converts incoming nutrients into useful substances and by-products • reactions proceed much faster when the right enzymes are available

DNA replicates during cell division Transcription performed by RNA polymerase Requires a promoter site Several genes bundled to one promoter = operon In higher organisms, mRNA is spliced Translation performed by Ribosomes Protein synthesis needs raw material The Central Dogma

Genes to proteins • Proteins are synthesized as chains of elementary proteins, amino-acids • They fold, giving rise to complicated 3d structures • Several molecules may be assembled into more complicated ‘machines’, such as RNAP, ribosomes, etc.

Very complex Structured Stoichiometry is more easily identified than rate laws Many networks available in databases, e.g. Kegg Reactions linked to individual genes Lots of feedback Metabolic network

Metabolic network has a lot of control • Feedback between • Metabolites • Genes and proteins • Continuous adjustment to external conditions • Signaling networks • Control is through rate laws, but also through stochastic mechanisms

Hybrid systems • much of the underlying dynamics is continuous, but.. • complexity and lack of detailed kinetic information require the use of hybrid abstractions

Hybrid systems • Two topics to be addressed: • How to build a good hybrid abstraction • How to analyze a network that includes hybrid abstractions

mRNA β-gal perm repressor Allo- Lactose Lactose External Lactose Using hybrid systems abstractions to build hybrid systems abstractions • The lac operon is a bistable genetic switch • Multiple positive feedback bistable • Input: external lactose • State: x={M,B,A,L,P}

HIGH I=1 LOW I=0 Using hybrid systems abstractions to build hybrid systems abstractions • May be abstracted to an automaton: • Input: external lactose • State: {I} • The characteristic still depends on the underlying kinetic parameters!

Reachability • The full lac model can be simulated to investigate induction, but that can be expensive • The question of whether induction is possible may be framed as a reachability problem • Many other situations with discrete outcomes are amenable to reachability Initial • Question 1: • which ones end up in a viable final state? • Question 2: • which ones survive? Final Irreversible damage

Example Kinetics 1 • Dynamic models have a special structure! • More generally,

y1 The vector field is a unique (affine) function of the vectors at the end points Kinetics 1 (continued) y x

Kinetics 1 (continued) The vector field is a unique function of the vectors at the vertices [Belta, Habets, Kumar 2002]

Hybrid System Rectangular partitions Affine dynamics Kinetics (2) Transcription rate Concentration of repressor Transcription rate Concentration of allolactose

Piecewise affine approximation Simplest approximation with two affine pieces Can use any number, to achieve any desired precision

Piecewise is hybrid Piecewise approximation has different equations in each interval Transitions occur as the variable switches intervals

Several substrates that saturate Piecewise approximation has different equations in each interval Transitions occur as the variable switches intervals Can continue in many dimensions

Abstraction • Model the biochemical network as a switched system with continuous multi-affine dynamics • Each mode has simple dynamics • More insight • Approximation may be refined as needed • Partition may be refined independently of dynamics • No additional computational difficulties • Traditional simulations are easier • Efficient reachability algorithms can be applied

Reachability analysis • Can the system reach a set of states starting from a set of initial conditions?

Hybrid System Analysis • Reachability • Cell A is reachable from cell B if there is at least one trajectory from B to A • Cell A is not reachable from cell B if there are no trajectories from B to A

Glucose-lactose system • The lactose metabolism is self-nourishing: • The cell needs enzymes for: • Inbound lactose transport (permease) • Lactose processing (ß-galactosidase) • Permease and ß-galactosidase are gene products of the lac operon • Lac operon is repressed in the absence of allolactose • Allolactose is produced when lactose is processed • Bistability: • a low and a high lactose metabolism state • induction needed to move into the high state

Lac system in E.coli mRNA β-gal perm repressor Allo- Lactose Lactose External Lactose

Lac system in E.coli • Crucial switching property, sensitive to basal rate • Can be framed in terms of reachability

Lac system in E.coli • Hybrid model constructed using a fine grained linearization of the nonlinear rate laws • Predictions of the two models are very similar • Hybrid model within 5% uncertainty of model parameters

Glucose-lactose system • Lactose is an alternative energy source • Glucose is the preferred nutrient; bacteria also grow on lactose, but only when glucose is absent • There are two mechanisms that ensure this: • Inducer exclusion • Catabolite repression

mRNA perm b-gal Lac repressor External Lactose Allo- Lactose Lactose Glucose inhibits the influx of lactose CAP cAMP External Glucose CAP competes with lac repressor, enhancing transcription cAMP is produced when glucose is absent

Induction and reachability • Expect the vicinity of zero to be confined when system is bi-stable … unless it is induced byincreasing Le, decreasing Ge, or both … unless it is induced byincreasing Le, decreasing Ge, or both Suppose initially the system is at zero allolactose. Then it will have to settle on the lower sheet..

Induction and reachability • Up-switching possible if (Le,Ge) outside the bistable region for some time Upward switching trajectories Final, induced state Initial state, close to zero

Induction and reachability B • Follow trajectories in state space • Induced trajectories leave the vicinity of the initial state A

Induction and reachability B • Induced trajectories leave the vicinity of the initial state • For reachability, only need to cover the vicinity • Verify those configurations that do not leave the grid A

Reachability results • Bistable regions are non-inducible, hence they reach only the lower A values • Calculate highest Allolactose (A) reached • Sweep for (Le,Ge)

Reachability results • Non-inducible region should match the footprint of bi-stability

Analyzing networks of hybrid abstractions • The lac switch is one piece in a potentially huge circuit, which has both discontinuous and continuous elements • A “true” of hybrid system: • Discontinuous dynamics • Different state variables • Filippov states! • Hierarchy of modes!

Networks of hybrid abstractions • Continuous part of state space is still a set of concentrations • Dynamics is still given by reaction rates • Reaction rates are given by discontinuous functions of the state variables:

Networks of hybrid abstractions • Partition of continuous part of state space along threshold values • Boundaries treated as separate modes • Discrete transition system • Model checking

Summary • Molecular biology offers many instances of ‘natural’ hybrid systems • Very large state spaces, thousands of substances • Complex networks, nonlinear equations • Switching and other discontinuous behavior • Genes on/off • Multistability, bifurcation • Hybrid abstractions • Two aspects: • Constructing hybrid abstractions • Analyzing networks a hybrid systems • Both directions work towards automated analysis

Reading • Calin Belta – Boston U. • Hidde de Jong – INRIA Rhone-Alpes, FR/EU • Claire Tomlin – Berkeley • Ashish Tiwari – SRI, Palo Alto, CA • Joao Hespanha – Santa Barbara • V. Kumar, O. Sokolsky, G. Pappas, A. Julius, A. Halasz – U. Penn

Hybrid systems, reachability • Piecewise affine approximations of biochemical systems • Example I: Glucose-lactose • Example II: Tetracyclin resistance