Algorithmic and Complexity Results for Decompositions of Biological Networks into Monotone Subsystems

A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson [Goemans, M., Williamson, D., 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42 (6), 1115-1145]. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model, and it was found to perform close to optimally.

[1]  Denis Thieffry,et al.  A description of dynamical graphs associated to elementary regulatory circuits , 2003, ECCB.

[2]  E D Sontag,et al.  Some new directions in control theory inspired by systems biology. , 2004, Systems biology.

[3]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.

[4]  Sorin Istrail,et al.  Statistical Mechanics, Three-Dimensionality and NP-Completeness: I. Universality of Intractability of the Partition Functions of the Ising Model Across Non-Planar Lattices , 2000, STOC 2000.

[5]  Ramesh Hariharan,et al.  Derandomizing semidefinite programming based approximation algorithms , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[6]  B. Cipra.  The Ising Model Is NP-Complete , 2000 .

[7]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[8]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[9]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[10]  David Angeli,et al.  Multi-stability in monotone input/output systems , 2003, Syst. Control. Lett..

[11]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[12]  David Angeli,et al.  A small-gain theorem for almost global convergence of monotone systems , 2004, Syst. Control. Lett..

[13]  David Angeli,et al.  Multistability in monotone I/O systems , 2004 .

[14]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[15]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[16]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[17]  J. Monod,et al.  Teleonomic mechanisms in cellular metabolism, growth, and differentiation. , 1961, Cold Spring Harbor symposia on quantitative biology.

[18]  E. D. Sontagc,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2005 .

[19]  H. Meinhardt,et al.  Space-dependent cell determination under the control of morphogen gradient. , 1978, Journal of theoretical biology.

[20]  M. Hirsch.  Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[21]  Eduardo Sontag,et al.  Global attractivity, I/O monotone small-gain theorems, and biological delay systems , 2005 .

[22]  D. DeAngelis,et al.  Positive Feedback in Natural Systems , 1986 .

[23]  R. Steele,et al.  Optimization , 2005, Encyclopedia of Biometrics.

[24]  G. Odell,et al.  The segment polarity network is a robust developmental module , 2000, Nature.

[25]  J. Demongeot,et al.  Positive and negative feedback: striking a balance between necessary antagonists. , 2002, Journal of theoretical biology.

[26]  M. Malisoff,et al.  Remarks on Monotone Control Systems with Multi-Valued Input-State Characteristics , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[27]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[28]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[29]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[30]  Karpinski Marek,et al.  Efficient Amplifiers and Bounded Degree Optimization , 2001 .

[31]  Eduardo Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional , 2001 .

[32]  Srinivasan Venkatesh,et al.  On the advantage over a random assignment , 2002, STOC '02.

[33]  Eduardo D Sontag,et al.  On the stability of a model of testosterone dynamics , 2004, Journal of mathematical biology.

[34]  Nicholas T Ingolia,et al.  Topology and Robustness in the Drosophila Segment Polarity Network , 2004, PLoS biology.

[35]  E. M.,et al.  Statistical Mechanics , 2021, On Complementarity.

[36]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[37]  Marek Karpinski,et al.  Efficient Amplifiers and Bounded Degree Optimization , 2001, Electron. Colloquium Comput. Complex..

[38]  R. Thomas,et al.  Logical analysis of systems comprising feedback loops. , 1978, Journal of theoretical biology.

[39]  Hal L. Smith.  Systems of ordinary differential equations which generate an order preserving flow. A survey of results , 1988 .

[40]  Andrew P. Sage,et al.  Feedback Loops , 1992, Concise Encyclopedia of Modelling & Simulation.

[41]  Michael Malisoff,et al.  A small-gain theorem for motone systems with multivalued input-state characteristics , 2005, IEEE Transactions on Automatic Control.

[42]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[43]  Edoardo Amaldi,et al.  On the approximability of some NP-hard minimization problems for linear systems , 1996, Electron. Colloquium Comput. Complex..

[44]  Eduardo D. Sontag,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2006 .

[45]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[46]  G. Rinaldi,et al.  Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm , 1995 .

[47]  H. Kitano,et al.  A comprehensive pathway map of epidermal growth factor receptor signaling , 2005, Molecular systems biology.

[48]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..

[49]  Eduardo D. Sontag,et al.  Monotone systems under positive feedback: multistability and a reduction theorem , 2005, Syst. Control. Lett..

[50]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[51]  David Angeli,et al.  On predator-prey systems and small-gain theorems. , 2004, Mathematical biosciences and engineering : MBE.

[52]  Thomas Mestl,et al.  FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS , 1995 .

[53]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .

[54]  L Wolpert,et al.  Thresholds in development. , 1977, Journal of theoretical biology.

[55]  Eduardo D. Sontag,et al.  Molecular Systems Biology and Control , 2005, Eur. J. Control.

[56]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[57]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[58]  El Houssine Snoussi.  Necessary Conditions for Multistationarity and Stable Periodicity , 1998 .

[59]  Carsten Lund,et al.  The Approximation of Maximum Subgraph Problems , 1993, ICALP.

[60]  E.D. Sontag,et al.  An analysis of a circadian model using the small-gain approach to monotone systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[61]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[62]  S. Klamt,et al.  Generalized concept of minimal cut sets in biochemical networks. , 2006, Bio Systems.

[63]  Eduardo D. Sontag,et al.  Oscillation in multi-stable monotone system with slowly varying positive feedback , 2005 .