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Oscillatory morphodynamics provides necessary mechanical cues for many multicellular processes. Owing to their collective nature, these processes require robustly coordinated dynamics of individual cells, which are often separated too distantly to communicate with each other through biomaterial transportation. Although it is known that the mechanical balance generally plays a significant role in the systems' morphologies, it remains elusive whether and how the mechanical components may contribute to the systems' collective morphodynamics. Here, we study the collective oscillations in the Drosophila amnioserosa tissue to elucidate the regulatory roles of the mechanical components. We identify that the tensile stress is the key activator that switches the collective oscillations on and off. This regulatory role is shown analytically using the Hopf bifurcation theory. We find that the physical properties of the tissue boundary are directly responsible for synchronizing the oscillatory intensity and polarity of all inner cells and for orchestrating the spatial oscillation patterns inthe tissue.

In the epithelium of Drosophila during tissue elongation, contractile forces in follicle cells can oscillate. These oscillations correlate with increasing tension in the epithelium from egg chamber growth. A mathematical model is proposed to explain the observed oscillations, together with a mechanism of active regulation of cellular contractile forces., During tissue elongation from stage 9 to stage 10 in Drosophila oogenesis, the egg chamber increases in length by ∼1.7-fold while increasing in volume by eightfold. During these stages, spontaneous oscillations in the contraction of cell basal surfaces develop in a subset of follicle cells. This patterned activity is required for elongation of the egg chamber; however, the mechanisms generating the spatiotemporal pattern have been unclear. Here we use a combination of quantitative modeling and experimental perturbation to show that mechanochemical interactions are sufficient to generate oscillations of myosin contractile activity in the observed spatiotemporal pattern. We propose that follicle cells in the epithelial layer contract against pressure in the expanding egg chamber. As tension in the epithelial layer increases, Rho kinase signaling activates myosin assembly and contraction. The activation process is cooperative, leading to a limit cycle in the myosin dynamics. Our model produces asynchronous oscillations in follicle cell area and myosin content, consistent with experimental observations. In addition, we test the prediction that removal of the basal lamina will increase the average oscillation period. The model demonstrates that in principle, mechanochemical interactions are sufficient to drive patterning and morphogenesis, independent of patterned gene expression.

The rotation of a bacterial flagellar motor (BFM) is driven by multiple stators tethered to the cell wall. Here, we extend a recently proposed power-stroke model to study the BFM dynamics under different biophysical conditions. Our model explains several key experimental observations and reveals their underlying mechanisms. 1), The observed independence of the speed at low load on the number of stators is explained by a force-dependent stepping mechanism that is independent of the strength of the stator tethering spring. Conversely, without force-dependent stepping, an unrealistically weak stator spring is required. 2), Our model with back-stepping naturally explains the observed absence of a barrier to backward rotation. Using the same set of parameters, it also explains BFM behaviors in the high-speed negative-torque regime. 3), From the measured temperature dependence of the maximum speed, our model shows that stator-stepping is a thermally activated process with an energy barrier. 4), The recently observed asymmetry in the torque-speed curve between counterclockwise- and clockwise-rotating BFMs can be quantitatively explained by the asymmetry in the stator-rotor interaction potentials, i.e., a quasilinear form for the counterclockwise motor and a quadratic form for the clockwise motor.

Summary Background Cytokinesis in bacteria is mediated by a cytokinetic ring, termed the Z ring, which forms a scaffold for recruitment of other cell-division proteins. The Z ring is composed of FtsZ filaments, but their organization in the Z ring is poorly understood. In Escherichia coli , the Min system contributes to the spatial regulation of cytokinesis by preventing the assembly of the Z ring away from midcell. The effector of the Min system, MinC, inhibits Z ring assembly by a mechanism that is not clear. Results Here, we report that MinC controls the scaffolding function of FtsZ by antagonizing the mechanical integrity of FtsZ structures. Specifically, MinC antagonizes the ability of FtsZ filaments to be in a solid-like gel state. MinC is a modular protein whose two domains (MinC C and MinC N ) synergize to inhibit FtsZ function. MinC C interacts directly with FtsZ polymers to target MinC to Z rings. MinC C also prevents lateral interactions between FtsZ filaments, an activity that seems to be unique among cytoskeletal proteins. Because MinC C is inhibitory in vivo, it suggests that lateral interactions between FtsZ filaments are important for the structural integrity of the Z ring. MinC N contributes to MinC activity by weakening the longitudinal bonds between FtsZ molecules in a filament leading to a loss of polymer rigidity and consequent polymer shortening. On the basis of our results, we develop the first computational model of the Z ring and study the effects of MinC. Conclusions Control over the scaffolding activity of FtsZ probably represents a universal regulatory mechanism of bacterial cytokinesis.

The life cycle of bacterial cells consists of repeated elongation, septum formation, and division. Before septum formation, a division ring called the Z-ring, which is made of a filamentous tubulin analog, FtsZ, is seen at the mid cell. Together with several other proteins, FtsZ is essential for cell division. Visualization of strains with GFP-labeled FtsZ shows that the Z-ring contracts before septum formation and pinches the cell into two equal halves. Thus, the Z-ring has been postulated to act as a force generator, although the magnitude of the contraction force is unknown. In this article, we develop a mathematical model to describe the process of growth and Z-ring contraction in rod-like bacteria. The elasticity and growth of the cell wall is incorporated in the model to predict the contraction speed, the cell shape, and the contraction force. With reasonable parameters, the model shows that a small force from the Z-ring (8 pN in Escherichia coli ) is sufficient to accomplish division.

We present here the analytical relation between the gain of eukaryotic gradient sensing network and the associated thermodynamic cost. By analyzing a general incoherent type-1 feed-forward loop, we derive the gain function (G) through the reaction network and explicitly show that G depends on the nonequilibrium factor (0≤γ≤1 with γ=0 and 1 representing irreversible and equilibrium reaction systems, respectively), the Michaelis constant (K_{M}), and the turnover ratio (r_{cat}) of the participating enzymes. We further find the maximum possible gain is intrinsically determined by K_{M}/G_{max}=(1/K_{M}+2)/4. Our model also indicates that the dissipated energy (measured by -lnγ), from the intracellular energy-bearing bioparticles (e.g., ATP), is used to generate a force field F_{γ}∝(1-sqrt[γ]) that reshapes and disables the effective potential around the zero gain region, which leads to the ultrasensitive response to external chemical gradients.

Skeletal muscle contraction is a canonical example of motor-driven force generation. Despite the long history of research in this topic, a mechanistic explanation of the collective myosin force generation is lacking. We present a theoretical model of muscle contraction based on the conformational movements of individual myosins and experimentally measured chemical rate constants. Detailed mechanics of the myosin motor and the geometry of the sarcomere are taken into account. Two possible scenarios of force generation are examined. We find only one of the scenarios can give rise to a plausible contraction mechanism. We propose that the synchrony in muscle contraction is due to a force-dependent ADP release step. Computational results of a half sarcomere with 150 myosin heads can explain the experimentally measured force-velocity relationship and efficiency data. We predict that the number of working myosin motors increases as the load force is increased, thus showing synchrony among myosin motors during muscle contraction. We also find that titin molecules anchoring the thick filament are passive force generators in assisting muscle contraction.

Stochastic methods have been a staple for understanding complex systems in chemistry and physics. In the biological context, they are useful for understanding phenomena ranging from molecular-level fluctuations to cellular movement. We review the basic formalism behind stochastic methods and outline how they can be implemented for quantifying gene expression, movement of molecular motors, and the dynamics of cytoplasmic components. We show that stochastic methods are quantitative checks for proposed molecular mechanisms and can pose new questions for experiments. Structural information of cellular components can be incorporated into stochastic models to reveal new biological insights.

The incoherent type-1 feed-forward loop (I1-FFL) is ubiquitous in biological regulatory circuits. Although much is known about the functions of the I1-FFL motif, the energy cost incurred in the network and how it affects the performance of the network have not been investigated. Here, we study a generic I1-FFL enzymatic reaction network modelled after the GEF–GAP–Ras pathway responsible for chemosensory adaptation in eukaryotic cells. Our analysis shows that the I1-FFL network always operates out of equilibrium. Continuous energy dissipation is necessary to drive an internal phosphorylation–dephosphorylation cycle that is crucial in achieving strong short-time response and accurate long-time adaptation. In particular, we show quantitatively that the energy dissipated in the I1-FFL network is used (i) to increase the system's initial response to the input signals; (ii) to enhance the adaptation accuracy at steady state; and (iii) to expand the range of such accurate adaptation. Moreover, we find that the energy dissipation rate, the catalytic speed and the maximum adaptation accuracy in the I1-FFL network satisfy the same energy–speed–accuracy relationship as in the negative-feedback-loop (NFL) networks. Because the I1-FFL and NFL are the only two basic network motifs that enable accurate adaptation, our results suggest that a universal cost–performance trade-off principle may underlie all cellular adaptation processes independent of the detailed biochemical circuit architecture.

In bacterial chemotaxis, several types of receptors form mixed clusters. Receptor adaptation is shown to depend on the receptor's own conformational state rather than on the cluster's global activity, enabling cells to differentiate stimuli in complex environments., We develop here a model for mixed chemoreceptor clusters in which receptors interact directly with their nearest neighbors. A local adaptation scheme is used to describe the methylation kinetics of individual receptors. Predictions made by this model were tested by direct measurements of the receptor methylation dynamics for both Tar and Tsr in response to ligands sensed by either receptor. We show that the local adaptation mechanism tunes each receptor in the mixed cluster to its most responsive state, to maintain the cell's high sensitivity in complex environments with multiple cues. This mechanism also prevents the saturation of the whole receptor cluster with exposure to environments with extreme level of one type of stimulus., In environments with multiple cues, organisms need to sense different signals and to respond accordingly to enhance their chances of survival (Adler and Tso, 1974). In bacterial chemotaxis systems (see Hazelbauer et al (2008) for a recent review), different chemical stimuli are sensed by different types of chemoreceptors. In Escherichia coli cells, different types of chemoreceptors, e.g., the aspartate-sensing Tar receptor and the serine-sensing Tsr receptor, form mixed receptor cluster (Maddock and Shapiro, 1993; Ames et al, 2002), within which different types of chemoreceptors interact with each other cooperatively to amplify external signals (Bray et al, 1998; Sourjik and Berg, 2002; Mello and Tu, 2003b). The bacterial chemosensory system also adapts to prolonged stimuli by covalently modifying the chemoreceptors (methylation and demethylation). However, despite strong interactions between different receptors in the mixed cluster, the adaptive covalent modifications of individual receptors are observed to be insulated from each other. At moderate stimulation, only the receptors that bind the respective ligand adjust their methylation levels significantly in the adapted state (Silverman and Simon, 1977; Sanders and Koshland, 1988; Antommattei et al, 2004). This observed ligand-specific receptor methylation pattern challenges all-or-none allosteric models, such as the Monod–Wyman–Changdeux model (Monod et al, 1965; Sourjik and Berg, 2004; Mello and Tu, 2005; Keymer et al, 2006), that are commonly used to describe behavior of chemoreceptor clusters. It prompts the fundamental questions of how a highly cooperative mixed chemoreceptor complex adapts to multiple stimuli and whether it can distinguish different signals. In this paper, we combine theoretical and experimental methods to understand the adaptation mechanism of mixed chemoreceptor clusters. We propose a local adaptation mechanism (model) for the mixed receptor cluster (Figure 1). In our model, receptors interact with their neighboring receptors in the mixed cluster (Figure 1D) and thus act collectively to generate strong response to small external signals. However, the adaptation of an individual receptor in the mixed cluster depends predominantly on its own local conformational state (Figure 1B), rather than the activity of the entire cluster (Figure 1A and C). Much to our surprise, despite strong interaction between different receptors, our model predicts that only the receptor which binds with the external ligand changes its methylation level in steady state when the system adapts, while other types of receptors only change their methylation levels transiently during adaptation. Our model also predicts that permanent (steady state) methylation crosstalk occurs only when the system fails to adapt accurately, and there exists a direct connection between the adaptation error and the degree of permanent methylation crosstalk. Both predictions are verified by direct quantitative measurements of the dynamics of the Tar and Tsr methylation levels in response to MeAsp and serine (Figure 5). These experimental results cannot be explained by the existing models, such as the MWC-type model and the recently proposed independent receptor model by Goldman et al (2009). The predicted transient adaptation dynamics for a mixed receptor cluster also provides a mechanistic explanation for the previously observed overshoot of activity when E. coli cells adapt to a large step stimulus (Berg and Tedesco, 1975). After establishing the validity of the local adaptation mechanism in E. coli, we next explore the possible advantages of this adaptation mechanism for bacterial chemotaxis. We show that while the previously proposed global adaptation mechanisms compress different external environmental information, e.g., concentrations of different types of attractant ligands into one quantity (the overall activity of the cluster), the local adaptation mechanism preserves environmental information. The concentrations of different chemoeffectors are encoded (‘remembered') by the specific receptor methylation levels in the local adaptation model. These ligand-specific information can then be used by bacterial cells to precisely tune each type of receptor in the mixed cluster to its most responsive state, therefore maintaining high sensitivity and responsiveness in complex environments with multiple stimuli. The local adaptation mechanism, by effectively preventing methylation crosstalk, also prevents the poisoning effect by methylation contamination when bacterial cells are exposed to environments with extreme level of one type of stimulus. In summary, we have developed a modeling framework for understanding how a mixed chemoreceptor cluster adapts to complex environments with multiple cues. The local adaptation and global activation mechanism of the mixed receptor cluster proposed in this paper resolves the seemingly conflicting observations between strong receptor–receptor interactions and the absence of permanent receptor methylation crosstalk within a unified and predictive model. Direct quantitative measurements of the receptor methylation dynamics have confirmed the model predictions. The proposed model also predicts several characteristic consequences of the local adaptation mechanism, e.g., elimination of sensory poisoning, which may be tested in future experiments. Bacterial chemosensory machinery is a paradigm for studying adaptive sensory systems for detecting and adapting to environmental changes and signals, and we expect that the strategy of ‘adapting locally (individually) and acting globally (collectively)' may be used by other sensory systems that utilize multiple receptors to respond complex environmental changes., In bacterial chemotaxis, several types of ligand-specific receptors form mixed clusters, wherein receptor–receptor interactions lead to signal amplification and integration. However, it remains unclear how a mixed receptor cluster adapts to individual stimuli and whether it can differentiate between different types of ligands. Here, we combine theoretical modeling with experiments to reveal the adaptation dynamics of the mixed chemoreceptor cluster in Escherichia coli. We show that adaptation occurs locally and is ligand-specific: only the receptor that binds the external ligand changes its methylation level when the system adapts, whereas other types of receptors change methylation levels transiently. Permanent methylation crosstalk occurs when the system fails to adapt accurately. This local adaptation mechanism enables cells to differentiate individual stimuli by encoding them into the methylation levels of corresponding types of chemoreceptors. It tunes each receptor to its most responsive state to maintain high sensitivity in complex environments and prevents saturation of the cluster by one signal.