Effect of magnitude and variability of energy of activation in multisite ultrasensitive biochemical processes.

Protein activity is often regulated by ligand binding or by post-translational modifications such as phosphorylation. Moreover, proteins that are regulated in this way often contain multiple ligand binding sites or modification sites, which can operate to create an ultrasensitive dose response. Here, we consider the contribution of the individual modification/binding sites to the activation process, and how their individual values affect the ultrasensitive behavior of the overall system. We use a generalized Monod-Wyman-Changeux (MWC) model that allows for variable conformational free energy contributions from distinct sites, and associate a so-called activation parameter to each site. Our analysis shows that the ultrasensitivity generally increases as the conformational free energy contribution from one or more sites is strengthened. Furthermore, ultrasensitivity depends on the mean of the activation parameters and not on their variability. In some cases, we find that the best way to maximize ultrasensitivity is to make the contribution from all sites as strong as possible. These results provide insights into the performance objectives of multiple modification/binding sites and thus help gain a greater understanding of signaling and its role in diseases.

Replication-dependent size reduction precedes differentiation in Chlamydia trachomatis.

Chlamydia trachomatis is the most common cause of bacterial sexually transmitted infection. It produces an unusual intracellular infection in which a vegetative form, called the reticulate body (RB), replicates and then converts into an elementary body (EB), which is the infectious form. Here we use quantitative three-dimensional electron microscopy (3D EM) to show that C. trachomatis RBs divide by binary fission and undergo a sixfold reduction in size as the population expands. Conversion only occurs after at least six rounds of replication, and correlates with smaller RB size. These results suggest that RBs only convert into EBs below a size threshold, reached by repeatedly dividing before doubling in size. A stochastic mathematical model shows how replication-dependent RB size reduction produces delayed and asynchronous conversion, which are hallmarks of the Chlamydia developmental cycle. Our findings support a model in which RB size controls the timing of RB-to-EB conversion without the need for an external signal.

Optimal Proliferation and Differentiation of Chlamydia Trachomatis.

Chlamydia trachomatis is a bacterium that causes eye infection and blindness in humans. It has an unusual life cycle involving two developmental forms. Within a cytoplasmic inclusion, the reticulate body (RB) repeatedly divides by binary fission and asynchronously differentiates into the infectious elementary body (EB). Upon the death of the mammalian cell that host many such inclusions, only the EB form of the bacteria survive and proceed to infect other cells. Given the bacteria's fast spreading infection, conventional wisdom would have the few initial EB turn into RB, divide and proliferate first, and then eventually start converting in order to maximize the terminal EB population upon host cell lysis. Several biological processes are seen as possible mechanisms for implementing such a conversion strategy. However, the optimality of an instinctual strategy with a period of proliferate without conversion prior to the onset of differentiation has never been substantiated theoretically or justified mathematically. This paper formulates three relatively simple models that capture the essential features of the Chlamydia life cycle. When the initial infection is caused by the endocytosis of a small EB population well below the carrying capacity of the host cell, the Maximum Principle requires for these models an optimal conversion strategy that confirms and rigorously justifies the prevailing view of no conversion at the early stage of the host cell infection. However, the conventional supposition is found to be inappropriate for an initial EB (-to-RB) population near or above the carrying capacity. Previously suggested and new biological mechanisms are examined for their role in implementing the different optimal conversion strategies associated with models investigated herein.

The effect of site-to-site variability in ultrasensitive dose responses.

In this paper we study the ultrasensitive behavior of multisite phosphorylation or ligand binding systems, under site-to-site variations in the modification rates. Using computational methods and mathematical analysis, we prove that the Hill coefficient reaches its maximum value when all sites are identical to each other. This is shown for a non-cooperative multisite system with arbitrary activation function as well as for the well known MWC model. We also show that the Hill coefficient of the dose response is locally robust to variations in individual modification rates. The results suggest that maximal ultrasensitivity is reached when sites are similar to each other but not necessarily identical, a conformation found in unstructured modification domains present in many experimental systems.

Transient absolute robustness in stochastic biochemical networks.

Absolute robustness allows biochemical networks to sustain a consistent steady-state output in the face of protein concentration variability from cell to cell. This property is structural and can be determined from the topology of the network alone regardless of rate parameters. An important question regarding these systems is the effect of discrete biochemical noise in the dynamical behaviour. In this paper, a variable freezing technique is developed to show that under mild hypotheses the corresponding stochastic system has a transiently robust behaviour. Specifically, after finite time the distribution of the output approximates a Poisson distribution, centred around the deterministic mean. The approximation becomes increasingly accurate, and it holds for increasingly long finite times, as the total protein concentrations grow to infinity. In particular, the stochastic system retains a transient, absolutely robust behaviour corresponding to the deterministic case. This result contrasts with the long-term dynamics of the stochastic system, which eventually must undergo an extinction event that eliminates robustness and is completely different from the deterministic dynamics. The transiently robust behaviour may be sufficient to carry out many forms of robust signal transduction and cellular decision-making in cellular organisms.

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.

Ultrasensitivity in independent multisite systems.

Multisite modifications are widely recognized as an essential feature of many switch-like responses in signal transduction. It is usually assumed that the modification of one site directly or indirectly increases the rate of modification of neighboring sites. In this paper we provide a new set of assumptions for a multisite system to become highly ultrasensitive even in the absence of cooperativity or allostery. We assume that the individual sites are modified independently of each other, and that protein activity is an ultrasensitive function of the fraction of modified sites. These assumptions are particularly useful in the context of multisite systems with a large (8+) number of sites. We estimate the apparent Hill coefficient of the dose responses in the sequential and nonsequential cases, highlight their different qualitative properties, and discuss a formula to approximate dose responses in the nonsequential case. As an example we describe a model of bacterial chemotaxis that features robust ultrasensitivity and perfect adaptation over a wide range of ligand concentrations, based on non-allosteric multisite behavior at the level of receptors and flagella. We also include a model of the inactivation of the yeast pheromone protein Ste5 by cell cycle proteins.

A model of direction selectivity in the starburst amacrine cell network.

Displaced starburst amacrine cells (SACs) are retinal interneurons that exhibit GABA( A ) 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.