Robustness of Intracellular Oscillators and Bistable Switches Elling

Robustness of Intracellular Oscillators and Bistable Switches Elling W. Jacobsen Automatic Control Lab School of Electrical Engineering KTH

Outline • The role of autonomous oscillations and multistability in cellular biology – some examples • Models and uncertainty / perturbations - parametric vs structural perturbations • Quantifying robustness and identifying fragilities using distance to structural instability • Applications - two minimal gene regulatory networks - metabolism of activated white blood cells - mammalian circadian clock - B cell differentiation • Conclusions February 26, 2021

Circadian Clocks • provides periodic control of biological activity • autonomous oscillations in mammalian SCN generated by a network of interacting genes

The Segmentation Clock • Formation of somites in vertebrate segmentation are driven by an autonomous gene regulatory oscillator February 26, 2021

Oscillatory Calcium Signaling • Amplitude and frequency modulated Ca 2+ oscillatory signals controls a wide variety of intra- and intercellular processes February 26, 2021

Metabolic Oscillations in Activated Neutrophils • Neutrophils make up first line of defense against bacterial infections • oscillatory production of lethal chemicals. Hypothesis: provide peaks that kill bacteria without harming the cell itself February 26, 2021

The lac operon and phage lambda switches • lac genes turned on when glucose low and lactose available • positive feedback loop • first experimental evidence of intracellular bistability (1957) • Phage ¸ virus infecting a cell switches between Lytic and Lysogenic states through a bistable switch • A double negative feedback generates switch February 26, 2021

Apoptosis – Programmed Cell Death • The death decision in programmed cell death corresponds to a bistable switch February 26, 2021 • ~50 billion cells/day undergo apoptosis in a healthy adult human

Cell Cycle Control • Gene regulatory networks provide bistable switches that control transitions between the various phases of the cell cycle February 26, 2021

Blood Stem Cell Differentiation • Antigens stimulate differentiation of B cells into antibody secreting cells through a bistable switch

Intracellular Oscillators and Switches • Autonomous oscillators and multi-/bistable switches are frequently observed in the cell; biology utilizes strongly nonlinear phenomena • An important aim of systems biology is to determine the architecture of the underlying biochemical networks • Modeling: typically bottom-up, ODEs • Robustness analysis important for - model (in)validation - determining essential/non-essential components and interactions - identifying fragilities - ultimately elucidating the principles behind biological robustness February 26, 2021

A minimal gene oscillator Bifurcation diagram February 26, 2021

A minimal gene switch State space February 26, 2021 Bifurcation diagram

Robustness Analysis • How robust are these behaviors? - quantitative properties - persistence of behavior • Perturbations? - reflecting real biological uncertainty - model uncertainty; parametric, structural February 26, 2021

Perturbations / Uncertainty • For analysis of biological robustness, perturbations should reflect true biological uncertainty - external disturbances (environment) - internal perturbations (gene mutations) - internal noise (stochastic fluctuations) • For modeling purposes, perturbations should - reflect model uncertainty (uncertain kinetic models, uncertain components, …) - provide mechanistic insight February 26, 2021

Parametric vs Structural Perturbations • Parametric uncertainty: • Structural uncertainty: R - “standard” uncertainty description in biological models - mainly motivated by the fact that parameters are fitted to experimental data February 26, 2021 Pc Pn - ¢ represents a dynamic perturbation of the direct interactions between biochemical components - allows for uncertain strength/dynamics of interactions, uncertain number of nodes and edges - motivated by the fact that involved components usually highly uncertain, standardized description

Structural Robustness Analysis R Pc Pn • Consider persistence of qualitative behavior ) determine smallest perturbation that makes system structurally unstable • Assume nominal model possesses a steady-state and consider perturbations that translates it to a bifurcation point - sustained oscillations: translate steady-state to Hopf point - bistability: translate steady-state to saddle-node February 26, 2021 • Linear problem: write network on feedback form and apply Small Gain Theorem

Network on Feedback Form • Taylor expansion - Jacobian A determines structural stability - 2 nd and 3 rd order terms B and C determines type of bifurcation, e. g. , sub- or supercritical • Write the linear part on feedback form where February 26, 2021 is stable

Perturbing the Network The smallest distance to a structurally unstable steady-state is given by the robustness radius Where and ¢ in general will be a structured matrix • The corresponding dynamic perturbation February 26, 2021

Perturbing the activity of all network nodes • ¢ is a diagonal n £ n matrix • The robustness radius quantifies overall robustness of network behavior February 26, 2021

Perturbing the activity of single nodes • ¢ is a scalar • February 26, 2021 provides information on the role of single components

Perturbing individual edges • ¢ij is a scalar • February 26, 2021 provides information on the role of specific interactions

Comments • All perturbations are relative • Complex perturbations ¢ ) tight bounds for in general be computed (for moderate n) can • There may exist smaller perturbations that induce bifurcations away from considered steady-state. • Thus, can in principle only identify non-robust features February 26, 2021

Minimal Gene Oscillator – overall robustness • , i. e. , a 2% change in the component activities sufficient to translate steady-state into HB point

Minimal Gene Oscillator – perturbing single nodes and edges • Perturbing single components: • Perturbing single interactions: February 26, 2021

Minimal Gene Oscillator - impact of perturbing an edge on limit cycle • Impact of perturbing effect of transcription factor on gene activity R February 26, 2021 Pc Pn

Minimal Gene Oscillator – impact on bifurcation diagram February 26, 2021

Minimal gene switch – perturbing effect of gene A on gene B February 26, 2021

But the main use is of course for …. . more complex networks February 26, 2021

Application to Modeling of Metabolic Oscillations in Activated Neutrophils Olsen et al, Bio. Phys. J • two compartments: cytosol and phagosome • model: 7 metabolites in cytosol, 9 metabolites in phagosome • 14 known chemical reactions, 5 transport eqs • lumped model with 14 ODEs • 25 parameters, fitted to experimental data

Neutrophil Network • The 14 state model corresponds to a highly connected network • predicts experimentally observed period and amplitudes • robust to parameter variations up to 20% (Olsen et al, Bio. Phys. J)

Spatial Waves – extending the model • The Olsen model assumes perfectly mixed cell environment, i. e. , no spatial gradients • Experimental observations: oscillations correspond to spatiotemporal traveling waves • Cedersund (2008): oscillations in Olsen model disappear even for very small spatial gradients (large diffusion rates) ) appears unrobust to structural perturbations February 26, 2021

Robust Instability of Steady-state From Nyquist: poor robustness margins ) some small perturbation of network will make underlying steady-state structurally unstable

Perturbing all nodes Overall robustness: can only tolerate up to 0. 2% simultaneous change in all component activities

Perturbing Individual Components Perturbing the activity of metabolite 1 (per 3+) or 4 (H 2 O 2) by approx 1% completely removes oscillatory behavior February 26, 2021

Perturbing Single Edges The most significant fragilities involve interactions between metabolites 1, 2 and 4 February 26, 2021

Perturbing the Nonlinear Model • Most severe fragility involves effect of metabolite 1 on metabolite 2 • Can be fitted to a time-delay • Add delay to corresponding reaction kinetics of original model February 26, 2021

Effect of Delay on Robust Instability of Steady-State Nyquist Locus February 26, 2021

Effect of Delay on Limit Cycle Behavior • Effect of delay µ in reaction R 2 on bifurcation behavior of full model - linear analysis yields good prediction of nonlinear behavior - the added delay should be compared to oscillation period T>20 s February 26, 2021

Robustification • Robustness analysis reveals unrobust features of model and identifies main fragilities • Robustification: optimize robustness by adding generic perturbations to network edges ) provides experimentally testable hypotheses on plausible model modificiations Neutrophil model: increase robustness significantly through small modifications of kinetics for reactions involving metabolites 2 and 4 (Nenchev and Jacobsen, to appear) February 26, 2021

Mammalian Circadian Clock – nonessential interactions Leloup and Goldbeter (PNAS, 100, 12) • What is purpose of the apparent excessive complexity? • Common hypothesis: improved robustness February 26, 2021

Perturbing Individual Nodes • Nodes with robustness radius can be removed without inducing a qualitative change in network behavior ) nonessential • 11 out of 16 components in Leloup Goldbeter model non-essential February 26, 2021

Perturbing Individual Edges February 26, 2021 • 29 out of 34 edges (interactions) non-essential • Only interactions related to Per gene appear essential for oscillatory behavior

A single loop generates the oscillations February 26, 2021 • Reduced model with 5 states predicts full model oscillations well • Most of the network apparently not required for generating circadian oscillations

What about robustness? February 26, 2021 • Robustness almost unaffected by non-essential nodes • Purpose of network complexity?

Bistable Switch in B Cell Differentiation • Antigens stimulate differentiation of B cells into antibody secreting cells through a bistable switch • Robustness for perturbation of all nodes

Bistable Switch in B Cell Differentiation • Perturbing individual components - all components except APIp (10), TA (11) and TAA (12) appear essential February 26, 2021

Bistable Switch in B Cell Differentation • Perturbing single edges - most interactions needed to provide bistability February 26, 2021

Perturbing the nonlinear model • Perturbing effect of Blimp 1 m. RNA on Blimp 1 protein by 30% February 26, 2021

Summary • Nonlinear phenomena frequently utilized to create critical functionality in cellular biology • Analyzing the robustness of bistable switches and autonomous oscillators important for modeling and elucidating the mechanisms of the underlying biochemical networks • We considered robustness in terms of persistence of behavior and proposed to determine smallest perturbation that makes a steady-state structurally unstable (bifurcation) • By strategically adding perturbations to nodes and edges fragilities can be identified and key mechanisms unravelled February 26, 2021