Germán A. Enciso

Monotone Systems Under Positive Feedback: Multistability and a Reduction Theorem

For feedback loops involving single input, single output monotone systems with well-defined I/O characteristics, a recent paper by Angeli and Sontag provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a consequence of the technique. The present paper extends the approach to multiple inputs and outputs. A key idea is the introduction of a reduced system which preserves local stability properties.

A phosphatase threshold sets the level of Cdk1 activity in early mitosis in budding yeast

The Wee1 kinase inhibits cyclin-dependent kinase 1 (Cdk1) during early mitosis. A low level of Cdk1 activity must escape Wee1 inhibition to initiate early mitotic events, but the underlying mechanisms have remained unknown. In this paper, we show that a specific form of protein phosphatase 2A opposes activation of Wee1, which allows low-level activation of Cdk1 in early mitosis.

Nonessential sites improve phosphorylation switch.

Multisite phosphorylation is a common form of posttranslational protein regulation which has been used to increase the switchlike behavior of the protein response to increasing kinase concentrations. In this letter, we show that the switchlike response of multisite phosphoproteins is strongly enhanced by nonessential phosphorylation sites, a mechanism that is robust to parameter changes and easily implemented in nature. We obtained analytic estimates for the Hill exponent (or coefficient) of the switchlike response, and we observed that a tradeoff exists between the switch and the kinase threshold for activation. This also suggests a possible evolutionary mechanism for the relatively large numbers of phosphorylation sites found in various proteins.

Protein scaffolds can enhance the bistability of multisite phosphorylation systems.

The phosphorylation of a substrate at multiple sites is a common protein modification that can give rise to important structural and electrostatic changes. Scaffold proteins can enhance protein phosphorylation by facilitating an interaction between a protein kinase enzyme and its target substrate. In this work we consider a simple mathematical model of a scaffold protein and show that under specific conditions, the presence of the scaffold can substantially raise the likelihood that the resulting system will exhibit bistable behavior. This phenomenon is especially pronounced when the enzymatic reactions have sufficiently large KM, compared to the concentration of the target substrate. We also find for a closely related model that bistable systems tend to have a specific kinetic conformation. Using deficiency theory and other methods, we provide a number of necessary conditions for bistability, such as the presence of multiple phosphorylation sites and the dependence of the scaffold binding/unbinding rates on the number of phosphorylated sites.

A model of direction selectivity in the starburst amacrine cell network

Displaced starburst amacrine cells (SACs) are retinal interneurons that exhibit GABAA receptor-mediated and Cl− cotransporter-mediated, directionally selective (DS) light responses in the rabbit retina. They depolarize to stimuli that move centrifugally through the receptive field surround and hyperpolarize to stimuli that move centripetally through the surround (Gavrikov et al, PNAS 100(26):16047–16052, 2003, PNAS 103(49):18793–18798, 2006). They also play a key role in the activity of DS ganglion cells (DS GC; Amthor et al, Vis Neurosci 19:495–509 2002; Euler et al, Nature 418:845–852, 2002; Fried et al, Nature 420:411– 414, 2002; Gavrikov et al, PNAS 100(26):16047–16052, 2003, PNAS 103(49):18793–18798, 2006; Lee and Zhou, Neuron 51:787–799 2006; Yoshida et al, Neuron 30:771–780, 2001). In this paper we present a model of strong DS behavior of SACs which relies on the GABA-mediated communication within a tightly interconnected network of these cells and on the glutamate signal that the SACs receive from bipolar cells (a presynaptic cell that receives input from cones). We describe how a moving light stimulus can produce a large, sustained depolarization of the SAC dendritic tips that point in the direction that the stimulus moves (i.e., centrifugal motion), but produce a minimal depolarization of the dendritic tips that point in the opposite direction (i.e., centripetal motion). This DS behavior, which is quantified based on the relative size and duration of the depolarizations evoked by stimulus motion at dendritic tips pointing in opposite directions, is robust to changes of many different parameter values and consistent with experimental data. In addition, the DS behavior is strengthened under the assumptions that the Cl− cotransporters Na + -K + -Cl − and K + -Cl − are located in different regions of the SAC dendritic tree (Gavrikov et al, PNAS 103(49):18793–18798, 2006) and that GABA evokes a long-lasting response (Gavrikov et al, PNAS 100(26):16047–16052, 2003, PNAS 103(49):18793–18798, 2006; Lee and Zhou, Neuron 51:787–799, 2006). A possible mechanism is discussed based on the generation of waves of local glutamate and GABA secretion, and their postsynaptic interplay as the waves travel between cell compartments.

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

It has recently been shown that structural conditions on the reaction network, rather than a ‘fine-tuning’ of system parameters, often suffice to impart ‘absolute concentration robustness’ (ACR) on a wide class of biologically relevant, deterministically modelled mass-action systems. We show here that fundamentally different conclusions about the long-term behaviour of such systems are reached if the systems are instead modelled with stochastic dynamics and a discrete state space. Specifically, we characterize a large class of models that exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space, i.e. the system undergoes an extinction event. If the time to extinction is large relative to the relevant timescales of the system, the process will appear to settle down to a stationary distribution long before the inevitable extinction will occur. This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with ACR are recovered by showing that the quasi-stationary distribution of the robust species approaches a Poisson distribution.

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.

A Remark on Multistability for Monotone Systems II

A recent paper by Angeli and Sontag presented a stability criterium for feedback loops involving single input, single output models which admit a well-defined I/O characteristic and satisfy a monotonicity condition. This paper generalizes the argument to arbitrary strongly monotone systems. The main idea is to study the asymptotic behavior of a strongly monotone system by relating it to a graph on the plane which contains information about the equilibria of the system and their stability properties.

Monotone bifurcation graphs

Abstract In recent work by Angeli and the authors, it was shown that the stability and global behaviour of strongly monotone dynamical systems may be profitably studied using a technique that involves feedback decompositions into ‘well-behaved’ subsystems. The present paper generalizes the approach, so that it applies to a far larger class of systems. As an illustration, the techniques are used in the analysis of a nine-variable autoregulatory transcription network. Also, extensions to delay and reaction diffusion systems are considered.

A small-gain result for orthant-monotone systems under mixed feedback

Abstract This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders. A linear example is provided showing that the small-gain iteration used for the negative feedback case is not sufficient for global attractivity under mixed feedback. An algebraic characterization is given of the new small-gain condition, generalizing a result known in the negative feedback case. An application is given to nonlinear protein networks with one positive and one negative feedback loop.