During the blood clotting process, fibrinogen is cleaved by thrombin to produce fibrin monomers, which polymerize into a fibrin gel that stabilizes the clot. Although fibrinogen cannot react with itself, fibrinogen can interact with fibrin and affect clot structure. Motivated by these fibrin-fibrinogen interactions, we introduce a kinetic gelation model of polymer growth with two general types of monomers that have differing functionalities (reaction sites). The heterotypic copolymerization of two monomer types is modeled using a Ziff-Stell approach by tracking the temporal evolution of the concentrations of both types of free reaction sites on oligomers. We further investigate a special case of our kinetic gelation model in which no homogeneous interactions occur and look for conditions for which finite time blow-up occurs.

Changes in calcium concentration along the sperm flagellum regulates sperm motility and hyperactivation, characterized by an increased flagellar bend amplitude and beat asymmetry, enabling the sperm to reach and to penetrate the oocyte. However, the exact mechanisms of calcium signaling are yet unknown and under investigation. We develop a fluid-structure interaction model that couples the three-dimensional motion of the flagellum in a highly viscous Newtonian fluid with the calcium dynamics in the flagellum. The flagellum is modeled as a Kirchhoff-rod: an elastic rod with intrinsic curvature and twist. The calcium dynamics are represented as a one-dimensional reaction-diffusion model on a moving domain. The two models are coupled assuming that the sperm flagellum preferred curvature depends on the evolving calcium concentration in time. To investigate the effect of calcium on sperm motility, we compare model results of flagellar bend amplitude and swimming speed for different phenomenological functions representing the dependence of curvature on the calcium concentration.

Articular cartilage has a complex structure composed of a dense extracellular matrix (ECM), which includes fluid, a collagen network, and other proteins. Distributed in the matrix there are chondrocytes (cells) that synthesize the building blocks of the ECM. Pathologies such as osteoarthritis, injuries and normal wear and tear can cause the erosion and damage of articular cartilage. Tissue engineering represents a promising path towards the treatment of damaged cartilage. In this work, a hybrid mathematical model is used to investigate the phenomena of cartilage growth in a tissue-engineered construct (gel + cells) to elucidate and clarify the influence of different biological factors and conditions involved in the process. This hybrid model couples a discrete modeling approach for the chondrocytes, with a continuous approach for the remaining components of the matrix, modeled via a time dependent diffusion-reaction equation. We investigate the influence of gel porosity and cell seeding on the synthesis of new ECM. The insight provided by the model will be used to elucidate some of the outcomes of laboratory experiments involving tissue-engineered articular cartilage.

Our work has previously shown that interaction between repulsive intercellular signals (contact inhibition of locomotion, CIL) and attractive intercellular signals (co-attraction, COA) can allows for the directed migration of cell groups down a migratory corridor even when no chemoattractant is present. However, given certain corridor configurations, this effect breaks down when group sizes are large. Even if directed migration does not emerge due to interaction between CIL and COA in such setups, we investigate if the interaction can allow a large group to effectively interpret chemoattractant gradients too weak for a single cell to sense.

Recent advances in microscopy and computation have allowed for unprecedented quantitative study of cellular dynamics in developing organisms. We are focusing on the syncytial development of insect embryos, in which nuclei divide and move in a single shared cytoplasm. Specifically, we are interested in how nuclei move, how nuclei know where to move, and how nuclei know what to become. We are working to answer these questions using lightsheet microscopy and a transgenic line of the cricket Gryllus bimaculatus, whose nuclei are fluorescently labeled. We collect several terabytes of live-imaging data and then automatically segment and track the nuclei.  This allows us to gain insight in key unanswered questions in early embryogenesis. Axial expansion is characterized by nuclei dividing and expanding throughout the embryo. Following axial expansion is embryonic coalescence, when select nuclei move together to form the embryonic rudiment.  We have quantitatively described the motion of nuclei throughout both axial expansion and coalescence of healthy cricket embryos. In addition to answering questions related to the movement of specific nuclei, we have shown that some nuclei are excluded from the embryonic rudiment much early than previously thought. We also use tools from computational geometry to study the motion of nuclei relative to one another, after cellularization has occurred.

In recent years there has been an increased focus on understanding the molecular basis of bleeding disorders as well as the observed variability in levels of key coagulation factors. This has motivated the development of mathematical models that integrate platelet function, coagulation and blood flow. Such models can predict bleeding risk based on measurable biochemical and biophysical factors. In this research, we develop a well-mixed ODE model of bleeding that incorporates platelet deposition and the effects of blood flow though an H-shaped microfluidic device. The in vitro microfluidic model of bleeding provides model inputs and serves as an experimental validation.

The design and fabrication of microscale pumps using magnetically actuated bacterial flagella opens the door for many applications such as the pumping and regulation of chemicals. Here, we discuss simulations for a pump consisting of a regular two-dimensional array of rigid helices. Recent work investigating the flows above a small, finite array by numerically calculating the full dynamics showed that having random phase differences between helices seems essential to produce the flow patterns observed in experiments. We developed a model which allows us to treat random phase differences in an infinite array. Using a mean-field approach we define pumping as the existence of a self-consistent tilt angle of the array. Pumping is then examined numerically as a function of several parameters in the magnetic actuation and helical geometry. We demonstrate how this pumping flow may be mechanically halted by way of magnetic actuation or autonomously halted by the polymorphic transformation of bacterial flagella in response to environmental stimuli.

TBD

In bacteria, a flexible hook transmits torque from the rotary motor at the cell body to the flagellum. Previously, the hook has been modeled as a Kirchhoff rod between the cell body and rotating flagellum. To study effects of the hook's flexibility on the bacteria's dynamics for wide range hook stiffnesses, we develop an efficient simplified spring model for the hook by linearizing the Kirchhoff rod. We treat the hydrodynamics of the cell body and helical flagellum using resistance matrices calculated by the method of regularized Stokeslets. We investigate flagellar and swimming dynamics for a range of hook flexibilities and flagellar orientations relative to the cell body using parameters corresponding to Vibrio alginolyticus. For stiff hooks, the flagellum changes orientation relative to the cell body, undergoing an orbit with the period of the motor rotation. We find that as the hook stiffness decreases, steady-state orbits of the flagellum become unstable before the hook buckles, which may suggest a new mechanism of flick initiation in run-reverse-flick motility.

We present a mathematical model for the initiation of venous thrombosis (VT) due to slow flow and the consequent activation of the endothelial cells (ECs) lining the vein, in the absence of overt mechanical disruption of the EC layer. It includes all reactions of the tissue factor (TF) pathway of coagulation through fibrin formation, incorporates the accumulation of blood cells on activated ECs, accounts for the flow-mediated delivery and removal of coagulation proteins and blood cells from the locus of the reactions, and accounts for the activity of major inhibitors including heparan-sulfate-accelerated antithrombin and activated protein C. The model reveals that the occurrence of robust thrombin generation (a thrombin burst) depends in a threshold manner on the density of TF on the activated ECs and on the concentration of thrombomodulin and the degree of heparan-sulfate accelerated antithrombin activity on those cells. Small changes in any of these in appropriate narrow ranges switches the response between ‘‘no burst’’ and ‘‘burst.’’ The model predicts synergies among the inhibitors, both in terms of each inhibitor’s multiple targets, and in terms of interactions between the different inhibitors. The model strongly suggests that the rate and extent of accumulation of activated monocytes, platelets, and MPs that can support the coagulation reactions has a powerful influence on whether a thrombin burst occurs and the thrombin response when it does. The slow rate of accumulation of cells supporting coagulation is one reason that the progress of VT is so much slower than that of arterial thrombosis initiated by subendothelial exposure.

Sloshing dynamics refers to the study of the movement of a fluid inside a container. We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in a container, including the effects of surface tension on the free surface. An interesting setting is the microgravity environment, where the effects of surface tension dominates the effect of gravity. We restrict ourselves to a constant contact angle and we seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and prove a domain monotonicity result for the fundamental sloshing frequency. We provide the first-order perturbation formula for a simple eigenvalue in the limit of zero surface tension. We propose a finite-element numerical method for computing the sloshing modes of the coupled system, which works for arbitrary containers.