Quantitative Biology

Studying the development of malignant tumours, it is important to know and predict the proportions of different cell types in tissue samples. Knowing the expected temporal evolution of the proportion of normal tissue cells, compared to stem-like and non-stem like cancer cells, gives an indication about the progression of the disease and indicates the expected response to interventions with drugs. Such processes have been modeled using Markov processes. An essential step for the simulation of such models is then the determination of state transition probabilities. We here consider the experimentally more realistic scenario in which the measurement of cell population sizes is noisy, leading to a particular hidden Markov model. In this context, randomness in measurement is related to noisy measurements, which are used for the estimation of the transition probability matrix. Randomness in sampling, on the other hand, is here related to the error in estimating the state probability from small cell populations. Using aggregated data of fluorescence-activated cell sorting (FACS) measurement, we develop a minimum mean square error estimator (MMSE) and maximum likelihood (ML) estimator and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes using a transition probability matrix estimated from noisy data. We analyze the properties of two estimators for different noise distributions and prove an optimal solution for Gaussian distributions with the MMSE. Our numerical results show, that for noisy measurements the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach in which measurements are assumed to be noise free.

A living cell senses its environment and responds to external signals. In this work, we study theoretically, the precision at which cells can determine the position of a spatially localized transient extracellular signal. To this end, we focus on the case, where the stimulus is converted into the release of a small molecule that acts as a second messenger, for example, Ca 2+ , and activates kinases that change the activity of enzymes by phosphorylating them. We analyze the spatial distribution of phosphorylation events using stochastic simulations as well as a mean-field approach. Kinases that need to bind to the cell membrane for getting activated provide more accurate estimates than cytosolic kinases. Our results could explain why the rate of Ca 2+ detachment from the membrane-binding conventional Protein Kinase C α is larger than its phosphorylation rate.

In-vivo, cells are frequently exposed to multiple mechanical stimuli arising from the extracellular microenvironment, with deep impact on many biological functions. On the other hand, current methods for mechanobiology do not allow to easily replicate in-vitro the complex spatio-temporal behavior of such mechanical signals. Here, we introduce a new platform for studying the mechanical coupling between the extracellular environment and the nucleus in living cells, based on active substrates for cell culture made of Fe-coated polymeric micropillars. Under the action of quasi-static external magnetic fields, each group of pillars produces synchronous nano-mechanical stimuli at different points of the cell membrane, thanks to the highly controllable pillars' deflection. This method enables to perform a new set of experiments for the investigation of cellular dynamics and mechanotransduction mechanisms in response to a periodic nano-pinching, with tunable strength and arbitrary temporal shape. We applied this methodology to NIH3T3 cells, demonstrating how the nano-mechanical stimulation affects the actin cytoskeleton, nuclear morphology, H2B core-histone dynamics and MKL transcription-cofactor nuclear to cytoplasmic translocation.

The eukaryotic flagellum beats periodically, driven by the oscillatory dynamics of molecular motors, to propel cells and pump fluids. Small, but perceivable fluctuations in the beat of individual flagella have physiological implications for synchronization in collections of flagella as well as for hydrodynamic interactions between flagellated swimmers. Here, we characterize phase and amplitude fluctuations of flagellar bending waves using shape mode analysis and limit cycle reconstruction. We report a quality factor of flagellar oscillations, Q=38.0±16.7 (mean ± s.e.). Our analysis shows that flagellar fluctuations are dominantly of active origin. Using a minimal model of collective motor oscillations, we demonstrate how the stochastic dynamics of individual motors can give rise to active small-number fluctuations in motor-cytoskeleton systems.

We present a continuum level analytical model of a droplet of active contractile fluid consisting of filaments and motors. We calculate the steady state flows that result from a splayed polarisation of the filaments. We account for the interaction with an arbitrary external medium by imposing a viscous friction at the fixed droplet boundary. We then show that the droplet has non-zero force dipole and quadrupole moments, the latter of which is essential for self-propelled motion of the droplet at low Reynolds' number. Therefore, this calculation describes a simple mechanism for the motility of a droplet of active contractile fluid embedded in a 3D environment, which is relevant to cell migration in confinement (for example, embedded within a gel or tissue). Our analytical results predict how the system depends on various parameters such as the effective friction coefficient, the phenomenological activity parameter and the splay of the imposed polarisation.

Acute lymphoblastic leukemia (ALL) and chronic lymphocytic leukemia (CLL) are two major forms of leukemia that arise from lymphoid cells (LCs). ALL occurs mostly in children and CLL occurs mainly in old people. However, the Philadelphia-chromosome-positive ALL (Ph+-ALL) and the Ph-like ALL occur in both children and adults. To understand childhood leukemia/lymphoma, we have recently proposed two hypotheses on the causes and the mechanism of cell transformation of a LC. Hypothesis A is: repeated bone-remodeling during bone-growth and bone-repair may be a source of cell injuries of marrow cells including hematopoietic stem cells (HSCs), myeloid cells, and LCs. Hypothesis B is: a LC may have three pathways on transformation: a slow, a rapid, and an accelerated. We discuss in the present paper the developing mechanisms of ALL and CLL by these hypotheses. Having a peak incidence in young children, ALL may develop mainly as a result of rapid cell transformation of a lymphoblast (or pro-lymphocyte). Differently, Ph+-ALL and Ph-like ALL may develop as results of transformation of a lymphoblast via accelerated pathway. Occurring mainly in adults, CLL may be a result of transformation of a memory B-cell via slow pathway. By causing cell injuries of HSCs and LCs, repeated bone-remodeling during bone-growth and bone-repair may be related to the cell transformation of a LC. In conclusion, ALL may develop as a result of cell transformation of a lymphoblast via rapid or accelerated pathway; and repeated bone-remodeling during bone-growth may be a trigger for the cell transformation of a lymphoblast in a child.

Cell-based, mathematical modeling of collective cell behavior has become a prominent tool in developmental biology. Cell-based models represent individual cells as single particles or as sets of interconnected particles, and predict the collective cell behavior that follows from a set of interaction rules. In particular, vertex-based models are a popular tool for studying the mechanics of confluent, epithelial cell layers. They represent the junctions between three (or sometimes more) cells in confluent tissues as point particles, connected using structural elements that represent the cell boundaries. A disadvantage of these models is that cell-cell interfaces are represented as straight lines. This is a suitable simplification for epithelial tissues, where the interfaces are typically under tension, but this simplification may not be appropriate for mesenchymal tissues or tissues that are under compression, such that the cell-cell boundaries can buckle. In this paper we introduce a variant of VMs in which this and two other limitations of VMs have been resolved. The new model can also be seen as on off-the-lattice generalization of the Cellular Potts Model. It is an extension of the open-source package VirtualLeaf, which was initially developed to simulate plant tissue morphogenesis where cells do not move relative to one another. The present extension of VirtualLeaf introduces a new rule for cell-cell shear or sliding, from which T1 and T2 transitions emerge naturally, allowing application of VirtualLeaf to problems of animal development. We show that the updated VirtualLeaf yields different results than the traditional vertex-based models for differential-adhesion-driven cell sorting and for the neighborhood topology of soft cellular networks.

We demonstrate how concepts of statistical mechanics of interacting particles can have important implications in the choice of interaction potentials to model qualitative properties of cell aggregates in theoretical biology. We illustrate this by showing cell sorting phenomena for cell groups with different adhesiveness parameters: ranging from well-mixed cells aggregates to full segregation of cell type passing through engulfment via adhesiveness tuning.

Cancer cells exhibit increased motility and proliferation, which are instrumental in the formation of tumours and metastases. These pathological changes can be traced back to malfunctions of cellular signalling pathways, and calcium signalling plays a prominent role in these. We formulate a new model for cancer cell movement which for the first time explicitly accounts for the dependence of cell proliferation and cell-cell adhesion on calcium. At the heart of our work is a non-linear, integro-differential (non-local) equation for cancer cell movement, accounting for cell diffusion, advection and proliferation. We also employ an established model of cellular calcium signalling with a rich dynamical repertoire that includes experimentally observed periodic wave trains and solitary pulses. The cancer cell density exhibits travelling fronts and complex spatial patterns arising from an adhesion-driven instability (ADI). We show how the different calcium signals and variations in the strengths of cell-cell attraction and repulsion shape the emergent cellular aggregation patterns, which are a key component of the metastatic process. Performing a linear stability analysis, we identify parameter regions corresponding to ADI. These regions are confirmed by numerical simulations, which also reveal different types of aggregation patterns and these patterns are significantly affected by \ca. Our study demonstrates that the maximal cell density decreases with calcium concentration, while the frequencies of the calcium oscillations and the cell density oscillations are approximately equal in many cases. Furthermore, as the calcium levels increase the speed of the travelling fronts increases, which is related to a higher cancer invasion potential. These novel insights provide a step forward in the design of new cancer treatments that may rely on controlling the dynamics of cellular calcium.

In population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this Letter, we study a reaction-diffusion (RD) model of population growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortality rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e., extinction. The early signatures of the transition to extinction at a bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of travelling wave solutions of the population density is demonstrated. The parameter regimes in which the travelling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.