Using Mathematics To Understand Biological Complexity

This volume tackles a variety of biological and medical questions using mathematical models to understand complex system dynamics. Working in collaborative teams of six, each with a senior research mentor, researchers developed new mathematical models to address questions in a range of application areas. Topics include retinal degeneration, biopolymer dynamics, the topological structure of DNA, ensemble analysis, multidrug-resistant organisms, tumor growth modeling, and geospatial modeling of malaria. The work is the result of newly formed collaborative groups begun during the Collaborative Workshop for Women in Mathematical Biology hosted by the Institute of Pure and Applied Mathematics at UCLA in June 2019. Previous workshops in this series have occurred at IMA, NIMBioS, and MBI.

This volume examines a variety of biological and medical problems using mathematical models to understand complex system dynamics. Featured topics include autism spectrum disorder, ectoparasites and allogrooming, argasid ticks dynamics, super-fast nematocyst firing, cancer-immune population dynamics, and the spread of disease through populations. Applications are investigated with mathematical models using a variety of techniques in ordinary and partial differential equations, difference equations, Markov-chain models, Monte-Carlo simulations, network theory, image analysis, and immersed boundary method. Each article offers a thorough explanation of the methodologies used and numerous tables and color illustrations to explain key results. This volume is suitable for graduate students and researchers interested in current applications of mathematical models in the biosciences. The research featured in this volume began among newly-formed collaborative groups at the 2017 Women Advancing Mathematical Biology Workshop that took place at the Mathematical Biosciences Institute in Columbus, Ohio. The groups spent one intensive week working at MBI and continued their collaborations after the workshop, resulting in the work presented in this volume.

Mathematical biology also known as biomathematics is the sub-field of biology which uses analysis, abstractions and mathematical models to study the principles that influence structure, development and behavior of living organisms. It uses techniques and tools of mathematics to understand complex, non-linear mechanisms in biology. It aims to create models and representations of biological processes which can be used in practical as well as theoretical research. Tools and techniques of applied mathematics are commonly used for creating such models. Some of the areas of research in this field are algebraic biology, complex systems biology, computational neuroscience, abstract relational biology and evolutionary biology. This book explores all the important aspects of this field in the present day scenario. It also strives to provide a fair idea about mathematical biology and to help develop a better understanding of the latest developments within the field. In this book, using case studies and examples, constant effort has been made to make the understanding of the difficult concepts of this discipline as easy and informative as possible, for the readers.

This book develops the mathematical tools essential for students in the life sciences to describe interacting systems and predict their behavior. From predator-prey populations in an ecosystem, to hormone regulation within the body, the natural world abounds in dynamical systems that affect us profoundly. Complex feedback relations and counter-intuitive responses are common in nature; this book develops the quantitative skills needed to explore these interactions. Differential equations are the natural mathematical tool for quantifying change, and are the driving force throughout this book. The use of Euler’s method makes nonlinear examples tractable and accessible to a broad spectrum of early-stage undergraduates, thus providing a practical alternative to the procedural approach of a traditional Calculus curriculum. Tools are developed within numerous, relevant examples, with an emphasis on the construction, evaluation, and interpretation of mathematical models throughout. Encountering these concepts in context, students learn not only quantitative techniques, but how to bridge between biological and mathematical ways of thinking. Examples range broadly, exploring the dynamics of neurons and the immune system, through to population dynamics and the Google PageRank algorithm. Each scenario relies only on an interest in the natural world; no biological expertise is assumed of student or instructor. Building on a single prerequisite of Precalculus, the book suits a two-quarter sequence for first or second year undergraduates, and meets the mathematical requirements of medical school entry. The later material provides opportunities for more advanced students in both mathematics and life sciences to revisit theoretical knowledge in a rich, real-world framework. In all cases, the focus is clear: how does the math help us understand the science?

This book describes the evolution of several socio-biological systems using mathematical kinetic theory. Specifically, it deals with modeling and simulations of biological systems whose dynamics follow the rules of mechanics as well as rules governed by their own ability to organize movement and biological functions. It proposes a new biological model focused on the analysis of competition between cells of an aggressive host and cells of a corresponding immune system. Proposed models are related to the generalized Boltzmann equation. The book may be used for advanced graduate courses and seminars in biological systems modeling.

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include: the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools, recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology, an increase in computing power which enables calculations and simulations to be performed that were not previously possible, and an increasing interest in in-silico experimentation due to the complications involved in human and animal research. This new book presents the latest leading-edge research in the field.

The textbook is designed to provide a "non-intimidating" entry to the field of mathematical biology. It is also useful for those wishing to teach an introductory course. Although there are many good mathematical biology texts available, most books are too advanced mathematically for most biology majors. Unlike undergraduate math majors, most biology major students possess a limited math background. Given that computational biology is a rapidly expanding field, more students should be encouraged to familiarize themselves with this powerful approach to understand complex biological phenomena. Ultimately, our goal with this undergraduate textbook is to provide an introduction to the interdisciplinary field of mathematical biology in a way that does not overly terrify an undergraduate biology major, thereby fostering a greater appreciation for the role of mathematics in biology

The branch of biology which employs mathematical models and abstractions of the living organisms is known as mathematical biology. These models and abstractions are employed for the purpose of investigating the principles which govern the development, behavior and structure of the systems. This discipline uses techniques and tools of applied mathematics for the modeling and mathematical representation of biological processes. Mathematical models are primarily used for describing systems in a quantitative manner. This facilitates the simulation of their behavior which in turn helps in predicting their properties, which might not have been visible otherwise. Some of the various areas of research under this discipline are abstract relational biology, algebraic biology, complex systems biology and evolutionary biology. While understanding the long-term perspectives of the topics, the book makes an effort in highlighting their impact as a modern tool for the growth of the discipline. The various studies that are constantly contributing towards advancing technologies and evolution of mathematical biology are examined in detail. This book will serve as a reference to a broad spectrum of readers