Cybergenetics: Control theory for living cells

Department of Biosystems Science and Engineering, ETH-Zürich Cybergenetics: Control theory for living cells Corentin Briat Joint work with Ankit Gupta and Mustafa Khammash

Introduction Overview Cybergenetics: Control theory for living cells at the genetic/molecular level Endogenous (analysis) Synthetic (design)

Introduction Overview Cybergenetics: Control theory for living cells at the genetic/molecular level Endogenous (analysis) Synthetic (design) System: single-cell vs. cell population Single-cell control: control of the abundance of a molecule of interest inside a given cell Population control: control of the abundance of a molecule of interest across a cell population; e.g. the mean value

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 2 / 18 Introduction Overview Cybergenetics: Control theory for living cells at the genetic/molecular level Endogenous (analysis) Synthetic (design) System: single-cell vs. cell population Single-cell control: control of the abundance of a molecule of interest inside a given cell Population control: control of the abundance of a molecule of interest across a cell population; e.g. the mean value Controllers: in-silico controllers vs. in-vivo controllers in-silico controller: the controller is implemented inside a computer/microcontroller in-vivo controller: the controller is implemented inside the cell These are different problems with different challenges, especially theoretical ones!

Contents 1 Biology, models in biology and noise 2 Regulation in biology 3 Cybergenetics

A bit of (simplified) biology... Central Dogma of Biology DNA: support of genetic information (ATGC) Transcription Messenger RNA (mrna): convey genetic information to the ribosomes Translation Protein: functional 3D chain of amino acids; e.g. insulin A control problem How to act on transcription in order to regulate the protein abundance? }{{}}{{} control input controlled output

Reaction networks A reaction network consists of... A set of distinct species X state A set of reactions R involving the species in X dynamics Examples (Bio)chemistry: X = {distinct molecular species} and R = {reaction rules} Ecology: X = {animal/vegetal species and resources} R = {predation and reproduction rules} Epidemiology: X = {illness state of individuals} R = {disease propagation and recovery rules} Multi-agent systems: X = {all the distinct agents} R = {communication and update rules} Communication networks: X = {users, links and queues} R = {communication, transmission and protocol update rules}

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 5 / 18 Reaction networks A reaction network consists of... A set of distinct species X state A set of reactions R involving the species in X dynamics Gene expression model X = {Gene, mrna, P} R = {R 1, R 2, R 3, R 4 } The reaction network is given by R 1 : Gene R 2 : mrna R 3 : mrna R 4 : P k m Gene + mrna (Transcription) γ m φ (Degradation/dilution reaction) k p mrna + P (Translation) γ p φ (Degradation/dilution reaction) where the parameters k m, γ m, k p, γ p are the reaction rates. In particular, k m is the transcription rate and k p is the translation rate.

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 6 / 18 Models for reaction networks Deterministic models Works well when the populations are in high-copy number; e.g. in Avogadro number 6.02 10 23 (in chemistry for instance) Concentration of the molecular species well-defined continuous real-valued Model formulated as systems of ODEs, DDEs or PDEs Analysis and control can be done using well-established methods Simulation tools are well-developed, accurate and cheap (at least for moderate sizes)

Models for reaction networks Deterministic models Works well when the populations are in high-copy number; e.g. in Avogadro number 6.02 10 23 (in chemistry for instance) Concentration of the molecular species well-defined continuous real-valued Model formulated as systems of ODEs, DDEs or PDEs Analysis and control can be done using well-established methods Simulation tools are well-developed, accurate and cheap (at least for moderate sizes) Stochastic models When the population are in low-copy number randomness in the interactions cannot be neglected anymore noisy dynamics Concentrations are ill-defined molecular counts discrete integer-valued variable Stochastic models are then necessary: jump Markov processes (CTMC) (+ some approximations) Analysis are control techniques are scarcer, yet some exist In the context of reaction networks, novel analysis and control tools are needed Simulation tools exist but large models (yet small for biology) are intractable

Noise in biology Some clarifications Randomness (Math) = Noise (Biology)

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 7 / 18 Noise in biology Some clarifications Randomness (Math) = Noise (Biology) Noise is intrinsic to living organisms and their evolution Mutations are random (at least as understood so far) Phenotypical randomness two genetically identical individuals can look different Molecular counts within a single cell can be low ( 10) intrinsic noise High cell-to-cell variability extrinsic noise We focus here on noise in the firing of the reactions (and no extrinsic noise) Is noise detrimental or is it beneficial? Or even both?

Noise in biology Some clarifications Randomness (Math) = Noise (Biology) Noise is intrinsic to living organisms and their evolution Mutations are random (at least as understood so far) Phenotypical randomness two genetically identical individuals can look different Molecular counts within a single cell can be low ( 10) intrinsic noise High cell-to-cell variability extrinsic noise We focus here on noise in the firing of the reactions (and no extrinsic noise) Is noise detrimental or is it beneficial? Or even both? In summary Noise allows to achieve certain difficult functions in a robust way; e;g. oscillations, switching, amplification, entrainment, etc. We refer to them as noise-induced properties as they are absent from deterministic counterparts Is there any such noise-induced property in regulation? Sounds rather paradoxical...

Contents 1 Biology, models in biology and noise 2 Regulation in biology 3 Cybergenetics

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 9 / 18 Regulation, Integral action, Homeostasis and Perfect adaptation Regulation in control systems Regulation in control theory is the ability of a controlled system to remain at a pre-specified constant steady-state despite the presence of (constant) disturbance It is known that integral action is necessary to ensure such a property Homeostasis Homeostasis is the ability of living organisms to regulate their internal state Perfect adaptation is the property that molecular levels will return back to their pre-stimuli value after the appearance of a perturbing stimulus Regulation motifs hence naturally exist in living cells How do they relate to integral control?

Endogenous regulation motifs for perfect adaptation Negative feedback loop motif Incoherent feedforward motif Comments They can both ensure (perfect) adaptation In the perfect adaptation case integral control Several endogenous circuits have been identified Negative feedback: Bacterial chemotaxis, Incoherent FF: But... Paradigm problem: regulation circuits are indissociable from the system Unclear whether this would work for controlling more general pathways Does not work if we change the input node or if the input acts on the edges Stochastic setting?

Contents 1 Biology, models in biology and noise 2 Regulation in biology 3 Cybergenetics

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 12 / 18 Antithetic integral feedback Open-loop network A reaction network (X, R) X 1 is the actuated species Y = X l is the controlled species

Antithetic integral feedback Open-loop network A reaction network (X, R) X 1 is the actuated species Y = X l is the controlled species Control objective (roughly) Find a set of additional species Z and additional reactions R c that acts on the actuated species X 1 in a way that the controlled species Y robustly tracks a desired set-point µ/θ (in a certain way). Antithetic integral controller R c θ 1 : Y Y + Z 2 (Measurement reaction) R c µ 2 : φ Z 1 (Reference reaction) R c η 3 : Z 1 + Z 2 φ (Comparison reaction) R c k 4 : Z 1 Z 1 + X 1 (Actuation reaction) where θ, µ, η, k > 0 are the controller parameters.

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 13 / 18 Antithetic integral feedback Standard integral control System

Antithetic integral feedback Standard integral control System Antithetic integral control (AIC)

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 14 / 18 Results Deterministic case Works if the open-loop network is (locally) asymptotically stable and (locally) output controllable, the set-point µ/θ is achievable, and the rate parameters k, η > 0 are chosen appropriately Tracking is at the level of concentrations at both the single-cell and population levels Mathematically proven for a wide-class of biochemical systems

Results Deterministic case Works if the open-loop network is (locally) asymptotically stable and (locally) output controllable, the set-point µ/θ is achievable, and the rate parameters k, η > 0 are chosen appropriately Tracking is at the level of concentrations at both the single-cell and population levels Mathematically proven for a wide-class of biochemical systems Stochastic case Works if the open-loop network is ergodic and globally output controllable, and the set-point µ/θ is achievable Works for all k, η > 0 noise-induced property!! Tracking is performed at the level of mean protein numbers at both the single-cell and the population levels. Mathematically proven for the class of unimolecular reaction networks, the more general case has yet to be proved (not easy)

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 15 / 18 Example - Control of gene expression Deterministic control problem Under what conditions the (unique) equilibrium point of the closed-loop reaction network is locally asymptotically stable and the protein concentration exhibits tracking and perfect adaptation? Stochastic control problem Under what conditions the closed-loop reaction network is ergodic and the protein mean over the cell-population exhibits tracking and perfect adaptation?

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 16 / 18 Simulation results - Gene expression network 15 10 15 10 15 10 40 30 20 5 5 5 10 0 0 20 40 60 0 0 10 20 30 40 50 0 0 10 20 30 40 50 0 0 200 400 600 800 1000

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 17 / 18 Discussion One of the first results having this Cybergenetic flavor/spirit Reaction-network implementation Modularity Genericity This network has a noise-induced property that is (almost) independent of the controlled system Extremely important because exactly implementing reactions is difficult in practice A lot of things remain to be done/improved Model: modeling, parameter estimation, model discrimination, sensitivity analysis, optimization, simulation Analysis: mathematical and numerical techniques Control: different classes of controllers, different objectives, etc. Filtering: noise filtering/compensation, observation of hidden variables Implementation...

Corentin Briat, D-BSSE@ETH-Zürich ACCESS Alumni Day 2016 KTH, Stockholm, Sweden 18 / 18 Thanks everyone for listening Any questions?