# Applied Math Seminar

## Applied Math Seminar

## Applied Math Seminar

**Title:**A Mathematical Model for the Force and Energetics in Competitive Running

**Abstract:**Competitive running has been around for thousands of years and many people have wondered what the optimal form and strategy is for running a race. In his paper, Behncke develops a simple mathematical model that focuses on the relationships and dynamics between the forces and energetics at play in order to find an optimal strategy for racing various distances. In this talk, I will describe the biomechanics, energetics, and optimization of running in Behncke's model and present his findings. Note: you do not have to like running to come to this talk :)

## Applied Math Seminar

**Title:**Mathematical deep learning for drug discovery

**Abstract:**Designing efficient drugs for curing diseases is of essential importance for the 21

^{st}century's life science. Computer-aided drug design and discovery has obtained a significant recognition recently. However, the geometric complexity of protein-drug complexes remains a grand challenge to conventional computational methods, including machine learning algorithms. We assume that the physics of interest of protein-drug complexes lies on low-dimensional manifolds or subspaces embedded in a high-dimensional data space. We devise topological abstraction, differential geometry reduction, graph simplification, and multiscale modeling to construct low-dimensional representations of biomolecules in massive and diverse datasets. These representations are integrated with various deep learning algorithms for the predictions of protein-ligand binding affinity, drug toxicity, drug solubility, drug partition coefficient and mutation induced protein stability change, and for the discrimination of active ligands from decoys. I will briefly discuss the working principle of various techniques and their performance in D3R Grand Challenges,a worldwide competition series in computer-aided drug design and discovery (http://users.math.msu.edu/users/wei/D3R_GC3.pdf).

## Applied Math Seminar

**Title:** Mathematics for Breast Cancer Research: investigating the role of iron.

**Abstract:** Breast cancer cells are addicted to iron. The mechanisms by which malignant cells acquire and contain high levels of iron are not completely understood. Furthermore, other cell types in a tumor, such as immune cells, can either aid or inhibit cancer cells from acquiring high levels of iron. In order to shed light in the question of how iron affects breast cancer growth, we are applying mathematical tools including polynomial dynamical systems over finite fields and 3D multiscale mathematical modeling. In this talk we will survey how mathematics is aiding in understanding the mechanisms of this addictive iron behavior of malignant cells, and present some preliminary work.

## Applied Math Seminar

**Title:** Preconditioning for Accurate Solutions of the Biharmonic Eigenvalue Problem

**Abstract:** Solving ill-conditioned systems poses two basic problems: convergence and accuracy. Preconditioning can overcome slow convergence, but this is only practical if the preconditioned system can be formed sufficiently accurately. In fact, for a fourth order operator, existing eigenvalue algorithms may compute smaller eigenvalues with little or no accuracy in standard double precision. In this talk, we combine standard matrix eigenvalue solvers with an *accurate *preconditioning scheme in order to compute the smallest eigenvalue of the biharmonic operator to machine precision in spite of ill-conditioning. The results on various domains are compared with the best known computations from the literature to demonstrate the accuracy and applicability of the method.

## Applied Math Seminar

**Title:**Disease ecology meets economics

**Abstract:**

**Understanding why some human populations remain extremely poor despite current development trends around the world remains a mystery to the natural, social and mathematical sciences. The poor rely on their immediate natural environment for subsistence and suffer from high burdens of infectious diseases. We present a general framework for modeling the ecology of poverty and disease, focusing on infectious diseases and renewable resources. Interactions between these ecological drivers of poverty and economics create reinforcing feedbacks resulting in three possible development regimes: 1) globally stable wealthy/healthy development, 2) globally stable unwealthy/unhealthy development, and 3) bistability. We show that the proportion of parameters leading to poverty is larger than that resulting in healthy/wealthy development; bistability consistently emerges as a general property of generalized disease-economic systems and that the systems under consideration are most sensitive to human disease parameters. The framework highlights feedbacks, processes and parameters that are important to measure in future studies of development, to identify effective and sustainable pathways out of poverty.**

## Applied Math Seminar

**Title:**Simulating Within-Vector Generation of the Malaria Parasite Diversity

**Abstract:**Plasmodium falciparum, the malaria parasite causing the most severe disease in humans, undergoes an asexual stage within the human host, and a sexual stage within the vector host, Anopheles mosquitoes. Because mosquitoes may be superinfected with parasites of different genotypes, this sexual stage of the parasite life-cycle presents the opportunity to create genetically novel parasites. To investigate the role that mosquitoes’ biology plays on the generation of parasite diversity, which introduces bottlenecks in the parasites’ development, we first constructed a stochastic model of parasite development within-mosquito, generating a distribution of parasite densities at five parasite life-cycle stages: gamete, zygote, ookinete, oocyst, and sporozoite, over the lifespan of a mosquito. We then coupled a model of sequence diversity generation via recombination between genotypes to the stochastic parasite population model. Our model framework shows that bottlenecks entering the oocyst stage decrease diversity from the initial gametocyte population in a mosquito’s blood meal, but diversity increases with the possibility for recombination and proliferation in the formation of sporozoites. Furthermore, when we begin with only two distinct parasite genotypes in the initial gametocyte population, the probability of transmitting more than two unique genotypes from mosquito to human is over 50% for a wide range of initial gametocyte densities.

## Applied Math Seminar

**Title:** Finding cycles in discrete dynamical systems.

**Abstract:** Discrete dynamical systems often exhibit chaotic behavior, and as a result finding cycles can be computationally expensive. I present a new approach to this problem, based on adding a nonlinear feedback that stabilizes the cycles. We are then able to find cycles numerically in polynomial time. The main theoretical new insight is casting the problem in the language of complex analysis, and finding new complex polynomials that generalize work of Ted Suffridge that optimize the number of steps one needs in order to stabilize the system. This is joint work with D. Dmitrishin, A. Khamitova and A. Stokolos.

## Applied Math Seminar

**Title:** Modeling RNA secondary structure with auxiliary information

**Abstract:** The secondary structure of an RNA sequence plays an important role in determining its function, but directly observing RNA secondary structure is costly and difficult. Therefore, researchers have developed computational tools to predict the secondary structure of RNAs. One of the most popular methods is the Nearest Neighbor Thermodynamic Model (NNTM). More recently, high-throughput data that correlates with the state of a nucleotide being paired or unpaired has been developed. This data, called SHAPE for `selective 2'-hydroxyl acylation analyzed by primer extension', has been incorporated as auxiliary information into the objective function of NNTM with the goal of improving the accuracy of the predictions. This type of prediction is referred to as SHAPE-directed RNA secondary structure modeling. The addition of auxiliary information usually improves the accuracy of the predictions of NNTM. This talk will discuss challenges in RNA secondary structure modeling using NNTM and will provide ideas for developing synthetic auxiliary information that can be incorporated into NNTM to improve the accuracy of the predictions.

## Applied Math Seminar

**Title:**Investigating the structure of Earth's interior

**Abstract:**This talk will focus on the fluid dynamics of Earth and planetary mantles (interiors) and their surface manifestations. By necessity, convection in planetary mantles is largely studied using numerical models on supercomputers, though the right parameter range is still often out of reach. In order to solve the equations governing fluid dynamics inside the Earth, we need to know about the velocity, temperature density and general structure (such as viscosity) of the interior.

Over the past few decades, much work has been done to constrain the viscosity structure of the Earth's mantle using inverse techniques, viscoelastic modelling and post-glacial rebound data. Variations in the Earth's gravitational potential anomalies (geoid) provide constraints on the density structure in the mantle. Seismic tomography can be used to investigate radial viscosity variations on instantaneous flow models. By specifying a possible viscosity structure and predicting a synthetic geoid, we can compare with the observed geoid to see how well our viscosity structure matches the real Earth. Examining over 50 tomographic models we found 2 possible profiles for the viscosity structure inside the Earth.