Bounds on the Ultrasensitivity of Biochemical Reaction Cascades.

The ultrasensitivity of a dose response function can be quantifiably defined using the generalized Hill coefficient of the function. We examined an upper bound for the Hill coefficient of the composition of two functions, namely the product of their individual Hill coefficients. We proved that this upper bound holds for compositions of Hill functions, and that there are instances of counterexamples that exist for more general sigmoidal functions. Additionally, we tested computationally other types of sigmoidal functions, such as the logistic and inverse trigonometric functions, and we provided computational evidence that in these cases the inequality also holds. We show that in large generality there is a limit to how ultrasensitive the composition of two functions can be, which has applications to understanding signaling cascades in biochemical reactions.

Shadow enhancers mediate trade-offs between transcriptional noise and fidelity.

Enhancers are stretches of regulatory DNA that bind transcription factors (TFs) and regulate the expression of a target gene. Shadow enhancers are two or more enhancers that regulate the same target gene in space and time and are associated with most animal developmental genes. These multi-enhancer systems can drive more consistent transcription than single enhancer systems. Nevertheless, it remains unclear why shadow enhancer TF binding sites are distributed across multiple enhancers rather than within a single large enhancer. Here, we use a computational approach to study systems with varying numbers of TF binding sites and enhancers. We employ chemical reaction networks with stochastic dynamics to determine the trends in transcriptional noise and fidelity, two key performance objectives of enhancers. This reveals that while additive shadow enhancers do not differ in noise and fidelity from their single enhancer counterparts, sub- and superadditive shadow enhancers have noise and fidelity trade-offs not available to single enhancers. We also use our computational approach to compare the duplication and splitting of a single enhancer as mechanisms for the generation of shadow enhancers and find that the duplication of enhancers can decrease noise and increase fidelity, although at the metabolic cost of increased RNA production. A saturation mechanism for enhancer interactions similarly improves on both of these metrics. Taken together, this work highlights that shadow enhancer systems may exist for several reasons: genetic drift or the tuning of key functions of enhancers, including transcription fidelity, noise and output.

Stochastic models of nucleosome dynamics reveal regulatory rules of stimulus-induced epigenome remodeling.

The genomic positions of nucleosomes are a defining feature of the cell's epigenomic state, but signal-dependent transcription factors (SDTFs), upon activation, bind to specific genomic locations and modify nucleosome positioning. Here we leverage SDTFs as perturbation probes to learn about nucleosome dynamics in living cells. We develop Markov models of nucleosome dynamics and fit them to time course sequencing data of DNA accessibility. We find that (1) the dynamics of DNA unwrapping are significantly slower in cells than reported from cell-free experiments, (2) only models with cooperativity in wrapping and unwrapping fit the available data, (3) SDTF activity produces the highest eviction probability when its binding site is adjacent to but not on the nucleosome dyad, and (4) oscillatory SDTF activity results in high location variability. Our work uncovers the regulatory rules governing SDTF-induced nucleosome dynamics in live cells, which can predict chromatin accessibility alterations during inflammation at single-nucleosome resolution.

Non-cooperative mechanism for bounded and ultrasensitive chromatin remodeling.

Chromatin remodeling is an essential form of gene regulation that is involved in a variety of biological processes. We develop a theoretical model that takes advantage of percolation effects at the level of nucleosome interactions, which allows for ultrasensitive chromatin expansion. This model is non-cooperative and readily provides spatial bounds to the expansion region, preventing uncontrolled remodeling events. We explore different chromatin architectures and the ultrasensitivity of the chromatin density as a function of transcription factor concentration. We also compare our model with experimental data involving an inhibitor of nucleosome acetylation. These results suggest a novel mechanism for spatially-bounded chromatin remodeling and they provide means for quantitative comparisons between proposed models of chromatin architecture.

Stochastic Chlamydia Dynamics and Optimal Spread.

Chlamydia trachomatis is an important bacterial pathogen that has an unusual developmental switch from a dividing form (reticulate body or RB) to an infectious form (elementary body or EB). RBs replicate by binary fission within an infected host cell, but there is a delay before RBs convert into EBs for spread to a new host cell. We developed stochastic optimal control models of the Chlamydia developmental cycle to examine factors that control the number of EBs produced. These factors included the probability and timing of conversion, and the duration of the developmental cycle before the host cell lyses. Our mathematical analysis shows that the observed delay in RB-to-EB conversion is important for maximizing EB production by the end of the intracellular infection.

Shadow enhancers can suppress input transcription factor noise through distinct regulatory logic.

Shadow enhancers, groups of seemingly redundant enhancers, are found in a wide range of organisms and are critical for robust developmental patterning. However, their mechanism of action is unknown. We hypothesized that shadow enhancers drive consistent expression levels by buffering upstream noise through a separation of transcription factor (TF) inputs at the individual enhancers. By measuring the transcriptional dynamics of several Kruppel shadow enhancer configurations in live Drosophila embryos, we showed that individual member enhancers act largely independently. We found that TF fluctuations are an appreciable source of noise that the shadow enhancer pair can better buffer than duplicated enhancers. The shadow enhancer pair is also uniquely able to maintain low levels of expression noise across a wide range of temperatures. A stochastic model demonstrated the separation of TF inputs is sufficient to explain these findings. Our results suggest the widespread use of shadow enhancers is partially due to their noise suppressing ability.

In all higher organisms, life begins with a single cell. During the early stages of development, this single cell grows and divides multiple times to develop into the many different kinds of cells that make up an organism. This is a highly regulated process during which cells receive instructions telling them what kind of cell to become. These instructions are relayed via genes, and a particular combination of activated genes determines the cell's fate. Specific pieces of DNA, known as enhancers, act as switches that control when and where genes are active, while so-called shadow enhancers are found in groups and work together to turn on the same gene in a similar way. Shadow enhancers are often active during the early stages of life to direct the formation of specialized cells in different parts of the body. But so far, it has been unclear why it is beneficial to the divide the role of activating genes across several shadow enhancers rather than a single one. Here, Waymack et al. examined shadow enhancers around a gene called Kruppel in embryos of the fruit fly Drosophila melanogaster. Manipulating the shadow enhancers showed that they help to make gene activity more resistant to changes. Factors such as fluctuations in temperature have different effects on each shadow enhancer. Having several shadow enhancers working together ensures that, whatever happens, the right genes still get activated. For genes like Kruppel, which are key for healthy development, the ability to withstand unexpected changes is a valuable evolutionary benefit. The study of Waymack et al. reveals why shadow enhancers are involved in the regulation of many genes, which may help to better understand developmental defects. Many conditions caused by such defects are influenced by both genetics and the environment. Genetic illnesses can vary in severity, which may be related to the roles of shadow enhancers. As such, studying shadow enhancers could lead to new approaches for treating genetic diseases.

Slack reactants: A state-space truncation framework to estimate quantitative behavior of the chemical master equation.

State space truncation methods are widely used to approximate solutions of the chemical master equation. While most methods of this kind focus on truncating the state space directly, in this work, we propose modifying the underlying chemical reaction network by introducing slack reactants that indirectly truncate the state space. More specifically, slack reactants introduce an expanded chemical reaction network and impose a truncation scheme based on desired mass conservation laws. This network structure also allows us to prove inheritance of special properties of the original model, such as irreducibility and complex balancing. We use the network structure imposed by slack reactants to prove the convergence of the stationary distribution and first arrival times. We then provide examples comparing our method with the stationary finite state projection and finite buffer methods. Our slack reactant system appears to be more robust than some competing methods with respect to calculating first arrival times.

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.

Absolutely robust controllers for chemical reaction networks.

In this work, we design a type of controller that consists of adding a specific set of reactions to an existing mass-action chemical reaction network in order to control a target species. This set of reactions is effective for both deterministic and stochastic networks, in the latter case controlling the mean as well as the variance of the target species. We employ a type of network property called absolute concentration robustness (ACR). We provide applications to the control of a multisite phosphorylation model as well as a receptor-ligand signalling system. For this framework, we use the so-called deficiency zero theorem from chemical reaction network theory as well as multiscaling model reduction methods. We show that the target species has approximately Poisson distribution with the desired mean. We further show that ACR controllers can bring robust perfect adaptation to a target species and are complementary to a recently introduced antithetic feedback controller used for stochastic chemical reactions.

Embracing Noise in Chemical Reaction Networks.

We provide a short review of stochastic modeling in chemical reaction networks for mathematical and quantitative biologists. We use as case studies two publications appearing in this issue of the Bulletin, on the modeling of quasi-steady-state approximations and cell polarity. Reasons for the relevance of stochastic modeling are described along with some common differences between stochastic and deterministic models.

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.

Compact modeling of allosteric multisite proteins: application to a cell size checkpoint.

We explore a framework to model the dose response of allosteric multisite phosphorylation proteins using a single auxiliary variable. This reduction can closely replicate the steady state behavior of detailed multisite systems such as the Monod-Wyman-Changeux allosteric model or rule-based models. Optimal ultrasensitivity is obtained when the activation of an allosteric protein by its individual sites is concerted and redundant. The reduction makes this framework useful for modeling and analyzing biochemical systems in practical applications, where several multisite proteins may interact simultaneously. As an application we analyze a newly discovered checkpoint signaling pathway in budding yeast, which has been proposed to measure cell growth by monitoring signals generated at sites of plasma membrane growth. We show that the known components of this pathway can form a robust hysteretic switch. In particular, this system incorporates a signal proportional to bud growth or size, a mechanism to read the signal, and an all-or-none response triggered only when the signal reaches a threshold indicating that sufficient growth has occurred.

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.

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 K(M), 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 phosphatase threshold sets the level of Cdk1 activity in early mitosis in budding yeast.

Entry into mitosis is initiated by synthesis of cyclins, which bind and activate cyclin-dependent kinase 1 (Cdk1). Cyclin synthesis is gradual, yet activation of Cdk1 occurs in a stepwise manner: a low level of Cdk1 activity is initially generated that triggers early mitotic events, which is followed by full activation of Cdk1. Little is known about how stepwise activation of Cdk1 is achieved. A key regulator of Cdk1 is the Wee1 kinase, which phosphorylates and inhibits Cdk1. Wee1 and Cdk1 show mutual regulation: Cdk1 phosphorylates Wee1, which activates Wee1 to inhibit Cdk1. Further phosphorylation events inactivate Wee1. We discovered that a specific form of protein phosphatase 2A (PP2A(Cdc55)) opposes the initial phosphorylation of Wee1 by Cdk1. In vivo analysis, in vitro reconstitution, and mathematical modeling suggest that PP2A(Cdc55) sets a threshold that limits activation of Wee1, thereby allowing a low constant level of Cdk1 activity to escape Wee1 inhibition in early mitosis. These results define a new role for PP2A(Cdc55) and reveal a systems-level mechanism by which dynamically opposed kinase and phosphatase activities can modulate signal strength.

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.