Models of vimentin organization under actin-driven transport.

Intermediate filaments form an essential structural network, spread throughout the cytoplasm, and play a key role in cell mechanics, intracellular organization, and molecular signaling. The maintenance of the network and its adaptation to the cell's dynamic behavior relies on several mechanisms implicating cytoskeletal crosstalk which are not fully understood. Mathematical modeling allows us to compare several biologically realistic scenarios to help us interpret experimental data. In this study we observe and model the dynamics of the vimentin intermediate filaments in single glial cells seeded on circular micropatterns following microtubule disruption by nocodazole treatment. In these conditions, the vimentin filaments move towards the cell center and accumulate before eventually reaching a steady state. In the absence of microtubule-driven transport, the motion of the vimentin network is primarily driven by actin-related mechanisms. To model these experimental findings, we hypothesize that vimentin may exist in two states, mobile and immobile, and switch between the states at unknown (either constant or nonconstant) rates. Mobile vimentin is assumed to advect with either constant or nonconstant velocity. We introduce several biologically realistic scenarios using this set of assumptions. For each scenario, we use differential evolution to find the best parameter sets resulting in a solution that most closely matches the experimental data and then the assumptions are evaluated using the Akaike information criterion. This modeling approach allows us to conclude that our experimental data are best explained by a spatially dependent trapping of intermediate filaments or a spatially dependent speed of actin-dependent transport.

A simple in-host model for COVID-19 with treatments: model prediction and calibration.

In this paper, we provide a simple ODEs model with a generic nonlinear incidence rate function and incorporate two treatments, blocking the virus binding and inhibiting the virus replication to investigate the impact of calibration on model predictions for the SARS-CoV-2 infection dynamics. We derive conditions of the infection eradication for the long-term dynamics using the basic reproduction number, and complement the characterization of the dynamics at short-time using the resilience and reactivity of the virus-free equilibrium are considered to inform on the average time of recovery and sensitivity to perturbations in the initial virus free stage. Then, we calibrate the treatment model to clinical datasets for viral load in mild and severe cases and immune cells in severe cases. Based on the analysis, the model calibrated to these different datasets predicts distinct scenarios: eradication with a non reactive virus-free equilibrium, eradication with a reactive virus-free equilibrium, and failure of infection eradication. Moreover, severe cases generate richer dynamics and different outcomes with the same treatment. Calibration to different datasets can lead to diverse model predictions, but combining long- and short-term dynamics indicators allows the categorization of model predictions and determination of infection severity.

Dual-wavelength stopped-flow analysis of the lateral and longitudinal assembly kinetics of vimentin.

Vimentin is a highly charged intermediate filament protein that inherently forms extended dimeric coiled coils, which serve as the basic building blocks of intermediate filaments. Under low ionic strength conditions, vimentin filaments dissociate into uniform tetrameric complexes of two anti-parallel-oriented, half-staggered coiled-coil dimers. By addition of salt, vimentin tetramers spontaneously reassemble into filaments in a time-dependent process: 1) lateral assembly of tetramers into unit-length filaments, 2) longitudinal annealing of unit-length filaments, and 3) longitudinal assembly of filaments coupled with subsequent radial compaction. To independently determine the lateral and longitudinal assembly kinetics, we measure with a stopped-flow instrument the static light scattering signal at two different wavelengths (405 and 594 nm) with a temporal resolution of 3 ms and analyze the signals based on Rayleigh-Gans theory. This theory considers that the intensity of the scattered light depends not only on the molecular weight of the scattering object but also on its shape. This shape dependence is more pronounced at shorter wavelengths, allowing us to decompose the scattered light signal into its components arising from lateral and longitudinal filament assembly. We demonstrate that both the lateral and longitudinal filament assembly kinetics increase with salt concentration.

Using Fluorescence Recovery After Photobleaching data to uncover filament dynamics.

Fluorescence Recovery After Photobleaching (FRAP) has been extensively used to understand molecular dynamics in cells. This technique when applied to soluble, globular molecules driven by diffusion is easily interpreted and well understood. However, the classical methods of analysis cannot be applied to anisotropic structures subjected to directed transport, such as cytoskeletal filaments or elongated organelles transported along microtubule tracks. A new mathematical approach is needed to analyze FRAP data in this context and determine what information can be obtain from such experiments. To address these questions, we analyze fluorescence intensity profile curves after photobleaching of fluorescently labelled intermediate filaments anterogradely transported along microtubules. We apply the analysis to intermediate filament data to determine information about the filament motion. Our analysis consists of deriving equations for fluorescence intensity profiles and developing a mathematical model for the motion of filaments and simulating the model. Two closed forms for profile curves were derived, one for filaments of constant length and one for filaments with constant velocity, and three types of simulation were carried out. In the first type of simulation, the filaments have random velocities which are constant for the duration of the simulation. In the second type, filaments have random velocities which instantaneously change at random times. In the third type, filaments have random velocities and exhibit pausing between velocity changes. Our analysis shows: the most important distribution governing the shape of the intensity profile curves obtained from filaments is the distribution of the filament velocity. Furthermore, filament length which is constant during the experiment, had little impact on intensity profile curves. Finally, gamma distributions for the filament velocity with pauses give the best fit to asymmetric fluorescence intensity profiles of intermediate filaments observed in FRAP experiments performed in polarized migrating astrocytes. Our analysis also shows that the majority of filaments are stationary. Overall, our data give new insight into the regulation of intermediate filament dynamics during cell migration.

Impact of noise on the regulation of intracellular transport of intermediate filaments.

Noise affects all biological processes from molecules to cells, organisms and populations. Although the effect of noise on these processes is highly variable, evidence is accumulating which shows natural stochastic fluctuations (noise) can facilitate biological functions. Herein, we investigate the effect of noise on the transport of intermediate filaments in cells by comparing the stochastic and deterministic formalizations of the bidirectional transport of intermediate filaments, long elastic polymers transported along microtubules by antagonistic motor proteins (Dallon et al., 2019; Portet et al., 2019). By numerically exploring discrepancies in timescales and attractors between both formalizations, we characterize the impact of stochastic fluctuations on the individual and ensemble transport. Biologically, we find that noise promotes the collective movement of intermediate filaments and increases the efficiency of its regulation by the biochemical properties of motor-cargo interactions. While stochastic fluctuations reduce the impact of the initial distributions of motor proteins in cells, the number of binding sites and the affinity of motor-cargo interactions are the key parameters controlling transport efficiency and efficacy.

A general mathematical model for the in vitro assembly dynamics of intermediate filament proteins.

Intermediate filament (IF) proteins assemble into highly flexible filaments that organize into complex cytoplasmic networks: keratins in all types of epithelia, vimentin in endothelia, and desmin in muscle. Since IF elongation proceeds via end-to-end annealing of unit-length filaments and successively of progressively growing filaments, it is important to know how their remarkable flexibility, i.e., their persistence length lp, influences the assembly kinetics. In fact, their lp ranges between 0.3 µm (keratin K8/K18) and 1.0 µm (vimentin and desmin), and thus is orders of magnitude lower than that of microtubules and F-actin. Here, we present a unique mathematical model, which implements the semiflexible nature of the three IF types based on published semiflexible polymers theories and depends on a single free parameter k0. Calibrating this model to filament mean length dynamics of the three proteins, we demonstrate that the persistence length is indeed essential to accurately describe their assembly kinetics. Furthermore, we reveal that the difference in flexibility alone does not explain the significantly faster assembly rate of keratin filaments compared with that of vimentin. Likewise, desmin assembles approximately six times faster than vimentin, even though both their filaments exhibit the same lp value. These data strongly indicate that differences in their individual amino acid sequences significantly impact the assembly rates. Nevertheless, using a single k0 value for each of these three key representatives of the IF protein family, our advanced model does accurately describe the length distribution and mean length dynamics and provides effective filament assembly rates. It thus provides a tool for future investigations on the impact of posttranslational modifications or amino acid changes of IF proteins on assembly kinetics. This is an important issue, as the discovery of mutations in IF genes causing severe human disease, particularly for desmin and keratins, is steadily increasing.

Risk of COVID-19 variant importation - How useful are travel control measures?

We consider models for the importation of a new variant COVID-19 strain in a location already seeing propagation of a resident variant. By distinguishing contaminations generated by imported cases from those originating in the community, we are able to evaluate the contribution of importations to the dynamics of the disease in a community. We find that after an initial seeding, the role of importations becomes marginal compared to that of community-based propagation. We also evaluate the role of two travel control measures, quarantine and travel interruptions. We conclude that quarantine is an efficacious way of lowering importation rates, while travel interruptions have the potential to delay the consequences of importations but need to be applied within a very tight time window following the initial emergence of the variant.

Optogenetic Tuning Reveals Rho Amplification-Dependent Dynamics of a Cell Contraction Signal Network.

Local cell contraction pulses play important roles in tissue and cell morphogenesis. Here, we improve a chemo-optogenetic approach and apply it to investigate the signal network that generates these pulses. We use these measurements to derive and parameterize a system of ordinary differential equations describing temporal signal network dynamics. Bifurcation analysis and numerical simulations predict a strong dependence of oscillatory system dynamics on the concentration of GEF-H1, an Lbc-type RhoGEF, which mediates the positive feedback amplification of Rho activity. This prediction is confirmed experimentally via optogenetic tuning of the effective GEF-H1 concentration in individual living cells. Numerical simulations show that pulse amplitude is most sensitive to external inputs into the myosin component at low GEF-H1 concentrations and that the spatial pulse width is dependent on GEF-H1 diffusion. Our study offers a theoretical framework to explain the emergence of local cell contraction pulses and their modulation by biochemical and mechanical signals.

A simple model for COVID-19.

An S L 1 L 2 I 1 I 2 A 1 A 2 R epidemic model is formulated that describes the spread of an epidemic in a population. The model incorporates an Erlang distribution of times of sojourn in incubating, symptomatically and asymptomatically infectious compartments. Basic properties of the model are explored, with focus on properties important in the context of current COVID-19 pandemic.

A primer on model selection using the Akaike Information Criterion.

A powerful investigative tool in biology is to consider not a single mathematical model but a collection of models designed to explore different working hypotheses and select the best model in that collection. In these lecture notes, the usual workflow of the use of mathematical models to investigate a biological problem is described and the use of a collection of model is motivated. Models depend on parameters that must be estimated using observations; and when a collection of models is considered, the best model has then to be identified based on available observations. Hence, model calibration and selection, which are intrinsically linked, are essential steps of the workflow. Here, some procedures for model calibration and a criterion, the Akaike Information Criterion, of model selection based on experimental data are described. Rough derivation, practical technique of computation and use of this criterion are detailed.

Deciphering the transport of elastic filaments by antagonistic motor proteins.

Intermediate filaments are long elastic fibers that are transported by the microtubule-associated motor proteins kinesin and dynein inside the cell. How elastic filaments are efficiently transported by antagonistic motors is not well understood and is difficult to measure with current experimental techniques. Adapting the tug-of-war paradigm for vesiclelike cargos, we develop a mathematical model to describe the motion of an elastic filament punctually bound to antagonistic motors. As observed in cells, up to three modes of transport are obtained; dynein-driven retrograde, kinesin-driven anterograde fast motions, and a slow motion. Motor properties and initial conditions that depend on intracellular context regulate the transport of filaments. Filament elasticity is found to affect both the mode and the efficiency of transport. We further show that the coordination of motors along the filament emerges from the interplay between intracellular context and elastic properties of filaments.

Stochastic modeling reveals how motor protein and filament properties affect intermediate filament transport.

Intermediate filaments are a key component of the cytoskeleton. Their transport along microtubules plays an essential role in the control of the shape and structural organization of cells. To identify the key parameters responsible for the control of intermediate filament transport, we generated a model of elastic filament transport by microtubule-associated dynein and kinesin. The model is also applicable to the transport of any elastically-coupled cargoes. We investigate the effect of filament properties such as number of motor binding sites, length, and elasticity on motion of filaments. Additionally, we consider the effect of motor properties, i.e. off rates, on filament transport. When one motor has a catch bond off rate it dictates the motion, whereas when motors have the same type of off rate filaments can alternate between retrograde and anterograde motions. The elasticity of filaments optimizes the filament transport and the coordination of motors along the length of the filament.

Investigation of Novel Regulation of N-myristoyltransferase by Mammalian Target of Rapamycin in Breast Cancer Cells.

Breast cancer is the most common cancer in women worldwide. Hormone receptor breast cancers are the most common ones and, about 2 out of every 3 cases of breast cancer are estrogen receptor (ER) positive. Selective ER modulators, such as tamoxifen, are the first line of endocrine treatment of breast cancer. Despite the expression of hormone receptors some patients develop tamoxifen resistance and 50% present de novo tamoxifen resistance. Recently, we have demonstrated that activated mammalian target of rapamycin (mTOR) is positively associated with overall survival and recurrence free survival in ER positive breast cancer patients who were later treated with tamoxifen. Since altered expression of protein kinase B (PKB)/Akt in breast cancer cells affect N-myristoyltransferase 1 (NMT1) expression and activity, we investigated whether mTOR, a downstream target of PKB/Akt, regulates NMT1 in ER positive breast cancer cells (MCF7 cells). We inhibited mTOR by treating MCF7 cells with rapamycin and observed that the expression of NMT1 increased with rapamycin treatment over the period of time with a concomitant decrease in mTOR phosphorylation. We further employed mathematical modelling to investigate hitherto not known relationship of mTOR with NMT1. We report here for the first time a collection of models and data validating regulation of NMT1 by mTOR.

Keratin dynamics: modeling the interplay between turnover and transport.

Keratin are among the most abundant proteins in epithelial cells. Functions of the keratin network in cells are shaped by their dynamical organization. Using a collection of experimentally-driven mathematical models, different hypotheses for the turnover and transport of the keratin material in epithelial cells are tested. The interplay between turnover and transport and their effects on the keratin organization in cells are hence investigated by combining mathematical modeling and experimental data. Amongst the collection of mathematical models considered, a best model strongly supported by experimental data is identified. Fundamental to this approach is the fact that optimal parameter values associated with the best fit for each model are established. The best candidate among the best fits is characterized by the disassembly of the assembled keratin material in the perinuclear region and an active transport of the assembled keratin. Our study shows that an active transport of the assembled keratin is required to explain the experimentally observed keratin organization.

Epidemiological implications of mobility between a large urban centre and smaller satellite cities.

An SIR infectious disease propagation model is considered that incorporates mobility of individuals between a large urban centre and smaller satellite cities. Because of the difference in population sizes, the urban centre has standard incidence and satellite cities have mass action incidence. It is shown that the general basic reproduction number [Formula: see text] acts as a threshold between global asymptotic stability of the disease free equilibrium and disease persistence. The case of Winnipeg (MB, Canada) and some neighbouring satellite communities is then considered numerically to complement the mathematical analysis, highlighting the importance of taking into account not only [Formula: see text] but also other measures of disease severity. It is found that the large urban centre governs most of the behaviour of the general system and control of the spread is better achieved by targeting it rather than reducing movement between the units. Also, the capacity of a satellite city to affect the general system depends on its population size and its connectivity to the main urban centre.

Dynamics of in vitro intermediate filament length distributions.

An aggregation model with explicit expression of association rate constants is considered to study in vitro type III intermediate filament length distribution dynamics. Different assumptions on the properties of filaments and probability of aggregation are considered, leading to four models. Fitting of model responses to experimental data leads to the identification of the most appropriate model to represent each time point of the assembly. A combination of models allows the construction of a mixed model that represents well the complete assembly dynamics: it is found that the rate constants decrease with respect to filament size when the aggregation involves at least one short filament, whereas for longer filaments they are almost independent of size. The flexible nature of filaments is thus important in the assembly of intermediate filaments.

An in vivo intermediate filament assembly model.

A model is developed to study the in vivo intermediate filament organization in terms of repartition between four different structural states: soluble proteins, particles, short, and long filaments. An analysis is conducted, showing that the system has a unique, globally asymptotically stable equilibrium. By means of sensitivity analysis, the influence of parameters on the system is studied. It is shown that, in agreement with biological observations, posttranslational modifications of intermediate filament proteins resulting in filament solubilization are the main regulators of the intermediate filament organization. A high signalling-dependent solubilization of filaments favours the intermediate filament aggregation in particles.

Vimentin intermediate filament formation: in vitro measurement and mathematical modeling of the filament length distribution during assembly.

The salt-induced in vitro assembly of cytoplasmic intermediate filament (IF) proteins such as vimentin is characterized by a very rapid lateral association of soluble tetrameric subunits into 60-nm-long full-width "unit-length" filaments (ULFs). We have demonstrated for this prototype IF protein that filament elongation occurs by the longitudinal annealing of ULFs into short IFs. These IFs further longitudinally anneal and thus constitute a progressively elongating filament population that over time yields filaments of several microm in length. Previously, we provided a mathematical model for the kinetics of the assembly process based on the average length distribution of filaments as determined by time-lapse electron and atomic force microscopy. Thereby, we were able to substantiate the concept that end-to-end-annealing of both ULFs and short filaments is obligatory for the formation of long IFs (Kirmse, R.; Portet, S.; Mücke, N. Aebi, U.; Herrmann, H.; Langowski, J. J. Biol. Chem. 2007, 282, 18563-18572). As the next step in understanding the mechanics of IF formation, we have expanded our mathematical model to describe the quantitative aspects of IF assembly by taking into account geometry constraints as well as the diffusion properties of rodlike linear aggregates. Thereby, we have developed a robust model for the time-dependent filament length distribution of IFs under standard conditions.

Simulating the formation of keratin filament networks by a piecewise-deterministic Markov process.

Keratin intermediate filament networks are part of the cytoskeleton in epithelial cells. They were found to regulate viscoelastic properties and motility of cancer cells. Due to unique biochemical properties of keratin polymers, the knowledge of the mechanisms controlling keratin network formation is incomplete. A combination of deterministic and stochastic modeling techniques can be a valuable source of information since they can describe known mechanisms of network evolution while reflecting the uncertainty with respect to a variety of molecular events. We applied the concept of piecewise-deterministic Markov processes to the modeling of keratin network formation with high spatiotemporal resolution. The deterministic component describes the diffusion-driven evolution of a pool of soluble keratin filament precursors fueling various network formation processes. Instants of network formation events are determined by a stochastic point process on the time axis. A probability distribution controlled by model parameters exercises control over the frequency of different mechanisms of network formation to be triggered. Locations of the network formation events are assigned dependent on the spatial distribution of the soluble pool of filament precursors. Based on this modeling approach, simulation studies revealed that the architecture of keratin networks mostly depends on the balance between filament elongation and branching processes. The spatial distribution of network mesh size, which strongly influences the mechanical characteristics of filament networks, is modulated by lateral annealing processes. This mechanism which is a specific feature of intermediate filament networks appears to be a major and fast regulator of cell mechanics.