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Reduction of dynamical biochemical reaction networks in computational biology

Ovidiu Radulescu, Alexander N. Gorban, Andrei Zinovyev, Vincent Noel

arXiv:1205.2851v1q-bio.MNphysics.chem-ph

TL;DR

Large biochemical networks are difficult to analyze amid incomplete information, uncertainty, and multiscale dynamics. The paper revisits reduction through dominance and parameter order, relating it to tropical geometry and methods for linear and nonlinear networks. It concludes that separated timescales can be summarized by dominant reduced subsystems, while noting limits associated with parametrization and computation.

  • Problem

    Large dynamical biochemical models face incomplete mechanistic and parameter information, while stochastic simulation of unreduced large models can be computationally expensive.

    Source: §1 · pp. 1, 3

  • Method

    The paper develops order-based dominant-subsystem reduction for linear and nonlinear networks and uses tropical geometry to interpret invariant manifolds, QSS species, and QE reactions.

    Source: §5 · p. 8, §8 · p. 12

  • Results

    For networks with separated timescales, essential system features can be summarized by a dominant reduced subsystem determined by order relations among parameters or their combinations.

    Source: §8 · p. 12

  • Takeaways & Limitations

    The reduction strategies are presented as a way to study large dynamical network models and reveal critical features of complex organizations.

    Source: §8 · p. 12

    Most described reduction methods require full parametrization and have local validity, while solving the polynomial systems needed for new rates can have double-exponential complexity.

    Source: §1 · p. 3, §6 · p. 10

Abstract

from arXiv · show

Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques.

1 Introduction

Biochemical networks are large and intricate, while incomplete and uncertain parameters and interaction details constrain dynamical modeling. This review revisits reduction methods for multiscale networks, emphasizing parameter-order-based robustness and restricting its coverage to deterministic models.

1 Introduction

  • Incomplete parameter and mechanistic information, together with overfitting and non-identifiability, constrains larger dynamical models, making reduction a step toward extracting features for data-based identification.The introduction also cites orders-of-magnitude differences between in vivo and in vitro parameter values.Source: §1 · pp. 1–2
  • Parameter-independent approaches address combinatorial growth through exact lumping, graph-based coarse-graining, metabolite pooling, searches over lumps, and qualitative regulatory-graph reduction.The passage gives examples for receptor and scaffold signaling, rule-based modeling, metabolic networks, and logical regulatory graphs.Source: §1 · p. 2
  • Separated timescales motivate reductions using invariant manifolds and averaging, while identifying QSS species and QE reactions without full IM or CSP machinery remains unresolved.The introduction describes IM and CSP approaches and notes that large reaction constants alone do not identify QE reactions.Source: §1 · p. 2
  • Averaging obtains slow coarse-scale dynamics by time-averaging over a period longer than rapid fine-scale oscillations; its implementation depends on how coarse and fine variables are defined.Source: §1 · p. 2
  • Stochastic networks vary in species abundance and reaction rates; large unreduced models can make SSA expensive, prompting adaptations of deterministic reduction ideas and mixed species-reaction partitions.The review cites approaches spanning piecewise-deterministic, switched-diffusion, jump-diffusion, and averaged regimes.Source: §1 · p. 3
  • For biochemical networks with separated constants, reduced models can remain accurate across broad parameter and variable ranges because they depend on parameter order relations rather than precise values.Most other reduction methods described require full parametrization and have local validity.Source: §1 · p. 3
  • The review revisits fundamental reduction concepts toward future computational-biology tools, but limits its coverage to deterministic models and is not exhaustive.Source: §1 · p. 3

2 Deterministic dynamical networks

The section sets out the ingredients and general differential-equation description used for deterministic dynamical reaction networks. It identifies reaction stoichiometry and concentration-dependent rates as parts of that description.

2 Deterministic dynamical networks

  • A dynamic reaction network is specified by its component list A={A1, ..., An} and its reaction list, or reaction mechanism.Source: §2 · p. 4
  • For homogeneous systems at fixed volume, nonlinear reaction-network dynamics are described by a system of differential equations.Source: §2 · p. 4
  • Each reaction has a global stoichiometric vector νj=βj−αj, and its rate is a nonlinear function of concentrations; the mass-action example uses multivariate concentration polynomials.Source: §2 · p. 4

3 Multi-scale reduction of monomolecular reaction networks

With totally separated reaction constants, monomolecular networks admit an order-based multiscale approximation that can be constructed by graph rewriting. Its reduced dynamics relate each remaining timescale to an effective reaction step.

3 Multi-scale reduction of monomolecular reaction networks

  • For monomolecular networks, each eigenvalue is the inverse of a timescale, and the multiscale approximation uses eigenvalues and eigenvectors that approximate those of the original network.Source: §3 · p. 4
  • For arbitrary monomolecular networks with totally separated constants, multiscale approximations are acyclic and deterministic.Source: §3 · p. 4
  • Reduction requires only qualitative rate ordering: integer edge labels rank reaction constants, with the fastest assigned the lowest label, and indicate process timescales.Source: §3 · p. 4
  • Pruning retains the fastest outgoing edge and removes the slowest step in separated cycles, but pruning slow exits from cycles would give an incorrect long-time approximation.Slow exits cause total cycle mass to decay; removing them would instead keep that mass constant.Source: §3 · p. 5
  • Pooling glues each cycle into a vertex whose concentration sums its species; outgoing reaction rates are weighted by the normalized quasi-stationary concentrations within the cycle.Source: §3 · p. 5
  • On each timescale, the reduced acyclic digraph behaves as an effective single-step reaction transferring incoming mass between species selected by its left and right eigenvectors.Under total separation, the eigenvectors have an approximate 0–1 type property.Source: §3 · p. 5
  • The approach applies to linear subnetworks and pseudo-monomolecular reactions; broad support in the logarithmic timescale distribution is presented as an empirical check for total separation.The passage associates a log-uniform distribution with the Zipf law.Source: §3 · p. 5

4 Separation, dominance, and tropical geometry

Dominance among model elements can be expressed through max-plus algebra and tropical geometry. Tropicalization replaces polynomial dynamics with piecewise-smooth dynamics that retain dominant monomials, providing a good approximation for monomolecular networks with totally separated timescales.

4 Separation, dominance, and tropical geometry

  • When reactions compete for the same pool, pruning all but the dominant reaction applies the max-plus replacement of a sum of well-separated positive terms by its maximum.Tropical geometry also provides a framework for simplifying polynomial or rational ordinary differential equations with separated monomials.Source: §4 · p. 6
  • Under the logarithmic transformation, positive-coefficient monomials become linear functions of log concentrations.The transformation is u_i = log x_i.Source: §4 · p. 6
  • For polynomial reaction rates, tropicalization replaces a polynomial with a piecewise-smooth function equal to its largest monomial in absolute value.The dominant monomial can change across concentration-space domains; where at least two monomials are equal, the max-plus polynomial is nonsmooth, defining a tropical variety.Source: §4 · p. 6
  • With total separation, monomolecular networks behave on each timescale like a single reaction step, and tropicalization is a good approximation.The paper also discusses nonlinear networks as a broader setting in which tropicalization can usefully approximate smooth dynamics.Source: §4 · p. 6

5 Quasi-steady state and Quasi-equilibrium, revisited

The section revisits how quasi-steady-state (QSS) and quasi-equilibrium (QE) reductions differ, and presents trajectory-based and tropical-geometry methods for identifying them. The methods distinguish the reductions through slaving and dominant-term structure, with Michaelis–Menten dynamics illustrating their regimes and potential for global or hybrid reduction.

5 Quasi-steady state and Quasi-equilibrium, revisited

  • A general direct method for identifying QE reactions and QSS species is lacking, so the section presents singular-perturbation and tropical-geometry approaches.The first approach uses trajectories; the second uses tropicalization.Source: §5 · p. 7
  • The trajectory-based approach detects slaved species by comparing actual trajectories with imposed trajectories that solve a polynomial equation.It requires a fully parametrized model, although the identified QE reactions and QSS species are described as robust across broad parameter ranges when timescales are separated.Source: §5 · pp. 7–8
  • Dominant-term pruning separates low-concentration slaved species satisfying QSS conditions from large-concentration species governed by fast reversible QE reactions.For Michaelis–Menten, the complex ES is slaved in both cases; the pruned balance distinguishes QE when the reversible rates remain and the dominated term is removed.Source: §5 · pp. 7–8
  • A formal alternative identifies QE reactions and QSS species by checking for sliding modes on the tropical manifold when the pruned fast polynomial is a 2-nomial.Stability of the QE or QSS manifold implies a sliding mode along the corresponding tropical hyperplane.Source: §5 · p. 8
  • In the Michaelis–Menten example, k2 >> k−1 gives QSS behavior, whereas k2 << k−1 makes both S and ES rapid species satisfying QE conditions.The tropicalized dynamics has a sliding-mode set that is a global attractor; its arms correspond to linear and saturated regimes.Source: §5 · pp. 8–9
  • Tropicalization can yield global model reductions or hybrid piecewise-simpler models whose regimes change when trajectories meet tropical-manifold hyperplanes.The section notes cell-cycle modeling as an application of hybridization across successive phases.Source: §5 · p. 9

6 Graph rewriting for large nonlinear, deterministic, dynamical networks

This section recasts reduction of nonlinear dynamical networks as graph rewriting, using pruning and pooling to produce simpler reaction networks. It describes how pools and new rates are constructed, notes the algebraic complexity of recalculating rates, and illustrates modular reduction on NF-κB signaling.

6 Graph rewriting for large nonlinear, deterministic, dynamical networks

  • Nonlinear-network reduction prunes QSS species and QE reactions, pools reactions or species, and calculates kinetic laws for the resulting network.Reaction pooling is used with QSS conditions, while species pooling is used with QE conditions.Source: §6 · p. 9
  • Reaction pools replace reactions with a single reaction whose stoichiometry is the weighted sum of pool stoichiometries; intermediates are conserved within the pool.Species pools are linear combinations of species concentrations, and the relevant pools are derived from kernels of the fast-subsystem stoichiometry matrix.Source: §6 · pp. 9–10
  • After pooling, QE and QSS algebraic equations must be solved to recalculate reaction rates, but polynomial or rational rate systems can have double-exponential complexity.The passage therefore motivates approximate solutions, which can be derived formally when equation monomials are well separated.Source: §6 · p. 10
  • In the NF-κB application, a 39-species, 65-reaction, 90-parameter model was reduced using modular graph rewriting on tightly connected submodels.Submodel decomposition reduces the complexity of computing elementary modes and solving algebraic systems for recalculated rates.Source: §6 · p. 10
  • For the cytoplasmic NF-κB submodel, two elementary modes summarize inhibitor degradation and NF-κB release and translocation, with rates calculated as functions of terminal-species concentrations.The two rates are given in the passage as rational expressions involving the concentrations of IkBa@csl, IkBa:p50:p65@csl, and A20.Source: §6 · pp. 10–11

7 Model reduction and model identification

Model reduction can simplify parameter identification by pruning mechanistic detail and replacing groups of parameters with effective combinations that are more observable from data.

7 Model reduction and model identification

  • Complex mechanistic models can be pruned to simpler phenomenological models, avoiding overfitting and parameter-identifiability problems.The reduced model can combine critical parameters into a smaller set of effective parameters that can be identified.Source: §7 · p. 11
  • Reduction can reveal critical model features and parameters through qualitative order comparisons without exhaustive parameter scans.The paper contrasts this approach with numerical sensitivity studies, which it describes as more time-consuming and less insightful.Source: §7 · p. 11
  • Effective parameters formed as monomials can be more observable from experimental data and constrain parameters in the full model.Some full-model parameters may remain arbitrary and be fixed to generic values.Source: §7 · p. 11

8 Conclusion

The paper frames dominance and limitation as organizing principles for reducing multiscale biochemical networks, while noting that further algorithm development and tool implementation remain future work.

8 Conclusion

  • The mathematical strategies are intended to support study of large dynamical network models and reveal critical features of complex organizations.Source: §8 · p. 12
  • For systems with well-separated timescales, a dominant reduced subsystem summarizes essential features and depends on parameter orders or combinations.The paper describes methods for calculating such subsystems in linear and nonlinear networks.Source: §8 · p. 12
  • Tropicalization provides a geometric framework for identifying invariant manifolds, quasi-steady-state species, and quasi-equilibrium reactions.Source: §8 · p. 12
  • Further work is needed to extend the mathematical ideas and reduction algorithms and implement them in computational biology tools.Source: §8 · p. 12

Algorithm 1 : reduction of monomolecular networks with separation

The algorithm prunes reactions by dominance, repeatedly glues cycles and renormalizes their exits, then restores cycles to construct an acyclic dominant system.

Algorithm 1 : reduction of monomolecular networks with separation

  • At each branching node, pruning retains the reaction with the maximal kinetic constant, producing an auxiliary network without branching.The auxiliary network may still contain cycles and confluences.Source: Algorithm 1 : reduction of… · p. 18
  • Cycles are glued into single vertices, and their exit reactions are restored with rates renormalized using the cycle's quasi-stationary distribution.For a limiting step, an exit from A_j with rate k becomes an exit with rate k k_lim_i/k_j.Source: Algorithm 1 : reduction of… · p. 18
  • Pruning and cycle gluing are iterated until an acyclic network is obtained, after which cycles are restored in reverse through the hierarchy.The restored network approximates relaxation of the initial network and is called the dominant system.Source: Algorithm 1 : reduction of… · p. 19
  • The initial pruning needs only an ordering of kinetic constants, but renormalized monomials can admit multiple orderings and thus multiple reduced digraphs.Source: Algorithm 1 : reduction of… · p. 19

Algorithm 2 : reduction of nonlinear networks with separation

The nonlinear workflow identifies quasi-steady-state species and quasi-equilibrium reactions, then prunes and pools fast components; examples illustrate reductions in tropicalized and biochemical models.

Algorithm 2 : reduction of nonlinear networks with separation

  • The presented identification of quasi-steady-state species and quasi-equilibrium reactions currently uses trajectories rather than tropicalization.Source: Algorithm 2 : reduction of nonlinear… · p. 19
  • Trajectory comparison and clustering separate slaved from slow species; dominated monomials are pruned before slaved species are classified as quasi-steady-state or quasi-equilibrium components.Slaved species not involved in quasi-equilibrium reactions are classified as quasi-steady-state species.Source: Algorithm 2 : reduction of nonlinear… · p. 20
  • For quasi-steady-state reduction, intermediate species and their reactions are identified, and elementary modes satisfy zero fast-subsystem but nonzero terminal stoichiometry.Rates for the elementary modes are expressed in terms of terminal species; uniqueness requires independent terminal stoichiometries.Source: Algorithm 2 : reduction of nonlinear… · p. 20
  • The quasi-steady-state algebraic equations are not yet solved automatically; the authors propose tropical geometry as a future means of automating this step.Source: Algorithm 2 : reduction of nonlinear… · p. 20
  • For quasi-equilibrium reduction, species pools are derived from fast-subsystem conservation relations and concentrations are re-expressed through the pools.Rates of terminal reactions are then rewritten in terms of those pools.Source: Algorithm 2 : reduction of nonlinear… · pp. 20–21
  • For monomolecular networks with total separation, each timescale behaves as a single step, with species categorized by concentration behavior.The figure distinguishes practically constant species, rapid low-concentration intermediates, a consumed species, and a produced species.Source: Figure 2 · p. 23
  • In the cell-cycle example, y1, y2, and y5 are slaved, while y3 and y4 are quasi-stationary on intervals, reducing dynamics to one dimension on those intervals.The tropicalized and smooth limit-cycle dynamics stay within bounded distance one of each other, with relatively small distance during quasi-stationary intervals.Source: Algorithm 2 : reduction of nonlinear… · p. 25
  • For the NF-κB model, reduction produced a 14-species, 25-reaction model with the same stoichiometry as another model but different rate functions.The hierarchy also includes a simplest model with 5 species, 8 reactions, and 15 parameters.Source: Figure 6 · p. 27, Figure 7 · p. 28
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