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We present a systems-level analysis of end-stage kidney disease (ESKD) with a dynamical network analysis of 14 commonly measured blood-based biomarkers in patients undergoing regular haemodialysis. Utilizing a validated pipeline for declining homeostatic systems, our approach learns a dynamical model together with an invertible transformation that simplifies the behaviour of observed biomarkers into natural variables. Within the natural variables, we identified two distinct dynamical behaviours: (i) stochastic accumulation, the random accumulation of abnormal values, and (ii) mallostasis, a deterministic drift towards worse health. These behaviours are identified by persistent fluctuations indicating weak stability, or a gradual shift in homeostatic set point, respectively. Both lead to worsening natural variable values, making the natural variables salient survival predictors with preferred directions of increasing risk. When this worsening is transformed back into observable biomarkers, it generates a coherent spectrum of worsening medical signs characteristic of a medical syndrome. Specifically, we found that small modules of natural variables corresponded to two existing syndromes commonly afflicting ESKD patients: protein-energy wasting and sepsis. We also identified new prospective syndromes. Our findings suggest that natural variables are robust, systems-level biomarkers, capturing the complex, holistic changes in health associated with ESKD., Comment: 50 pages (15 for main, 35 supplemental), 10 figures in main

Naturally occurring protein fibers often undergo anisotropic swelling when hydrated. Within a tendon, a hydrated collagen fibril's radius expands by 40% but its length only increases by 5%. The same effect, with a similar relative magnitude, is observed for single hair shafts. Fiber hydration is known to affect elastic properties. Here we show that anisotropic swelling constrains the anisotropic linear elastic properties of fibers. First we show, using data from disparate previously reported studies, that anisotropic swelling can be described as an approximately linear function of water content. Then, under the observation that the elastic energy of swelling can be minimized by the anisotropic shape, we relate swelling anisotropy to elastic anisotropy -- assuming radial (transverse) symmetry within a cylindrical geometry. We find an upper bound for the commonly measured axial Poisson ratio $\nu_{zx}<1/2$. This is significantly below recently estimated values for collagen fibrils extracted from tissue-level measurements, but is consistent with both single hair shaft and single collagen fibril mechanical and hydration studies. Using $\nu_{zx}$, we can then constrain the product $\gamma \equiv (1-\nu_{xy}) E_z/E_x$ -- where $\nu_{xy}$ is the seldom measured transverse Poisson ratio and $E_z/E_x$ is the ratio of axial to radial Young's moduli., Comment: 13 pages, 5 figures. To be published in Journal of the Mechanical Behavior of Biomedical Materials

Widespread interest in non-destructive biomarkers of aging has led to a curse of plenty: a multitude of biological ages that each proffers a 'true' health-adjusted age of an individual. While each measure provides salient information on the aging process, they are each univariate, in contrast to the "hallmark" and "pillar" theories of aging which are explicitly multidimensional, multicausal and multiscale. Fortunately, multiple biological ages can be systematically combined into a multidimensional network representation. The interaction network between these biological ages permits analysis of the multidimensional effects of aging, as well as quantification of causal influences during both natural aging and, potentially, after anti-aging intervention. The behaviour of the system as a whole can then be explored using dynamical network stability analysis which identifies new, efficient biomarkers that quantify long term resilience scores on the timescale between measurements (years). We demonstrate this approach using a set of 8 biological ages from the longitudinal Swedish Adoption/Twin Study of Aging (SATSA). After extracting an interaction network between these biological ages, we observed that physiological age, a proxy for cardiometabolic health, serves as a central node in the network, implicating it as a key vulnerability for slow, age-related decline. We furthermore show that while the system as a whole is stable, there is a weakly stable direction along which recovery is slow - on the timescale of a human lifespan. This slow direction provides an aging biomarker which correlates strongly with chronological age and predicts longitudinal decline in health - suggesting that it estimates an important driver of age-related changes., Comment: 65 pages including supplemental

Using longitudinal study data, we dynamically model how aging affects homeostasis in both mice and humans. We operationalize homeostasis as a multivariate mean-reverting stochastic process. We hypothesize that biomarkers have stable equilibrium values, but that deviations from equilibrium of each biomarker affects other biomarkers through an interaction network - this precludes univariate analysis. We therefore looked for age-related changes to homeostasis using dynamic network stability analysis, which transforms observed biomarker data into independent "natural" variables and determines their associated recovery rates. Most natural variables remained near equilibrium and were essentially constant in time. A small number of natural variables were unable to equilibrate due to a gradual drift with age in their homeostatic equilibrium, i.e. allostasis. This drift caused them to accumulate over the lifespan course and makes them natural aging variables. Their rate of accumulation was correlated with risk of adverse outcomes: death or dementia onset. We call this tendency for aging organisms to drift towards an equilibrium position of ever-worsening health "mallostasis". We demonstrate that the effects of mallostasis on observed biomarkers are spread out through the interaction network. This could provide a redundancy mechanism to preserve functioning until multi-system dysfunction emerges at advanced ages., Comment: 42 pages including supplmenetal

We model the effects of disease and other exogenous damage during human aging. Even when the exogenous damage is repaired at the end of acute disease, propagated secondary damage remains. We consider both short-term mortality effects due to (acute) exogenous damage and long-term mortality effects due to propagated damage within the context of a generic network model (GNM) of individual aging that simulates a U.S. population. Across a wide range of disease durations and severities we find that while excess short-term mortality is highest for the oldest individuals, the long-term years of life lost are highest for the youngest individuals. These appear to be universal effects of human disease. We support this conclusion with a phenomenological model coupling damage and mortality. Our results are consistent with previous lifetime mortality studies of atom bomb survivors and post-recovery health studies of COVID-19. We suggest that short-term health impact studies could complement lifetime mortality studies to better characterize the lifetime impacts of disease on both individuals and populations.

We investigate the elastic properties of anisotropic elastomers with a double-twist director field, which is a model for collagen fibrils or blue phases. We observe a significant Poynting-like effect, coupling torsion (fibril twist) and extension. For freely-rotating boundary conditions, we identify a structural bistability at very small extensional strains which undergoes a saddle-node bifurcation at a critical strain -- at approximately 1% strain for a parameterization appropriate for collagen fibrils. With clamped boundary conditions appropriate for many experimental setups, the bifurcation is not present. We expect significant helical shape effects when fixed torsion does not equal the equilibrium torsion of freely-rotating boundary conditions, due to residual torques., Comment: 11 pages, 9 figures

Single file diffusion (SFD) exhibits anomalously slow collective transport when particles are able to immobilize by binding and unbinding to the one-dimensional channel within which the particles diffuse. We have explored this system for short pore-like channels using a symmetric exclusion process (SEP) with fully stochastic dynamics. We find that for shorter channels, a non-Fickian regime emerges for slow binding kinetics. In this regime the average flux $\langle \Phi \rangle \sim 1/L^3$, where $L$ is the channel length in units of the particle size. We find that a two-state model describes this behavior well for sufficiently slow binding rates, where the binding rates determine the switching time between high-flux bursts of directed transport and low-flux leaky states. Each high-flux burst is Fickian with $\langle \Phi \rangle \sim 1/L$. Longer systems are more often in a low flux state, leading to the non-Fickian behavior.

We have built a computational model for individual aging trajectories of health and survival, which contains physical, functional, and biological variables, and is conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with an interpretable interaction network, where health variables are coupled by explicit pair-wise interactions within a stochastic dynamical system. Our dynamic joint interpretable network (DJIN) model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival from baseline health states, and infers an interpretable network of directed interactions between the health variables. The network identifies plausible physiological connections between health variables as well as clusters of strongly connected health variables. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than multiple dedicated linear models for health outcomes and survival. We compare our model with flexible lower-dimensional latent-space models to explore the dimensionality required to accurately model aging health outcomes. Our DJIN model can be used to generate synthetic individuals that age realistically, to impute missing data, and to simulate future aging outcomes given arbitrary initial health states.

Collagen fibrils are the main structural component of load-bearing tissues such as tendons, ligaments, skin, the cornea of the eye, and the heart. The D-band of collagen fibrils is an axial periodic density modulation that can be easily characterized by tissue-level X-ray scattering. During mechanical testing, D-band strain is often used as a proxy for fibril strain. However, this approach ignores the coupling between strain and molecular tilt. We examine the validity of this approximation using an elastomeric collagen fibril model that includes both the D-band and a molecular tilt field. In the low strain regime, we show that the D-band strain substantially underestimates fibril strain for strongly twisted collagen fibrils -- such as fibrils from skin or corneal tissue., Comment: 12 pages, 1 figure

We adapt the theory of anisotropic rubber elasticity to model cross-linked double-twist liquid crystal cylinders such as exhibited in biological systems. In mechanical extension we recover strain-straightening, but with an exact expression in the small twist-angle limit. In compression, we observe coexistence between high and low twist phases. Coexistence begins at small compressive strains and is robustly observed for any anisotropic cross-links and for general double-twist functions -- but disappears at large twist angles. Within the coexistence region, significant compression of double-twist cylinders is allowed at constant stress. Our results are qualitatively consistent with previous observations of swollen or compressed collagen fibrils, indicating that this phenomenon may be readily accessible experimentally., Comment: 7 pages, 5 figures

The lysyl oxidase (LOX) enzyme that catalyses cross-link formation during the assembly of collagen fibrils in vivo is too large to diffuse within assembled fibrils, and so is incompatible with a fully equilibrium mechanism for fibril formation. We propose that enzymatic cross-links are formed at the fibril surface during the growth of collagen fibrils; as a consequence no significant reorientation of previously cross-linked collagen molecules occurs inside collagen fibrils during fibril growth in vivo. By imposing local equilibrium only at the fibril surface, we develop a coarse-grained quantitative model of in vivo fibril structure that incorporates a double-twist orientation of collagen molecules and a periodic D-band density modulation along the fibril axis. Radial growth is controlled by the density of available collagen molecules around the fibril. In contrast with earlier equilibrium models of fibril structure, we find that all fibrils can exhibit a core-shell structure that is controlled only by the fibril radius. At small radii a core is developed with a linear double-twist structure as a function of radius. Within the core the double-twist structure is largely independent of the D-band. Within the shell at larger radii, the structure approaches a constant twist configuration that is strongly coupled with the D-band. We suggest a stable radius control mechanism that corneal fibrils can exploit near the edge of the linear core regime; while larger tendon fibrils can use a cruder version of growth control that does not select a preferred radius., Comment: 13 pages, 10 figures

The gradual accumulation of damage and dysregulation during the aging of living organisms can be quantified. Even so, the aging process is complex and has multiple interacting physiological scales -- from the molecular to cellular to whole tissues. In the face of this complexity, we can significantly advance our understanding of aging with the use of computational models that simulate realistic individual trajectories of health as well as mortality. To do so, they must be systems-level models that incorporate interactions between measurable aspects of age-associated changes. To incorporate individual variability in the aging process, models must be stochastic. To be useful they should also be predictive, and so must be fit or parameterized by data from large populations of aging individuals. In this perspective, we outline where we have been, where we are, and where we hope to go with such computational models of aging. Our focus is on data-driven systems-level models, and on their great potential in aging research.

To describe the interaction between longitudinal density modulations along collagen fibrils (the D-band) with the radial twist-field of molecular orientation (double-twist), we couple phase-field-crystal (PFC) with liquid-crystalline free-energies to obtain a hybrid model of equilibrium collagen fibril structure. We numerically compute the resulting axial and radial structure. We find two distinct fibrillar phases, `L' and `C', with a coexistence line that ends in an Ising-like critical point. We propose that coexistence between these phases can explain the bimodal distribution of fibril radii that has been widely reported within tendon tissues. Tensile strain applied to our model fibrils straightens the average fibrillar twist and flattens the D-band modulation. Our PFC approach should apply directly to other longitudinally-modulated chiral filaments, such as fibrin and intermediate filaments.

We computationally study the effects of binding kinetics to the channel wall, leading to transient immobility, on the diffusive transport of particles within narrow channels, that exhibit single-file diffusion (SFD). We find that slow binding kinetics leads to an anomalously slow diffusive transport. Remarkably, the scaled diffusivity $\hat{D}$ characterizing transport exhibits scaling collapse with respect to the occupation fraction $p$ of sites along the channel. We present a simple "cage-physics" picture that captures the characteristic occupation fraction $p_{scale}$ and the asymptotic $1/p^2$ behavior for $p/p_{scale} \gtrsim 1$. We confirm that subdiffusive behavior of tracer particles is controlled by the same $\hat{D}$ as particle transport.

We confront a network model of human aging and mortality in which nodes represent health attributes that interact within a scale-free network topology, with observational data that uses both clinical and laboratory (pre-clinical) health deficits as network nodes. We find that individual health attributes exhibit a wide range of mutual information with mortality and that, with a re- construction of their relative connectivity, higher-ranked nodes are more informative. Surprisingly, we find a broad and overlapping range of mutual information of laboratory measures as compared with clinical measures. We confirm similar behavior between most-connected and least-connected model nodes, controlled by the nearest-neighbor connectivity. Furthermore, in both model and observational data, we find that the least-connected (laboratory) nodes damage earlier than the most-connected (clinical) deficits. A mean-field theory of our network model captures and explains this phenomenon, which results from the connectivity of nodes and of their connected neighbors. We find that other network topologies, including random, small-world, and assortative scale-free net- works, exhibit qualitatively different behavior. Our disassortative scale-free network model behaves consistently with our expanded phenomenology observed in human aging, and so is a useful tool to explore mechanisms of and to develop new predictive measures for human aging and mortality., Comment: 16 pages, 16 figures

Collagen fibrils are versatile self-assembled structures that provide mechanical integrity within mammalian tissues. The radius of collagen fibrils vary widely depending on experimental conditions \textit{in vitro} or anatomical location \textit{in vivo}. Here we explore the variety of thermodynamically stable fibril configurations that are available. We use a liquid crystal model of radial collagen fibril structure with a double-twist director field. Using a numerical relaxation method we show that two dimensionless parameters, the ratio of saddle-splay to twist elastic constants $ k_{24}/K_{22}$ and the ratio of surface tension to chiral strength $\hat{\gamma} \equiv \gamma/(K_{22}q)$, largely specify both the scaled fibril radius and the associated surface twist of equilibrium fibrils. We find that collagen fibrils are the stable phase with respect to the cholesteric phase only when the reduced surface tension is small, $\hat{\gamma} \lesssim 0.2$. Within this stable regime, collagen fibrils can access a wide range of radii and associated surface twists. Remarkably, we find a maximal equilibrium surface twist of $0.33$ rad ($19^{\text{o}}$). Our results are compatible with corneal collagen fibrils, and we show how the large surface twist is needed to explain the narrow distribution of corneal fibril radii. Conversely, we show how small surface twist is required for the thermodynamic stability of tendon fibrils in the face of considerable polydispersity of radius.

Randomly rotating particles that have been isotropically labeled with rigidly linked fluorophores will undergo non-isotropic (patchy) photobleaching under illumination due to the dipole coupling of fluorophores with light. For a rotational diffusion rate $D$ of the particle and a photobleaching timescale $\tau$ of the fluorophores, the dynamics of this process are characterized by the dimensionless combination $D \tau$. We find significant interparticle fluctuations at intermediate $D \tau$. These fluctuations vanish at both large and small $D \tau$, or at small or large elapsed times $t$. Associated with these fluctuations between particles, we also observe transient non-monotonicities of the brightness of individual particles. These non-monotonicities can be as much 20\% of the original brightness. We show that these novel photobleach-fluctuations dominate over variability of single-fluorophore orientation when there are at least $10^3$ fluorophores on individual particles., Comment: 6 pages, 5 figures

To explore the mechanistic relationships between ageing, frailty and mortality, we developed a computational model in which possible health attributes are represented by the nodes of a complex network. Each node can be either damaged (i.e. a deficit) or undamaged. Damage of connected nodes facilitates further local damage. Our model recovers the known patterns of frailty and mortality without any programmed ageing. It helps us to understand how the observed maximum of the frailty index (FI) might arise, and allows us to start to understand how health deficits accumulate. Large model populations allow us to exploit new analytic tools, including information theory. This will allow us to systematically characterize the effects of sudden changes in the health trajectories of individuals and serve as a way to evaluate large clinical and population databases., Comment: 7 pages, 1 figure

Aging is associated with the accumulation of damage throughout a persons life. Individual health can be assessed by the Frailty Index (FI). The FI is calculated simply as the proportion $f$ of accumulated age related deficits relative to the total, leading to a theoretical maximum of $f \leq 1$. Observational studies have generally reported a much more stringent bound, with $f \leq f_{max} <1$. The value of $f_{max}$ in observational studies appears to be non-universal, but $f_{max} \approx 0.7$ is often reported. A previously developed network model of individual aging was unable to recover $f_{max}<1$ while retaining the other observed phenomenology of increasing $f$ and mortality rates with age. We have developed a computationally accelerated network model that also allows us to tune the scale-free network exponent $\alpha$. The network exponent $\alpha$ significantly affects the growth of mortality rates with age. However, we are only able to recover $f_{max}$ by also introducing a deficit sensitivity parameter $1-q$, which is equivalent to a false-negative rate $q$. Our value of $q=0.3$ is comparable to finite sensitivities of age-related deficits with respect to mortality that are often reported in the literature. In light of non-zero $q$, we use mutual information $I$ to provide a non-parametric measure of the predictive value of the FI with respect to individual mortality. We find that $I$ is only modestly degraded by $q<1$, and this degradation is mitigated when increasing number of deficits are included in the FI. We also find that the information spectrum, i.e. the mutual information of individual deficits vs connectivity, has an approximately power-law dependence that depends on the network exponent $\alpha$. Mutual information $I$ is therefore a useful tool for characterizing the network topology of aging populations., Comment: 12 pages

Selective autophagy must not only select the correct type of organelle, but also must discriminate between individual organelles of the same kind so that some but not all of the organelles are removed. We propose that physical clustering of autophagy receptor proteins on the organelle surface can provide an appropriate all-or-none signal for organelle degradation. We explore this proposal using a computational model restricted to peroxisomes and the relatively well characterized pexophagy receptor proteins NBR1 and p62. We find that larger peroxisomes nucleate NBR1 clusters first and lose them last through competitive coarsening. This results in significant size-selectivity that favors large peroxisomes, and can explain the increased catalase signal that results from siRNA inhibition of p62. Excess ubiquitin, resulting from damaged organelles, suppresses size-selectivity but not cluster formation. Our proposed selectivity mechanism thus allows all damaged organelles to be degraded, while otherwise selecting only a portion of organelles for degradation., Comment: 25 pages, 4 figures

Collagen fibril cross-sectional radii show no systematic variation between the interior and the periphery of fibril bundles, indicating an effectively constant rate of collagen incorporation into fibrils throughout the bundle. Such spatially homogeneous incorporation constrains the extracellular diffusion of collagen precursors from sources at the bundle boundary to sinks at the growing fibrils. With a coarse-grained diffusion equation we determine stringent bounds, using parameters extracted from published experimental measurements of tendon development. From the lack of new fibril formation after birth, we further require that the concentration of diffusing precursors stays below the critical concentration for fibril nucleation. We find that the combination of the diffusive bound, which requires larger concentrations to ensure homogeneous fibril radii, and lack of nucleation, which requires lower concentrations, is only marginally consistent with fully-processed collagen using conservative bounds. More realistic bounds may leave no consistent concentrations. Therefore, we propose that unprocessed pC-collagen diffuses from the bundle periphery followed by local C-proteinase activity and subsequent collagen incorporation at each fibril. We suggest that C-proteinase is localized within bundles, at fibril surfaces, during radial fibrillar growth. The much greater critical concentration of pC-collagen, as compared to fully-processed collagen, then provides broad consistency between homogeneous fibril radii and the lack of fibril nucleation during fibril growth., Comment: 10 pages, 2 figures

How long people live depends on their health, and how it changes with age. Individual health can be tracked by the accumulation of age-related health deficits. The fraction of age-related deficits is a simple quantitative measure of human aging. This quantitative frailty index (F) is as good as chronological age in predicting mortality. In this paper, we use a dynamical network model of deficits to explore the effects of interactions between deficits, deficit damage and repair processes, and the connection between the F and mortality. With our model, we qualitatively reproduce Gompertz's law of increasing human mortality with age, the broadening of the F distribution with age, the characteristic non-linear increase of the F with age, and the increased mortality of high-frailty individuals. No explicit time-dependence in damage or repair rates is needed in our model. Instead, implicit time-dependence arises through deficit interactions -- so that the average deficit damage rates increases, and deficit repair rates decreases, with age . We use a simple mortality criterion, where mortality occurs when the most connected node is damaged., Comment: 12 pages, 38 figures

Mammalian tissues contain networks and ordered arrays of collagen fibrils originating from the periodic self-assembly of helical 300 nm long tropocollagen complexes. The fibril radius is typically between 25 to 250 nm, and tropocollagen at the surface appears to exhibit a characteristic twist-angle with respect to the fibril axis. Similar fibril radii and twist-angles at the surface are observed in vitro, suggesting that these features are controlled by a similar self-assembly process. In this work, we propose a physical mechanism of equilibrium radius control for collagen fibrils based on a radially varying double-twist alignment of tropocollagen within a collagen fibril. The free-energy of alignment is similar to that of liquid crystalline blue phases, and we employ an analytic Euler-Lagrange and numerical free energy minimization to determine the twist-angle between the molecular axis and the fibril axis along the radial direction. Competition between the different elastic energy components, together with a surface energy, determines the equilibrium radius and twist-angle at the fibril surface. A simplified model with a twist-angle that is linear with radius is a reasonable approximation in some parameter regimes, and explains a power-law dependence of radius and twist-angle at the surface as parameters are varied. Fibril radius and twist-angle at the surface corresponding to an equilibrium free-energy minimum are consistent with existing experimental measurements of collagen fibrils. Remarkably, in the experimental regime, all of our model parameters are important for controlling equilibrium structural parameters of collagen fibrils., Comment: Accepted for publication in Soft Matter, DOI: 10.1039/c4sm01359j

The twin-arginine translocation (Tat) system transports folded proteins of various sizes across both bacterial and plant thylakoid membranes. The membrane-associated TatA protein is an essential component of the Tat translocon, and a broad distribution of different sized TatA-clusters is observed in bacterial membranes. We assume that the size dynamics of TatA clusters are affected by substrate binding, unbinding, and translocation to associated TatBC clusters, where clusters with bound translocation substrates favour growth and those without associated substrates favour shrinkage. With a stochastic model of substrate binding and cluster dynamics, we numerically determine the TatA cluster size distribution. We include a proportion of targeted but non-translocatable (NT) substrates, with the simplifying hypothesis that the substrate translocatability does not directly affect cluster dynamical rate constants or substrate binding or unbinding rates. This amounts to a translocation model without specific quality control. Nevertheless, NT substrates will remain associated with TatA clusters until unbound and so will affect cluster sizes and translocation rates. We find that the number of larger TatA clusters depends on the NT fraction $f$. The translocation rate can be optimized by tuning the rate of spontaneous substrate unbinding, $\Gamma_U$. We present an analytically solvable three-state model of substrate translocation without cluster size dynamics that follows our computed translocation rates, and that is consistent with {\em in vitro} Tat-translocation data in the presence of NT substrates., Comment: 20 pages, Accepted for publication in Physical Biology - This is not a copy edited version of the manuscript

We have modelled stress concentration around small gaps in anisotropic elastic sheets, corresponding to the peptidoglycan sacculus of bacterial cells, under loading corresponding to the effects of turgor pressure in rod-shaped bacteria. We find that under normal conditions the stress concentration is insufficient to mechanically rupture bacteria, even for gaps up to a micron in length. We then explored the effects of stress-dependent smart-autolysins, as hypothesised by Arthur L Koch [Advances in Microbial Physiology 24, 301 (1983); Research in Microbiology 141, 529 (1990)]. We show that the measured anisotropic elasticity of the PG sacculus can lead to stable circumferential propagation of small gaps in the sacculus. This is consistent with the recent observation of circumferential propagation of PG-associated MreB patches in rod-shaped bacteria. We also find a bistable regime of both circumferential and axial gap propagation, which agrees with behavior reported in cytoskeletal mutants of B. subtilis. We conclude that the elastic anisotropies of a bacterial sacculus, as characterised experimentally, may be relevant for maintaining rod-shaped bacterial growth., Comment: 6 pages, 4 figures

When deprived of fixed nitrogen (fN), certain filamentous cyanobacteria differentiate nitrogen-fixing heterocysts. There is a large and dynamic fraction of stored fN in cyanobacterial cells, but its role in directing heterocyst commitment has not been identified. We present an integrated computational model of fN transport, cellular growth, and heterocyst commitment for filamentous cyanobacteria. By including fN storage proportional to cell length, but without any explicit cell-cycle effect, we are able to recover a broad and late range of heterocyst commitment times and we observe a strong indirect cell-cycle effect. We propose that fN storage is an important component of heterocyst commitment and patterning in filamentous cyanobacteria. The model allows us to explore both initial and steady-state heterocyst patterns. The developmental model is hierarchical after initial commitment: our only source of stochasticity is observed growth rate variability. Explicit lateral inhibition allows us to examine $\Delta$patS, $\Delta$hetN, and $\Delta$patN phenotypes. We find that $\Delta$patS leads to adjacent heterocysts of the same generation, while $\Delta$hetN leads to adjacent heterocysts only of different generations. With a shortened inhibition range, heterocyst spacing distributions are similar to those in experimental $\Delta$patN systems. Step-down to non-zero external fixed nitrogen concentrations is also investigated., Comment: This is an author-created, un-copyedited version of an article accepted for publication in Physical Biology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version will be available online

We have developed a 3D off-lattice stochastic polymerization model to study subcellular oscillation of Min proteins in the bacteria Escherichia coli, and used it to investigate the experimental phenomenon of Min oscillation stuttering. Stuttering was affected by the rate of immediate rebinding of MinE released from depolymerizing filament tips (processivity), protection of depolymerizing filament tips from MinD binding, and fragmentation of MinD filaments due to MinE. Each of processivity, protection, and fragmentation reduces stuttering, speeds oscillations, and reduces MinD filament lengths. Neither processivity or tip-protection were, on their own, sufficient to produce fast stutter-free oscillations. While filament fragmentation could, on its own, lead to fast oscillations with infrequent stuttering; high levels of fragmentation degraded oscillations. The infrequent stuttering observed in standard Min oscillations are consistent with short filaments of MinD, while we expect that mutants that exhibit higher stuttering frequencies will exhibit longer MinD filaments. Increased stuttering rate may be a useful diagnostic to find observable MinD polymerization in experimental conditions., Comment: 21 pages, 7 figures, missing unit for k_f inserted

Under conditions of limited fixed-nitrogen, some filamentous cyanobacteria develop a regular pattern of heterocyst cells that fix nitrogen for the remaining vegetative cells. We examine three different heterocyst placement strategies by quantitatively modelling filament growth while varying both external fixed-nitrogen and leakage from the filament. We find that there is an optimum heterocyst frequency which maximizes the growth rate of the filament; the optimum frequency decreases as the external fixed-nitrogen concentration increases but increases as the leakage increases. In the presence of leakage, filaments implementing a local heterocyst placement strategy grow significantly faster than filaments implementing random heterocyst placement strategies. With no extracellular fixed-nitrogen, consistent with recent experimental studies of Anabaena sp. PCC 7120, the modelled heterocyst spacing distribution using our local heterocyst placement strategy is qualitatively similar to experimentally observed patterns. As external fixed-nitrogen is increased, the spacing distribution for our local placement strategy retains the same shape while the average spacing between heterocysts continuously increases., Comment: This is an author-created, un-copyedited version of an article accepted for publication in Physical Biology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version will be available online

Filamentous cyanobacteria growing in media with insufficient fixed nitrogen differentiate some cells into heterocysts, which fix nitrogen for the remaining vegetative cells. Transport studies have shown both periplasmic and cytoplasmic connections between cells that could transport fixed-nitrogen along the filament. Two experiments have imaged fixed-nitrogen distributions along filaments. In 1974,Wolk et al found a peaked concentration of fixed-nitrogen at heterocysts using autoradiographic techniques. In contrast, in 2007, Popa et al used nanoSIMS to show large dips at the location of heterocysts, with a variable but approximately level distribution between them. With an integrated model of fixed-nitrogen transport and cell growth, we recover the results of both Wolk et al and of Popa et al using the same model parameters. To do this, we account for immobile incorporated fixed-nitrogen and for the differing durations of labeled nitrogen fixation that occurred in the two experiments. The variations seen by Popa et al are consistent with the effects of cell-by-cell variations of growth rates, and mask diffusive gradients. We are unable to rule out a significant amount of periplasmic fN transport., Comment: This is an author-created, un-copyedited version of an article accepted for publication in Physical Biology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version will be available online

The coarsening instability typically disrupts steady-state cluster-size distributions. We show that degradation coupled to the cluster size, such as arising from biological proteolysis, leads to a novel fixed-point cluster size. Stochastic evaporative and condensative fluxes determine the width of the fixed-point size distribution. At the fixed-point, we show how the peak size and width depend on number, interactions, and proteolytic rate. This proteolytic size-control mechanism is consistent with the phenomenology of pseudo-pilus length control in the general secretion pathway of bacteria., Comment: Physical Review E: Statistical, Nonlinear, and Soft Matter Physics (2011) in press

We model the self-organization of the MinE ring that is observed during subcellular oscillations of the proteins MinD and MinE within the rod-shaped bacterium {\it Escherichia coli}. With a steady-state approximation, we can study the MinE-ring generically -- apart from the other details of the Min oscillation. Rebinding of MinE to depolymerizing MinD filament tips controls MinE ring formation through a scaled cell shape parameter $\tilde{r}$. We find two types of E-ring profiles near the filament tip: a strong plateau-like E-ring controlled by 1D diffusion of MinE along the bacterial length, or a weak cusp-like E-ring controlled by 3D diffusion near the filament tip. While the width of a strong E-ring depends on $\tilde{r}$, the occupation fraction of MinE at the MinD filament tip is saturated and hence the depolymerization speed do not depend strongly on $\tilde{r}$. Conversely, for weak E-rings both $\tilde{r}$ and the MinE to MinD stoichiometry strongly control the tip occupation and hence the depolymerization speed. MinE rings {\em in vivo} are close to the threshold between weak and strong, and so MinD-filament depolymerization speed should be sensitive to cell shape, stoichiometry, and the MinE-rebinding rate. We also find that the transient to MinE-ring formation is quite long in the appropriate open geometry for assays of ATPase activity {\it in vitro}, explaining the long delays of ATPase activity observed for smaller MinE concentrations in those assays without the need to invoke cooperative MinE activity., Comment: 8 pages, 5 figures

Within individual bacteria, we combine force-dependent polymerization dynamics of individual MreB protofilaments with an elastic model of protofilament bundles buckled into helical configurations. We use variational techniques and stochastic simulations to relate the pitch of the MreB helix, the total abundance of MreB, and the number of protofilaments. By comparing our simulations with mean-field calculations, we find that stochastic fluctuations are significant. We examine the quasi-static evolution of the helical pitch with cell growth, as well as timescales of helix turnover and denovo establishment. We find that while the body of a polarized MreB helix treadmills towards its slow-growing end, the fast-growing tips of laterally associated protofilaments move towards the opposite fast-growing end of the MreB helix. This offers a possible mechanism for targeted polar localization without cytoplasmic motor proteins., Comment: 7 figures, 1 table

Ongoing sub-cellular oscillation of Min proteins is required to block minicelling in E. coli. Experimentally, Min oscillations are seen in newly divided cells and no minicells are produced. In model Min systems many daughter cells do not oscillate following septation because of unequal partitioning of Min proteins between the daughter cells. Using the 3D model of Huang et al., we investigate the septation process in detail to determine the cause of the asymmetric partitioning of Min proteins between daughter cells. We find that this partitioning problem arises at certain phases of the MinD and MinE oscillations with respect to septal closure and it persists independently of parameter variation. At most 85% of the daughter cells exhibit Min oscillation following septation. Enhanced MinD binding at the static polar and dynamic septal regions, consistent with cardiolipin domains, does not substantially increase this fraction of oscillating daughters. We believe that this problem will be shared among all existing Min models and discuss possible biological mechanisms that may minimize partitioning errors of Min proteins following septation., Comment: 17 pages, including 6 figures. Typo in captions of fig.2,5 corrected. Version which appears in Physical Biology

We present maximally-fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time-step $\Delta t=A t_s^{2/3}$. For non-conserved systems, only effectively finite timesteps are accessible for similar unconditionally stable algorithms. We compare the scaling structure obtained from our maximally-fast conserved systems directly against the standard fixed-timestep Euler algorithm, and find that the error scales as $\sqrt{A}$ -- so arbitrary accuracy can be achieved., Comment: 5 pages, 3 postscript figures, Latex

We characterize the late-time scaling state of dry, coarsening, two-dimensional froths using a detailed, force-based vertex model. We find that the slow evolution of bubbles leads to systematic deviations from 120degree angles at three-fold vertices in the froth, with an amplitude proportional to the vertex speed, v ~ sqrt(t), but with a side-number dependence that is independent of time. We also find that a significant number of T1 side-switching processes occur for macroscopic bubbles in the scaling state, though most bubble annihilations involve four-sided bubbles at microscopic scales., Comment: 7 pages, 7 figures

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre's theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. We show that accuracy can be controlled with an unbounded time step Delta-t that grows with time t as Delta-t ~ t^alpha. We develop a classification scheme for the step exponent alpha and demonstrate that a class of simple linear algorithms gives alpha=1/3. For this class the speed up relative to a fixed time step grows with the linear size of the system as N/log N, and we estimate conservatively that an 8192^2 lattice can be integrated 300 times faster than with the Euler method., Comment: 14 pages, 6 figures

We examine fluctuation effects due to the low copy number of proteins involved in pattern-forming dynamics within a bacterium. We focus on a stochastic model of the oscillating MinCDE protein system regulating accurate cell division in E. coli. We find that, for some parameter regions, the protein concentrations are low enough that fluctuations are essential for the generation of patterns. We also examine the role of fluctuations in constraining protein concentration levels., Comment: 4 pages, 3 figures, accepted for publication in Phys. Rev. Lett

Particles moving along curved trajectories will diffuse if the curvature fluctuates sufficiently in either magnitude or orientation. We consider particles moving at a constant speed with either a fixed or with a Gaussian distributed curvature magnitude. At small speeds the diffusivity is independent of the speed. At larger particle speeds, the diffusivity depends on the speed through a novel exponent. We apply our results to intracellular transport of vesicles. In sharp contrast to thermal diffusion, the effective diffusivity increases with vesicle size and so may provide an effective means of intracellular transport., Comment: 4 pages, 3 figures (first figure takes awhile to print)

Positioning of the midcell division plane within the bacterium E. coli is controlled by the min system of proteins: MinC, MinD and MinE. These proteins coherently oscillate from end to end of the bacterium. We present a reaction--diffusion model describing the diffusion of min proteins along the bacterium and their transfer between the cytoplasmic membrane and cytoplasm. Our model spontaneously generates protein oscillations in good agreement with experiments. We explore the oscillation stability, frequency and wavelength as a function of protein concentration and bacterial length., Comment: 4 pages, 4 figures, Latex2e, Revtex4

We solve a coarsening system with small but arbitrary anisotropic surface tension and interface mobility. The resulting size-dependent growth shapes are significantly different from equilibrium microcrystallites, and have a distribution of grain sizes different from isotropic theories. As an application of our results, we show that the persistence decay exponent depends on anisotropy and hence is nonuniversal., Comment: 4 pages (revtex), 2 eps figures