Phosphorelays provide tunable signal processing capabilities for the cell

Achieving a complete understanding of cellular signal transduction requires deciphering the relation between structural and biochemical features of a signaling system and the shape of the signal-response relationship it embeds. Using explicit analytical expressions and numerical simulations, we present here this relation for four-layered phosphorelays, which are signaling systems that are ubiquitous in prokaryotes and also found in lower eukaryotes and plants. We derive an analytical expression that relates the shape of the signal-response relationship in a relay to the kinetic rates of forward, reverse phosphorylation and hydrolysis reactions. This reveals a set of mathematical conditions which, when satisfied, dictate the shape of the signal-response relationship. We find that a specific topology also observed in nature can satisfy these conditions in such a way to allow plasticity among hyperbolic and sigmoidal signal-response relationships. Particularly, the shape of the signal-response relationship of this relay topology can be tuned by altering kinetic rates and total protein levels at different parts of the relay. These findings provide an important step towards predicting response dynamics of phosphorelays, and the nature of subsequent physiological responses that they mediate, solely from topological features and few composite measurements; measuring the ratio of reverse and forward phosphorylation rate constants could be sufficient to determine the shape of the signal-response relationship the relay exhibits. Furthermore, they highlight the potential ways in which selective pressures on signal processing could have played a role in the evolution of the observed structural and biochemical characteristic in phosphorelays

Unlimited multistability and Boolean logic in microbial signalling

The ability to map environmental signals onto distinct internal physiological states or programmes is critical for single-celled microbes. A crucial systems dynamics feature underpinning such ability is multistability. While unlimited multistability is known to arise from multi-site phosphorylation seen in the signalling networks of eukaryotic cells, a similarly universal mechanism has not been identified in microbial signalling systems. These systems are generally known as two-component systems comprising histidine kinase (HK) receptors and response regulator proteins engaging in phosphotransfer reactions. We develop a mathematical framework for analysing microbial systems with multi-domain HK receptors known as hybrid and unorthodox HKs. We show that these systems embed a simple core network that exhibits multistability, thereby unveiling a novel biochemical mechanism for multistability. We further prove that sharing of downstream components allows a system with n multi-domain hybrid HKs to attain 3n steady states. We find that such systems, when sensing distinct signals, can readily implement Boolean logic functions on these signals. Using two experimentally studied examples of two-component systems implementing hybrid HKs, we show that bistability and implementation of logic functions are possible under biologically feasible reaction rates. Furthermore, we show that all sequenced microbial genomes contain significant numbers of hybrid and unorthodox HKs, and some genomes have a larger fraction of these proteins compared with regular HKs. Microbial cells are thus theoretically unbounded in mapping distinct environmental signals onto distinct physiological states and perform complex computations on them. These findings facilitate the understanding of natural two-component systems and allow their engineering through synthetic biology

On-line optimal input design increases the efficiency and accuracy of the modelling of an inducible synthetic promoter

Synthetic biology seeks to design biological parts and circuits that implement new functions in cells. Major accomplishments have been reported in this field, yet predicting a priori the in vivo behaviour of synthetic gene circuits is major a challenge. Mathematical models offer a means to address this bottleneck. However, in biology, modelling is perceived as an expensive, time-consuming task. Indeed, the quality of predictions depends on the accuracy of parameters, which are traditionally inferred from poorly informative data. How much can parameter accuracy be improved by using model-based optimal experimental design (MBOED)? To tackle this question, we considered an inducible promoter in the yeast S. cerevisiae. Using in vivo data, we re-fit a dynamic model for this component and then compared the performance of standard (e.g., step inputs) and optimally designed experiments for parameter inference. We found that MBOED improves the quality of model calibration by ∼60%. Results further improve up to 84% when considering on-line optimal experimental design (OED). Our in silico results suggest that MBOED provides a significant advantage in the identification of models of biological parts and should thus be integrated into their characterisation.This research was partially supported by EC funding H2020 FET OPEN 766840-COSY-BIO and a Royal Society of Edinburgh-MoST grant (to F.M.), EPSRC funding EP/P017134/1-CONDSYC (to L.B.) and Spanish MINECO, grant ref. AGL2015-67504-C3-2-R (to E.B.-C.).Peer reviewe

Phosphate sink containing two-component signaling systems as tunable threshold devices

Synthetic biology aims to design de novo biological systems and reengineer existing ones. These efforts have mostly focused on transcriptional circuits, with reengineering of signaling circuits hampered by limited understanding of their systems dynamics and experimental challenges. Bacterial two-component signaling systems offer a rich diversity of sensory systems that are built around a core phosphotransfer reaction between histidine kinases and their output response regulator proteins, and thus are a good target for reengineering through synthetic biology. Here, we explore the signal-response relationship arising from a specific motif found in two-component signaling. In this motif, a single histidine kinase (HK) phosphotransfers reversibly to two separate output response regulator (RR) proteins. We show that, under the experimentally observed parameters from bacteria and yeast, this motif not only allows rapid signal termination, whereby one of the RRs acts as a phosphate sink towards the other RR (i.e. the output RR), but also implements a sigmoidal signal-response relationship. We identify two mathematical conditions on system parameters that are necessary for sigmoidal signal-response relationships and define key parameters that control threshold levels and sensitivity of the signal-response curve. We confirm these findings experimentally, by in vitro reconstitution of the one HK-two RR motif found in the Sinorhizobium meliloti chemotaxis pathway and measuring the resulting signal-response curve. We find that the level of sigmoidality in this system can be experimentally controlled by the presence of the sink RR, and also through an auxiliary protein that is shown to bind to the HK (yielding Hill coefficients of above 7). These findings show that the one HK-two RR motif allows bacteria and yeast to implement tunable switch-like signal processing and provides an ideal basis for developing threshold devices for synthetic biology applications

Unlimited multistability and Boolean logic in microbial signalling

The ability to map environmental signals onto distinct internal physiological states or programmes is critical for single-celled microbes. A crucial systems dynamics feature underpinning such ability is multistability. While unlimited multistability is known to arise from multi-site phosphorylation seen in the signalling networks of eukaryotic cells, a similarly universal mechanism has not been identified in microbial signalling systems. These systems are generally known as two-component systems comprising histidine kinase (HK) receptors and response regulator proteins engaging in phosphotransfer reactions. We develop a mathematical framework for analysing microbial systems with multi-domain HK receptors known as hybrid and unorthodox HKs. We show that these systems embed a simple core network that exhibits multistability, thereby unveiling a novel biochemical mechanism for multistability. We further prove that sharing of downstream components allows a system with n multi-domain hybrid HKs to attain 3n steady states. We find that such systems, when sensing distinct signals, can readily implement Boolean logic functions on these signals. Using two experimentally studied examples of two-component systems implementing hybrid HKs, we show that bistability and implementation of logic functions are possible under biologically feasible reaction rates. Furthermore, we show that all sequenced microbial genomes contain significant numbers of hybrid and unorthodox HKs, and some genomes have a larger fraction of these proteins compared with regular HKs. Microbial cells are thus theoretically unbounded in mapping distinct environmental signals onto distinct physiological states and perform complex computations on them. These findings facilitate the understanding of natural two-component systems and allow their engineering through synthetic biology

Mean, minimum and maximum of the ratio of forward to reverse rate constants based on parameter sets resulting in hyperbolic and sigmoidal signal-response relationship.

<p>The results shown are for topologies 14 and 30, assuming monofunctional HK, and sampling all parameters (with equal total protein concentrations at different layers). For additional results using alternative classification and sampling schemes (different total protein concentrations at different layers), assuming bifunctional HK, as well as for results from topologies 16 and 32, see <i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s008" target="_blank">Text S4</a></i>. Mean values of all parameters as found in parameter sets resulting in hyperbolic and sigmoidal signal-response relationships in topologies 14, 16, 30 and 32 are provided as <i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s009" target="_blank">Text S5</a></i>.</p

Cartoon representations of the four topologies that allowed sigmoidal signal-response relationships in a significant part (more than 2%) of the sampled parameter space when considering monofunctional HK and different total protein concentrations at different layers (see also <b>Table 1</b>, Figure S1 and <i>Supporting Text S3</i>).

<p>Panels <b>A</b>, <b>B</b>, <b>C</b> and <b>D</b> show topologies 14, 16, 30 and 32 respectively, each corresponding to a specific set of reverse phosphotransfer and hydrolysis reactions being present. Reactions are shown as directional arrows, where thickness of the arrow indicates the relative strength of the reaction. In other words, arrows are weighted by the mean reaction rate constant obtained from all sampled parameter sets producing sigmoidality. For each layer and a given topology, a gray (open) backdrop indicates that the mean of total protein concentration at that layer is high (low), based on all sampled parameter sets producing sigmoidal signal-response curves (see <i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s009" target="_blank">Text S5</a></i> for actual mean parameter values and concentrations).</p

The results of the signal-response relationship classification for the 18 responsive topologies.

<p>For each topology, we have sampled 1000 parameter sets (rate constants and total protein concentrations) from a biologically permissible range (<i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s010" target="_blank">Text S6</a></i>), derived the signal-response curve for each parameter set and classified this curve as hyperbolic or sigmoidal (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#s4" target="_blank"><i>Methods</i></a>). The topologies are indicated with a binary identification code that indicates the presence (1) or absence (0) of reverse phosphotransfer reactions along the relay, and the presence (1) or absence (0) of hydrolysis reactions at layers 2 and 4 (see <i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s005" target="_blank">Text S1</a></i> for all possible topologies). The classification results are given as the fraction of parameter sets resulting in a sigmoidal signal-response curve (the rest being classified as hyperbolic) and assuming a monofunctional HK. Classifications are based on the second derivative of the signal-response curve at zero and from sampling all parameters (with equal total protein concentrations at different layers) (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#s4" target="_blank"><i>Methods</i></a>). For additional results using alternative classification and sampling schemes (different total protein concentrations at different layers) or assuming bifunctional HK, see <i>Supporting <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#pcbi.1003322.s007" target="_blank">Text S3</a></i>.</p

Analysis of noise and response properties of topologies 14 and 30 assuming monofunctional HK. A & B.

<p>Signal-response curve for topologies 14 (A) and 30 (B). The x-axis corresponds to the signal input to the system, which in the model is approximated by varying the HK auto-phosphorylation rate constant, <i>k_s</i>. The y-axis corresponds to the simulated mean of the fraction of phosphorylated RR, calculated using PRISM model checker (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#s4" target="_blank"><i>Methods</i></a>). The solid and dashed curves show the results obtained from constraining the system parameters in the hyperbolic and sigmoidal regimes respectively. <b>C & D.</b> Noise levels in phosphorylated RR for topologies 14 (C) and 30 (D). The x-axis shows the signal input to the system, taken to be the HK auto-phosphorylation rate constant, <i>k_s</i>. The y-axis shows the standard deviation over mean of the fraction of phosphorylated RR at steady state, both calculated using PRISM model checker (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003322#s4" target="_blank"><i>Methods</i></a>). <b>E & F.</b> Box plots showing the distribution of response times for topologies 14 (E) and 30 (F) as measured from hyperbolic and sigmoidal regimes. For each topology, the response time is measured for 100 randomly selected parameter sets from the hyperbolic and sigmoidal regimes.</p

Effect of CheS on the signal-response curve.

<p>The x- and y-axis show the ATP level and the corresponding steady state level of phosphorylated CheY2, respectively. The experimentally measured values are shown in circles (absence of CheS) and squares (presence of CheS). The phosphorylated CheY2 levels predicted by the model are shown with a dashed line (absence of CheS) and with a solid line (presence of CheS; where the CheA-P to CheY1 phosphotransfer reaction rate constant (<i>k_s</i>) and CheY1-P dephosphorylation rate constant (<i>k_hs</i>) were optimized for best fit to the experimental data; <i>k_s</i> = 50 and <i>k_hs</i> = 0.067). See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003890#pcbi.1003890.s006" target="_blank">Figure S6</a> for alternative fits to these experimental data where we have individually modelled the effect of CheS altering only <i>k_s</i> or <i>k_hs</i>. Error bars show the standard error of the mean obtained from three independent experiments.</p