# Mathematical Modeling in Computational Biology

The beginning of computational biology as a field, dates back to the 1960s and the use of computers in protein structure studies, such as performing Fourier analysis to determine the three-dimensional structure of proteins. Although the initial focus was on modeling the structure of proteins, by measuring and collecting various data, this field now covers a wide range of biological topics. The development of high-throughput technologies as well as the increase in the processing power of computers, especially since the 1990s, have enabled biology to rely more on computational methods. Mathematical approaches can be helpful in this regard.

Fibonacci was one of the first to apply mathematics to biology by introducing his famous series to describe the population growth of rabbits in the 13th century. Also, Daniel Bernoulli, who used differential equations in the 18th century to analyze the mortality caused by smallpox and demonstrate the impact of inoculation.

Since the 18th century, mathematics has been applied in various scopes of biology and medicine, such as studying the flow of biofluids (such as blood circulation), microbial colonies, bacterial and viral life cycles, and their relations with human diseases, genetics and heredity, clinical experiments, systems biology, epidemiology, immunology, instrumentation, microbiology and molecular cell biology, neuroscience, and so on.

Using mathematical approaches in the study of complex phenomena, including biological phenomena, helps us to better understand their mechanisms and enables us to predict and anticipate their behavior under different conditions. Among the mathematical methods, we can mention quantification of observations, optimization, data analysis and processing, prediction, and modeling.

Mathematical modeling is the process by which a real-world problem is described by a mathematical formulation. So, the key issue in modeling is to find a suitable form of the phenomenon in mathematical language in the form of various functions, algebraic or differential equations, linear operators and matrices, combinatorial forms and graphs, etc.

After this formulation is done, algorithms, theorems, and other common or new mathematical tools can be used to further study the model. Almost all fields of applied mathematics are used in these tools. Areas such as: differential equations (ODE and PDE), discrete-time and continuous-time dynamical systems, information theory and coding, graph and network theory, integral transformations, numerical analysis and calculation, delay differential equations, statistics, probability, and time series analysis.

By modeling, a link between mathematics, computer science, statistics, and biology is established, which not only gives a clearer view of the phenomena, but also makes biology continuously benefit from the progress of these fields.

In addition, the use of computational and mathematical models in biology allows us to understand the mechanisms of complex diseases, improve treatment strategies and drug discovery, increase our sight of new molecular mechanisms, and reveal the rules that govern biological systems.

Among the topics that are studied in this field, the following can be mentioned:

- The spread of epidemics
- Macromolecular complexes
- The central nervous system’s neural connectivity
- Validation of drug efficacy and safety
- Estimation of the survival rate for patients under a specific treatment
- Gene Regulatory Networks and Metabolic Networks

For more information, please refer to the following sources:

- Faugeras, O., & Janin, J. (2012).
*Modeling in Computational Biology and Biomedicine: A Multidisciplinary Endeavor*. Springer Science & Business Media. - Ingalls, B. P. (2013).
*Mathematical modeling in systems biology: an introduction*. MIT press. - Kuttler, C. (2009).
*Mathematical models in biology*. Munchen: Technische Universitat Munchen. - Bacaër, N. (2011).
*A short history of mathematical population dynamics*(Vol. 618). London: Springer.