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Global entrainment of transcriptional systems to periodic inputs
Giovanni Russo, Mario di Bernardo, Eduardo D. Sontag
TL;DR
The paper asks when nonlinear transcriptional systems can be globally entrained by external periodic inputs despite the difficulty of proving periodic responses and handling uncertainty. It uses contraction theory, including non-Euclidean norms and matrix measures, to derive general conditions and prove their applicability to transcriptional models. Under these conditions, all solutions converge to unique globally attracting periodic solutions, while some assumptions remain difficult to satisfy or require structural restrictions.
Problem
The paper addresses how to formally establish global entrainment of nonlinear biological systems driven by periodic inputs, where periodic forcing does not guarantee a periodic response.
Method
The paper applies contraction theory and associated matrix measures, including non-Euclidean norms, to derive contractivity conditions for periodically driven transcriptional systems.
Results
Under the derived contraction conditions, all solutions of the analyzed transcriptional systems converge to unique globally attracting periodic solutions with the input period.
Takeaways & Limitations
The results provide a systematic basis for analyzing entrainment and potentially designing synthetic transcription modules within the stated nonlinear system classes.
Takeaways & Limitations
Lyapunov approaches are difficult for periodically forced non-autonomous systems, and the derivative hypothesis in Theorem 6 may be the hardest biological assumption to satisfy.
Abstract
from arXiv · showhide
This paper addresses the problem of giving conditions for transcriptional systems to be globally entrained to external periodic inputs. By using contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems. The basic mathematical results needed from contraction theory are proved in the paper, making it self-contained.
1 Introduction
The paper frames global entrainment of nonlinear biological systems as a difficult problem and develops contraction-theoretic conditions that guarantee convergence to periodic behavior. Its self-contained analysis connects infinitesimal contraction to global convergence and applies matrix measures to periodically forced systems.
- Periodic entrainment is biologically important because environmental rhythms regulate organismal and cellular processes.
- Simulations cannot prove entrainment for all parameter values and may suffer numerical errors, while experiments require robustness to noise and uncertainty.
- Nonlinear systems driven by periodic inputs may exhibit harmonic effects, chaos, or quasiperiodicity rather than periodic responses.
- Lyapunov-function approaches can be difficult for biological systems, may not ensure convergence under noise or uncertainty, and are harder to apply to non-autonomous systems.
- Contraction theory instead asks whether all trajectories converge toward one another, enabling global and noise-robust convergence results.
- The paper provides self-contained contraction proofs and uses norms, matrix measures, and Jacobians to analyze periodically forced systems.
- Infinitesimal contraction yields exponential convergence of neighboring trajectories and, under the stated periodicity assumptions, convergence of every solution to a unique periodic solution.
- For the simple transcriptional example, contractivity implies that solutions globally converge to a unique period-T solution dependent on system parameters.
2.1 Mathematical model and problem statement
The paper formulates a driven transcriptional module in which a time-dependent input produces X, which regulates a downstream promoter module. It seeks to show that periodic inputs make all solutions converge to a unique limit cycle, using contraction of the unforced system.
- Model formulation: The model assumes X production is proportional to u(t), with linear degradation or dilution, while X drives a downstream promoter module.The input may be external or represent an activating enzyme or second messenger.
- Model formulation: The coupled system tracks X concentration x and protein-promoter complex concentration y, with conserved total promoter concentration ET.The binding and dissociation rates are k1 and k2, respectively.
- Problem statement: The stated objective is to show that periodic u(t) causes all solutions to converge to a globally attracting limit cycle with the same period T.The paper presents this as a unique limit cycle for system (12).
- Problem statement: Contraction is the key analytical strategy: when no input is present, system (12) is shown to be infinitesimally and therefore globally contracting.The subsequent theorem states that every solution under a nonnegative periodic input converges exponentially to a periodic solution of period T.
- Extensions: The paper extends the discussion beyond the basic module to multiple driven subsystems and more general nonlinear systems with similar structure.These extensions are introduced after the main entrainment result.
2.2 Proof of Theorem 3
The proof establishes contraction by transforming the Jacobian and evaluating a weighted 1-norm matrix measure. Suitable positive weights make the measure uniformly negative, yielding a contraction rate and the entrainment conclusion.
- Weighted matrix measure: The proof studies the Jacobian of system (12) after transforming it with a diagonal nonsingular matrix P.The weighted measure is induced by the vector norm |Px|1.
- Contraction conclusion: The proof concludes that the system is contracting, so solutions globally converge to a unique T-periodic solution.Figure 3 illustrates entrainment of the output to sinusoidal and repeating binary inputs.
- Weighted matrix measure: The matrix measure μP,1 is computed from the 1-norm measure μ1 after applying the similarity transformation PJP^-1.Negativity, rather than the exact numerical value of the measure, is the proof target.
- Choice of weights: The weights p1 and p2 are chosen positive with p2/p1 = 1 + ε, ε > 0, while maintaining p2 > p1.This choice is used to satisfy the simultaneous inequalities required for μP,1(J) < 0.
- Choice of weights: The concentration constraint ET − y ≥ 0 helps establish uniform negativity of the transformed matrix measure for appropriate ε.The resulting contraction rate c2 depends on system parameters and on the selected metric weights p1 and p2.
2.3 Generalizations
The paper extends contraction-based entrainment results to enzyme-activated modules, multiple downstream modules, cascades, and broader nonlinear systems. These generalizations establish conditions under which periodic inputs produce globally attracting periodic behavior and allow flexible module design.
- 2.3.1 Assuming X activation by enzyme kinetics: For the enzyme-activation model, a positive periodic input yields a globally attracting limit cycle after a linear coordinate transformation establishes a negative matrix measure.The transformed dynamics use the infinity-induced matrix measure to prove contraction; simulations show a stable limit cycle with the input’s period.
- 2.3.1 Assuming X activation by enzyme kinetics: The enzyme-activation system’s contraction proof follows from negativity of the relevant matrix-measure bound under positive parameters and nonnegative x.The bound is upper-bounded by −k1, so contraction follows and Theorem 2 gives entrainment to any periodic input.
- 2.3.2 Multiple driven systems: Multiple independently acting downstream modules can be driven by the same transcription factor while differing in biochemical parameters.The construction uses weighted matrix measures with positive scalars chosen to satisfy the resulting inequalities.
- 2.3.2 Multiple driven systems: A single input–multiple-output module can produce periodic outputs with different mean values and settling times when the relevant retroactivity estimates hold.The paper presents this as a design possibility based on the general oscillation result.
- 2.3.3 Transcriptional cascades: A cascade of infinitesimally contracting systems is contracting, so transcriptional cascades entrain to periodic inputs.Each cascade element is modeled as the transcriptional module described by (12), with intermediate variables interpreted as transcription factors.
- 2.3.4 More abstract systems: Theorems 6 and 7 provide contracting conditions for a broad two-dimensional input-output system and support flexible choices of self-degradation and interaction functions.A sufficient relaxation is that ∂a/∂x ≥ 2|∂f/∂x|, which implies contractivity under the stated hypotheses.
- Remarks: The generalizations can include self-degradation of y and cases where the X-module drives more than one downstream transcriptional module.These extensions preserve the same proof strategy under their stated structural conditions.
3 Materials and Methods
Simulations were performed in MATLAB Simulink using the variable-step ODE23t solver.
- All simulations used MATLAB Simulink, Version 7.4, with the variable-step ODE23t solver.Simulink models were available upon request.
4 Conclusions
The paper develops a systematic contraction-theory methodology for deriving conditions under which transcriptional modules are globally entrained to periodic inputs. It applies non-Euclidean norms and matrix measures across increasingly general driven systems, including multiple driven systems and abstract nonlinear modules.
- The paper presents a systematic methodology for deriving global entrainment conditions for transcriptional modules driven by periodic inputs.
- Contraction theory, non-Euclidean norms, and matrix measures characterize module behavior under external periodic excitation.
- The analysis progresses from a simple bimolecular reaction to general externally driven transcriptional modules and multiple driven systems.
- For the analyzed modules, contraction conditions are derived under generic assumptions on their parameters.
- More abstract systems with generic nonlinear degradation and interaction terms are used to discuss the design significance of the results.
A K-reachable sets
K-reachable sets generalize convex state spaces by requiring smoothly connected paths whose lengths are controlled relative to Euclidean distance. This weaker geometric condition supports convergence results even in non-convex spaces with obstacles.
- A set is K-reachable when every pair of points can be joined by a continuously differentiable curve contained in the set.
- Convex sets are 1-reachable because straight line segments connect any two points within the set.
- K-reachability is equivalent to bounding geodesic path length by a multiple of the Euclidean distance between points.
- Finite-dimensional norm equivalence ensures that a suitable K can be obtained.
- Unlike convexity, K-reachability permits non-convex phase spaces with obstacles while retaining the basis for trajectory convergence.The paper illustrates this generality with a two-dimensional non-convex set defined outside a quarter-disk.
B Proof of Theorem 1
The proof extends contraction from infinitesimal behavior to global convergence by propagating a smooth path between initial conditions through the system flow. The convex case follows because convex sets are 1-reachable.
- The proof assumes the state space is K-reachable and establishes the main contracting-systems result under this geometric condition.
- For any two initial states, a smooth curve inside the state space is selected to connect them.
- The system flow is applied to every point on the connecting curve, producing a continuously differentiable family of trajectories.
- The theorem concludes that infinitesimal contraction yields global contractivity, with the appendix providing a self-contained generalized proof.
C Proof of Theorem 2
For a T-periodic vector field, contraction yields a unique T-periodic solution attracting every solution under a quantitative reachability condition. Convexity makes that condition automatic, while non-convex systems may converge to an orbit of period kT.
- C Proof of Theorem 2: T-periodicity makes the initial time relevant only modulo T, so ϕ(kT + t, kT, ξ) = ϕ(t, 0, ξ).The result follows by shifting the trajectory and using uniqueness of solutions.
- C Proof of Theorem 2: The period map satisfies P^k(ξ) = ϕ(kT, 0, ξ), established recursively from the flow composition property.This links discrete iteration of P to continuous-time evolution over k periods.
- C Proof of Theorem 2: If C is closed and K-reachable, f contracts at rate c2, and Ke−c2T < 1, a unique T-periodic solution attracts every solution from C.The proof applies the contraction mapping theorem to the period-T map.
- C Proof of Theorem 2: For convex C, K = 1, so an infinitesimally contracting system automatically satisfies Ke−c2T < 1.Thus Theorem 9 directly supplies the unique T-periodic orbit and global convergence conclusion.
- C Proof of Theorem 2: In the non-convex case, choosing sufficiently large integer k makes Ke−c2kT < 1, yielding global convergence to a unique orbit with period kT.The vector field is also kT-periodic for every integer k.
D Cascades
The cascade result shows that two contracting subsystems can form a contracting complete system when the downstream mixed Jacobian is bounded and suitable weighted norms are chosen. The proof bounds the combined Jacobian’s matrix measure using these weights.
- D Cascades: A cascade of contracting systems can remain contracting, and it is sufficient to establish the result inductively for two systems.The cascade is analyzed through the subsystem Jacobians A, B, and C.
- D Cascades: The upstream subsystem must contract in one norm, while the downstream subsystem contracts in another norm with its upstream state treated as a parameter.These assumptions are expressed through negative matrix measures for A and C.
- D Cascades: The mixed Jacobian B must be bounded by an operator-norm constant over all states and times.Because Euclidean norms are equivalent, the bound can be verified in any norm.
- D Cascades: The complete system uses the weighted product norm |(x1, x2)| = p1|x1|∗ + p2|x2|∗∗ for positive weights p1 and p2.The weights combine the potentially different subsystem norms into one norm for the cascade.
- D Cascades: Bounding the induced norm of I + hJ separates upstream contraction, mixed coupling, and downstream contraction before taking h → 0.The resulting inequality uses a convex-combination bound on the weighted component contributions.
E A counterexample to entrainment
A cited counterexample separates behavior under constant and periodic inputs: all solutions converge to a steady state for constant input, whereas sinusoidal input produces chaotic behavior. The example is explicitly non-contracting.
- E A counterexample to entrainment: A non-contracting system can converge to a steady state under constant external input yet become chaotic when driven by u(t) = sin t.This demonstrates that steady-state behavior for constant inputs does not ensure entrainment under periodic inputs.
- E A counterexample to entrainment: The counterexample uses a four-state model with p, ξ, ψ, and ζ, including a saturating activation α(y) = y^2/(K + y^2).The parameter in the activation function is K = 0.0001.
- E A counterexample to entrainment: Figure 8 compares ξ(t) under u(t) = sin t and u(t) = 5.13, showing chaotic-like response for the periodic input and steady state for the constant input.The inputs and initial conditions were randomly chosen for the simulation.