Rational social distancing policy during epidemics with limited healthcare capacity.

Epidemics of infectious diseases posing a serious risk to human health have occurred throughout history. During recent epidemics there has been much debate about policy, including how and when to impose restrictions on behaviour. Policymakers must balance a complex spectrum of objectives, suggesting a need for quantitative tools. Whether health services might be 'overwhelmed' has emerged as a key consideration. Here we show how costly interventions, such as taxes or subsidies on behaviour, can be used to exactly align individuals' decision making with government preferences even when these are not aligned. In order to achieve this, we develop a nested optimisation algorithm of both the government intervention strategy and the resulting equilibrium behaviour of individuals. We focus on a situation in which the capacity of the healthcare system to treat patients is limited and identify conditions under which the disease dynamics respect the capacity limit. We find an extremely sharp drop in peak infections at a critical maximum infection cost in the government's objective function. This is in marked contrast to the gradual reduction of infections if individuals make decisions without government intervention. We find optimal interventions vary less strongly in time when interventions are costly to the government and that the critical cost of the policy switch depends on how costly interventions are.

Environmental Path-Entropy and Collective Motion.

Inspired by the swarming or flocking of animal systems we study groups of agents moving in unbounded 2D space. Individual trajectories derive from a "bottom-up" principle: individuals reorient to maximize their future path entropy over environmental states. This can be seen as a proxy for keeping options open, a principle that may confer evolutionary fitness in an uncertain world. We find an ordered (coaligned) state naturally emerges, as well as disordered states or rotating clusters; similar phenotypes are observed in birds, insects, and fish, respectively. The ordered state exhibits an order-disorder transition under two forms of noise: (i) standard additive orientational noise, applied to the postdecision orientations and (ii) "cognitive" noise, overlaid onto each individual's model of the future paths of other agents. Unusually, the order increases at low noise, before later decreasing through the order-disorder transition as the noise increases further.

Intrinsically motivated collective motion.

Collective motion is found in various animal systems, active suspensions, and robotic or virtual agents. This is often understood by using high-level models that directly encode selected empirical features, such as coalignment and cohesion. Can these features be shown to emerge from an underlying, low-level principle? We find that they emerge naturally under future state maximization (FSM). Here, agents perceive a visual representation of the world around them, such as might be recorded on a simple retina, and then move to maximize the number of different visual environments that they expect to be able to access in the future. Such a control principle may confer evolutionary fitness in an uncertain world by enabling agents to deal with a wide variety of future scenarios. The collective dynamics that spontaneously emerge under FSM resemble animal systems in several qualitative aspects, including cohesion, coalignment, and collision suppression, none of which are explicitly encoded in the model. A multilayered neural network trained on simulated trajectories is shown to represent a heuristic mimicking FSM. Similar levels of reasoning would seem to be accessible under animal cognition, demonstrating a possible route to the emergence of collective motion in social animals directly from the control principle underlying FSM. Such models may also be good candidates for encoding into possible future realizations of artificial "intelligent" matter, able to sense light, process information, and move.

Measuring Gaussian Rigidity Using Curved Substrates.

The Gaussian (saddle splay) rigidity of fluid membranes controls their equilibrium topology but is notoriously difficult to measure. In lipid mixtures, typical of living cells, linear interfaces separate liquid ordered (LO) from liquid disordered (LD) bilayer phases at subcritical temperatures. Here, we consider such membranes supported by curved substrates that thereby control the membrane curvatures. We show how spectral analysis of the fluctuations of the LO-LD interface provides a novel way of measuring the difference in Gaussian rigidity between the two phases. We provide a number of conditions for such interface fluctuations to be both experimentally measurable and sufficiently sensitive to the value of the Gaussian rigidity, while remaining in the perturbative regime of our analysis.

Shear-Driven Instabilities of Membrane Tubes and Dynamin-Induced Scission.

Motivated by the mechanics of dynamin-mediated membrane tube fission, we analyze the stability of fluid membrane tubes subjected to shear flow in azimuthal direction. We find a novel helical instability driven by the membrane shear flow which results in a nonequilibrium steady state for the tube fluctuations. This instability has its onset at shear rates that may be physiologically accessible under the action of dynamin and could also be probed using in vitro experiments on membrane nanotubes, e.g., using magnetic tweezers. We discuss how such an instability may play a role in the mechanism for dynamin-mediated membrane tube fission.

Dynamics of passive and active membrane tubes.

Utilising Onsager's variational formulation, we derive dynamical equations for the relaxation of a fluid membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across the membrane and variations in surface tension due to membrane incompressibility. We compute the relaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making new predictions for higher order (azimuthal) m-modes. We analyse the long and short wavelength limits of these modes by making use of various asymptotic arguments. We incorporate stochastic terms to our dynamical equations suitable to describe both passive thermal forces and non-equilibrium active forces. We derive expressions for the fluctuation amplitudes, an effective temperature associated with active fluctuations, and the power spectral density for both the thermal and active fluctuations. We discuss an experimental assay that might enable measurement of these fluctuations to infer the properties of the active noise. Finally we discuss our results in the context of active membranes more generally and give an overview of some open questions in the field.

A topologically driven glass in ring polymers.

The static and dynamic properties of ring polymers in concentrated solutions remains one of the last deep unsolved questions in polymer physics. At the same time, the nature of the glass transition in polymeric systems is also not well understood. In this work, we study a novel glass transition in systems made of circular polymers by exploiting the topological constraints that are conjectured to populate concentrated solutions of rings. We show that such rings strongly interpenetrate through one another, generating an extensive network of topological interactions that dramatically affects their dynamics. We show that a kinetically arrested state can be induced by randomly pinning a small fraction of the rings. This occurs well above the classical glass transition temperature at which microscopic mobility is lost. Our work both demonstrates the existence of long-lived inter-ring penetrations and realizes a novel, topologically induced, glass transition.

Hydro-osmotic Instabilities in Active Membrane Tubes.

We study a membrane tube with unidirectional ion pumps driving an osmotic pressure difference. A pressure-driven peristaltic instability is identified, qualitatively distinct from similar tension-driven Rayleigh-type instabilities on membrane tubes. We discuss how this instability could be related to the function and biogenesis of membrane bound organelles, in particular, the contractile vacuole complex. The unusually long natural wavelength of this instability is in agreement with that observed in cells.

The role of optical projection in the analysis of membrane fluctuations.

The spectral analysis of thermal fluctuations, or flickering, is a simple and non-invasive method widely used to determine the mechanical properties of artificial and biological lipid membranes. In its most common implementation, the position of the edge of a cell or vesicle is tracked from optical microscopy videos. However, a systematic disagreement with X-ray scattering and micromechanical manipulation data has brought into question the validity of the method. We present an improved analysis protocol that resolves these discrepancies by accounting for the finite vertical resolution of the optics used to image fluctuations.

Role of projection in the control of bird flocks.

Swarming is a conspicuous behavioral trait observed in bird flocks, fish shoals, insect swarms, and mammal herds. It is thought to improve collective awareness and offer protection from predators. Many current models involve the hypothesis that information coordinating motion is exchanged among neighbors. We argue that such local interactions alone are insufficient to explain the organization of large flocks of birds and that the mechanism for the exchange of long-range information necessary to control their density remains unknown. We show that large flocks self-organize to the maximum density at which a typical individual still can see out of the flock in many directions. Such flocks are marginally opaque--an external observer also still can see a substantial fraction of sky through the flock. Although this seems intuitive, we show it need not be the case; flocks might easily be highly diffuse or entirely opaque. The emergence of marginal opacity strongly constrains how individuals interact with one another within large swarms. It also provides a mechanism for global interactions: an individual can respond to the projection of the flock that it sees. This provides for faster information transfer and hence rapid flock dynamics, another advantage over local models. From a behavioral perspective, it optimizes the information available to each bird while maintaining the protection of a dense, coherent flock.

Erratum: Mobility Measurements Probe Conformational Changes in Membrane Proteins due to Tension [Phys. Rev. Lett. 115, 198101 (2015)].

This corrects the article DOI: 10.1103/PhysRevLett.115.198101.

Mobility Measurements Probe Conformational Changes in Membrane Proteins due to Tension.

The function of membrane-embedded proteins such as ion channels depends crucially on their conformation. We demonstrate how conformational changes in asymmetric membrane proteins may be inferred from measurements of their diffusion. Such proteins cause local deformations in the membrane, which induce an extra hydrodynamic drag on the protein. Using membrane tension to control the magnitude of the deformations, and hence the drag, measurements of diffusivity can be used to infer-via an elastic model of the protein-how conformation is changed by tension. Motivated by recent experimental results [Quemeneur et al., Proc. Natl. Acad. Sci. U.S.A. 111, 5083 (2014)], we focus on KvAP, a voltage-gated potassium channel from Aeropyrum pernix. The conformation of KvAP is found to change considerably due to tension, with its "walls," where the protein meets the membrane, undergoing significant angular strains. The torsional stiffness is determined to be 26.8k(B)T per radian at room temperature. This has implications for both the structure and the function of such proteins in the environment of a tension-bearing membrane.

Membrane composition variation and underdamped mechanics near transmembrane proteins and coats.

We study the effect of transmembrane proteins on the shape, composition, and thermodynamic stability of the surrounding membrane. When the coupling between membrane composition and curvature is strong enough, the nearby membrane composition and shape both undergo a transition from overdamped to underdamped spatial variation, well before the membrane becomes unstable in the bulk. This transition is associated with a change in the sign of the thermodynamic energy and, hence, favors the early stages of coat assembly necessary for vesiculation (budding) and may suppress the activity of mechanosensitive membrane channels and transporters. Our results suggest an approach to obtain physical parameters of the membrane that are otherwise difficult to measure.

Rings in random environments: sensing disorder through topology.

In this paper we study the role of topology in DNA gel electrophoresis experiments via molecular dynamics simulations. The gel is modelled as a 3D array of obstacles from which half edges are removed at random with probability p, thereby generating a disordered environment. Changes in the microscopic structure of the gel are captured by measuring the electrophoretic mobility of ring polymers moving through the medium, while their linear counterparts provide a control system as we show they are insensitive to these changes. We show that ring polymers provide a novel, non-invasive way of exploiting topology to sense microscopic disorder. Finally, we compare the results from the simulations with an analytical model for the non-equilibrium differential mobility, and find a striking agreement between simulation and theory.

Dynamics of self-threading ring polymers in a gel.

We study the dynamics of ring polymers confined to diffuse in a background gel at low concentrations. We do this in order to probe the inter-play between topology and dynamics in ring polymers. We develop an algorithm that takes into account the possibility that the rings hinder their own motion by passing through themselves, i.e. "self-threading". Our results suggest that the number of self-threadings scales extensively with the length of the rings and that this is substantially independent of the details of the model. The slowing down of the rings' dynamics is found to be related to the fraction of segments that can contribute to the motion. Our results give a novel perspective on the motion of ring polymers in a gel, for which a complete theory is still lacking, and may help us to understand the irreversible trapping of ring polymers in gel electrophoresis experiments.

Mobility in geometrically confined membranes.

Lipid and protein lateral mobility is essential for biological function. Our theoretical understanding of this mobility can be traced to the seminal work of Saffman and Delbrück, who predicted a logarithmic dependence of the protein diffusion coefficient (i) on the inverse of the size of the protein and (ii) on the "membrane size" for membranes of finite size [Saffman P, Delbrück M (1975) Proc Natl Acad Sci USA 72:3111-3113]. Although the experimental proof of the first prediction is a matter of debate, the second has not previously been thought to be experimentally accessible. Here, we construct just such a geometrically confined membrane by forming lipid bilayer nanotubes of controlled radii connected to giant liposomes. We followed the diffusion of individual molecules in the tubular membrane using single particle tracking of quantum dots coupled to lipids or voltage-gated potassium channels KvAP, while changing the membrane tube radius from approximately 250 to 10 nm. We found that both lipid and protein diffusion was slower in tubular membranes with smaller radii. The protein diffusion coefficient decreased as much as 5-fold compared to diffusion on the effectively flat membrane of the giant liposomes. Both lipid and protein diffusion data are consistent with the predictions of a hydrodynamic theory that extends the work of Saffman and Delbrück to cylindrical geometries. This study therefore provides strong experimental support for the ubiquitous Saffman-Delbrück theory and elucidates the role of membrane geometry and size in regulating lateral diffusion.

Learning to swim efficiently in a nonuniform flow field.

Microswimmers can acquire information on the surrounding fluid by sensing mechanical queues. They can then navigate in response to these signals. We analyze this navigation by combining deep reinforcement learning with direct numerical simulations to resolve the hydrodynamics. We study how local and nonlocal information can be used to train a swimmer to achieve particular swimming tasks in a nonuniform flow field, in particular, a zigzag shear flow. The swimming tasks are (1) learning how to swim in the vorticity direction, (2) learning how to swim in the shear-gradient direction, and (3) learning how to swim in the shear-flow direction. We find that access to laboratory frame information on the swimmer's instantaneous orientation is all that is required in order to reach the optimal policy for tasks (1) and (2). However, information on both the translational and rotational velocities seems to be required to accomplish task (3). Inspired by biological microorganisms, we also consider the case where the swimmers sense local information, i.e., surface hydrodynamic forces, together with a signal direction. This might correspond to gravity or, for microorganisms with light sensors, a light source. In this case, we show that the swimmer can reach a comparable level of performance to that of a swimmer with access to laboratory frame variables. We also analyze the role of different swimming modes, i.e., pusher, puller, and neutral.

Rational social distancing in epidemics with uncertain vaccination timing.

During epidemics people may reduce their social and economic activity to lower their risk of infection. Such social distancing strategies will depend on information about the course of the epidemic but also on when they expect the epidemic to end, for instance due to vaccination. Typically it is difficult to make optimal decisions, because the available information is incomplete and uncertain. Here, we show how optimal decision-making depends on information about vaccination timing in a differential game in which individual decision-making gives rise to Nash equilibria, and the arrival of the vaccine is described by a probability distribution. We predict stronger social distancing the earlier the vaccination is expected and also the more sharply peaked its probability distribution. In particular, equilibrium social distancing only meaningfully deviates from the no-vaccination equilibrium course if the vaccine is expected to arrive before the epidemic would have run its course. We demonstrate how the probability distribution of the vaccination time acts as a generalised form of discounting, with the special case of an exponential vaccination time distribution directly corresponding to regular exponential discounting.

A bending rigidity parameter for stress granule condensates.

Interfacial tension plays an important role in governing the dynamics of droplet coalescence and determining how condensates interact with and deform lipid membranes and biological filaments. We demonstrate that an interfacial tension-only model is inadequate for describing stress granules in live cells. Harnessing a high-throughput flicker spectroscopy pipeline to analyze the shape fluctuations of tens of thousands of stress granules, we find that the measured fluctuation spectra require an additional contribution, which we attribute to elastic bending deformation. We also show that stress granules have an irregular, nonspherical base shape. These results suggest that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Furthermore, we observe that the measured interfacial tensions and bending rigidities span a range of several orders of magnitude. Hence, different types of stress granules (and more generally, other biomolecular condensates) can only be differentiated via large-scale surveys.

The mechanics of FtsZ fibers.

Inhibition of the Fts family of proteins causes the growth of long filamentous cells, indicating that they play some role in cell division. FtsZ polymerizes into protofilaments and assembles into the Z-ring at the future site of the septum of cell division. We analyze the rigidity of GTP-bound FtsZ protofilaments by using cryoelectron microscopy to sample their bending fluctuations. We find that the FtsZ-GTP filament rigidity is κ=4.7±1.0×10(-27) Nm(2), with a corresponding thermal persistence length of l(p)=1.15±0.25µm, much higher than previous estimates. In conjunction with other model studies, our new higher estimate for FtsZ rigidity suggests that contraction of the Z-ring may generate sufficient force to facilitate cell division. The good agreement between the measured mode amplitudes and that predicted by equipartition of energy supports our use of a simple mechanical model for FtsZ fibers. The study also provides evidence that the fibers have no intrinsic global or local curvatures, such as might be caused by partial hydrolysis of the GTP.