Perspective Chapter: Multiscale Mathematical Modeling of Biological Systems for Bioinformatics and Medical Informatics

Abstract

This chapter provides an overview of multiscale mathematical modeling techniques used for analyzing complex biological systems in the fields of bioinformatics and medical informatics. It emphasizes the significance of mathematical modeling in gaining insights into biological processes and understanding their underlying mechanisms. The chapter discusses several modeling techniques, such as stochastic simulations, continuum models, and molecular dynamics simulations, and explores their applications in the context of bioinformatics and medical informatics. Additionally, the chapter highlights the challenges associated with multiscale modeling, such as the need for precise parameter estimation and computational efficiency. The overall message of the chapter is to stress the importance of developing and refining multiscale modeling techniques to advance our comprehension of biological systems and ultimately improve human health.

1. Introduction

Mathematical modeling has become an indispensable tool in the study of complex biological systems, especially in the fields of bioinformatics and medical informatics (see, e.g., [1, 2]). These fields deal with massive amounts of data, and mathematical models provide a way to make sense of the data by identifying patterns and relationships that would be difficult or impossible to detect otherwise (see, e.g., [3, 4]). In this chapter, we will explore multiscale mathematical modeling techniques and their applications in the analysis of complex biological systems (see, e.g., [2, 5]). We will also discuss the challenges associated with mathematical modeling in this context and strategies for overcoming them (see, e.g., [6, 7]).

Multiscale modeling is a particularly important technique for analyzing complex biological systems (see, e.g., [8, 9]). This approach involves the use of multiple models at different levels of detail, with each model capturing a different aspect of the system (see, e.g., [10, 11]). For example, a multiscale model of a biological system might include a molecular-level model that describes the behavior of individual molecules, as well as a tissue-level model that describes the behavior of cells within a tissue (see, for e.g., [12, 13]). By integrating these models, researchers can gain a more comprehensive understanding of the system as a whole (see, e.g., [14, 15]).

Mathematical modeling is also important for predicting the behavior of biological systems under different conditions (see, e.g., [13, 16]). For example, a mathematical model might be used to predict how a drug will interact with a particular protein in the body, or how changes in the environment might affect the behavior of a particular organism (see, e.g., [15, 17]). These predictions can then be used to guide experimental design and drug development, ultimately leading to better treatments for diseases (see, e.g., [18, 19]).

Recent research has shown that multiscale mathematical modeling can be used to gain insights into the behavior of complex biological systems (see, e.g., [20, 21]). For example, a recent study used a multiscale model to investigate the dynamics of the immune response to a viral infection (see, e.g., [22, 23]). The model was able to capture the complex interactions between different components of the immune system and predict the outcome of the infection (see, e.g., [24, 25]). Another study used a multiscale model to investigate the role of mechanical forces in the development of cancer (see, e.g., [26, 27]). The model was able to predict the behavior of cancer cells under different mechanical conditions and identify potential targets for cancer therapy (see, e.g., [28]).

Overall, mathematical modeling is a powerful tool for studying complex biological systems, and multiscale modeling is an important technique for gaining insights into the behavior of these systems (see, e.g., [29]). Recent research has shown that multiscale modeling can be used to make accurate predictions about the behavior of biological systems under different conditions and to identify potential targets for drug development (see, e.g., [30]).

2. Multiscale modeling techniques

There are several multiscale modeling techniques that are used in bioinformatics and medical informatics, including stochastic simulations, continuum models, and molecular dynamics simulations (see, e.g., [8]). In this section, we will provide an overview of these techniques and their applications.

2.1 Stochastic simulations

Biological systems that have inherent randomness or uncertainty can be modeled using stochastic simulations. They are based on probability theory and can provide valuable insights into the behavior of complex systems. Stochastic simulations are used in various fields of biology, including gene expression, signal transduction, and population dynamics.

Stochastic simulations are a class of computational methods used to model biological systems that have inherent randomness or uncertainty. They are based on probability theory and can provide valuable insights into the behavior of complex systems. Stochastic simulations are used in various fields of biology, including gene expression, signal transduction, and population dynamics.

The basic idea behind stochastic simulations is to model the behavior of a system as a set of random events. Each event is associated with a probability and the simulation proceeds by randomly selecting events and updating the state of the system accordingly. The simulation can be run multiple times to generate statistical data on the behavior of the system.

One of the most common types of stochastic simulations is the Gillespie algorithm, which is used to model chemical reactions. The Gillespie algorithm is based on the idea of a “chemical master equation”, which describes the time evolution of the probability distribution of the system. The algorithm proceeds by randomly selecting a reaction and updating the state of the system accordingly.

Stochastic simulations can be used to model a wide range of biological systems, from simple gene expression networks to complex ecosystems. They are particularly useful for systems that have a large number of interacting components or that exhibit complex behavior. Stochastic simulations can provide insights into the behavior of these systems that would be difficult or impossible to obtain using other methods.

One of the key advantages of stochastic simulations is that they can capture the effects of randomness and variability in biological systems. This is important because many biological systems are inherently stochastic, meaning that they exhibit random fluctuations in behavior even under identical conditions. Stochastic simulations can help to quantify and understand these fluctuations, which can be critical for predicting the behavior of biological systems in real-world settings.

Another advantage of stochastic simulations is that they can be used to explore the effects of different parameters and assumptions on the behavior of a system. For example, a stochastic simulation can be used to explore the effects of different levels of noise or variability on the behavior of a gene expression network. This can help to identify key factors that drive the behavior of the system and to optimize the experimental design.

Overall, stochastic simulations are a powerful tool for modeling and understanding complex biological systems. They are particularly useful for systems that exhibit randomness or variability and can provide insights that would be difficult or impossible to obtain using other methods.

2.2 Continuum models

Continuum models are used to model biological systems that can be described by continuous variables, such as concentration or density. They are based on differential equations and can provide insights into the spatiotemporal dynamics of biological systems. Continuum models are used in various fields of biology, including cell biology, physiology, and ecology.

Continuum models are based on the assumption that biological systems can be described by continuous variables that vary smoothly in space and time. These variables can include concentrations of chemical species, densities of cells or organisms, or other quantities that can be measured continuously. The behavior of these variables is described by differential equations that relate the rate of change of the variable to its current value and the values of other variables in the system.

Continuum models can be used to study a wide range of biological phenomena, including the growth and development of cells and organisms, the spread of diseases, and the interactions among species in ecological communities. They can provide insights into the spatiotemporal dynamics of these systems, including the formation of patterns and the emergence of collective behavior.

Continuum models are often used in conjunction with experimental data to test hypotheses and make predictions about the behavior of biological systems. They can also be used to design experiments that test specific predictions of the model. Overall, continuum models are a powerful tool for understanding the behavior of biological systems and for making predictions about their behavior in different conditions.

2.3 Molecular dynamics simulations

Molecular dynamics simulations are used to model biological systems at the molecular level. They are based on classical mechanics and can provide insights into the structure and dynamics of biological molecules. Molecular dynamics simulations are used in various fields of biology, including protein folding, ligand binding, and drug design.

The basic principle of molecular dynamics simulations is to solve the equations of motion for a system of atoms or molecules using numerical methods. The equations of motion are derived from Newton’s second law:

$$m_i\frac{d^2{\mathbf{r}}_i}{{\mathit{dt}}^2}={\mathbf{F}}_i,$$E1

where $m_i$ is the mass of atom $i$, ${\mathbf{r}}_i$ is its position vector, and ${\mathbf{F}}_i$ is the net force acting on it. The net force can be calculated from the potential energy function $U\left(\mathbf{r}\right)$, which depends on the positions of all atoms in the system:

$${\mathbf{F}}_i=-{\nabla }_{i}U\left(\mathbf{r}\right).$$E2

The potential energy function can be decomposed into different types of interactions, such as bonded interactions (e.g., covalent bonds, angles, dihedrals), non-bonded interactions (e.g., van der Waals forces, electrostatic forces), and external forces (e.g., applied fields). The choice of potential energy function depends on the level of detail and accuracy required for the simulation.

To solve the equations of motion numerically, a discrete time step $\mathrm{\Delta }t$ is introduced and an integration algorithm is applied to update the positions and velocities of all atoms at each time step. The integration algorithm should conserve energy and momentum and be stable and accurate. A common choice of integration algorithm is the Verlet algorithm or its variants (e.g., velocity Verlet, leapfrog Verlet).

The output of a molecular dynamics simulation is a trajectory that records the positions and velocities of all atoms as a function of time. The trajectory can be analyzed to calculate various properties of interest, such as structural features (e.g., distances, angles, contacts), thermodynamic quantities (e.g., temperature, pressure, free energy), kinetic parameters (e.g., diffusion coefficients, reaction rates), or functional aspects (e.g., conformational changes, binding events).

3. Challenges associated with multiscale modeling

Multiscale modeling presents several challenges, including the need for accurate parameter estimation and computational efficiency. In this section, we will discuss these challenges and their impact on the accuracy and reliability of multiscale models.

3.1 Parameter estimation

Parameter estimation is the process of determining the values of model parameters that best fit experimental data. In multiscale modeling, accurate parameter estimation is crucial for obtaining reliable predictions. However, parameter estimation can be challenging due to the complexity and nonlinearity of multiscale models.

Parameter estimation can be formulated as an optimization problem, where the objective is to minimize the difference between the model predictions and the experimental data. This difference is often quantified using a cost function, which measures the discrepancy between the model predictions and the experimental data. The cost function can be defined in various ways, depending on the nature of the data and the model.

One common approach to parameter estimation is to use least-squares regression, which involves minimizing the sum of the squared differences between the model predictions and the experimental data. This approach can be formulated as follows:

$$\underset{\theta }{min}\sum _{i=1}^n{\left(y_i-f\left(x_i,\theta \right)\right)}^2$$E3

where $\theta$ is the vector of model parameters, $y_i$ is the $i$-th experimental observation, $x_i$ is the corresponding input variable, $f\left(x_i,\theta \right)$ is the model prediction for the $i$-th observation, and $n$ is the number of observations.

The optimization problem in Eq. (3) can be solved using various numerical methods, such as gradient descent, Newton’s method, or the Levenberg-Marquardt algorithm. These methods involve iteratively updating the parameter estimates until a minimum of the cost function is reached.

Parameter estimation can be challenging in multiscale models due to the large number of parameters and the complexity of the model. In some cases, it may be necessary to use advanced optimization techniques, such as Bayesian inference or Markov chain Monte Carlo methods, to obtain reliable parameter estimates.

Overall, parameter estimation is a critical step in multiscale modeling, and accurate parameter estimates are essential for obtaining reliable predictions.

3.2 Computational efficiency

Multiscale models are computationally intensive, and their simulation can require significant computational resources. Computational efficiency is essential for reducing the simulation time and enabling the simulation of large-scale systems. However, improving computational efficiency can be challenging due to the complexity of multiscale models.

4. Conclusion

In conclusion, multiscale modeling techniques are essential for gaining insights into complex biological systems in the fields of bioinformatics and medical informatics. Stochastic simulations, continuum models, and molecular dynamics simulations are valuable tools for modeling biological systems at different scales. However, multiscale modeling presents several challenges, including the need for accurate parameter estimation and computational efficiency. Overcoming these challenges is crucial for developing reliable multiscale models and advancing our understanding of biological systems.

Acknowledgments

This work was supported partly by a research grant from AFOSR.

Data availability statement

The author confirms that the data supporting the findings of this study are available within the chapter or its supplementary materials.

Classification

2000 mathematics subject classification: 65C05, 65C20, 92C42, 92C60, 62 M20