Tuesday, November 26, 2013

Biologists Swimming in a Mathematical Ocean: Modeling Gene Expression

Math is everywhere in today's society, and advanced mathematics have fueled the engineering efforts which underly almost all technology. In the past, biology may have been a refuge for scientists that were math-adverse. This is not longer true; instead, biologists should openly embrace higher mathematics and employ these techniques in a variety of ways. Perhaps the most relevant application to the synthetic biologists is for developing mathematical models of gene expression and cellular behavior.

Biologists make models all the time: any hypothesis, for example, is a reflection of some model held by the scientist which describes how a certain system works. Many times, this model may be implicit in a description for a biological process. More powerful models are explicitly formulated using the language of mathematics. Leveraging mathematics allows quantitative predictions of a cell's behavior or gene expression pattern to be made with great precision. Engineering the metabolism of a cell by modulating expression of many different genes requires sophisticated and (as much as possible) accurate models.

How then, is the a biologist with limited mathematical training to understand models of gene expression, let alone craft ones themselves? Diving into advanced literature on the subject can lead one to practically drown in a sea of differential equations, boolean logic, and matrices. I must even admit to having trouble staying afloat at first: my formal mathematics education stopped at 'Calculus II for Bio Majors" at Cook College, Rutgers (a course which covered matrices but not differential equations). Fortunately, my brother is quiet adept at this sort of math and has provided me with some excellent tutorials.

Even if you don't have a skilled mathematician in the family, there are resources to help you learn gene expression modeling (or any modeling of a biological system). Below I feature links to several of these resources. The resource which I have found to have the best blend of exhaustive yet accessible explanation is a thesis from 2010 written by Hosam Abdel Aleem. I'm sure there are other articles and publications out there that are as good, if not even better, than Dr. Aleem's work, but his is a delightful read and quiet approachable (it's also freely available).

An Algebraic Approach to Modelling the Regulation of Gene Expression

This thesis throughly explains the philosophy behind modeling, as well as how to construct mathematical models of gene expression in detail. The author covers not only modeling by differential equations (continuous), but also boolean models (binary) and his own methods (discrete but with multiple values). I highly recommend this read to anybody that is interested in modeling gene expression but doesn't know where to start.

In future posts, I will provide more detail of my own experience learning some of this material, including some step by step examples of how to create a model, solve or analyze the associated equations, and calculate the results. Until that time, I'd like to wish all of my readers a very Happy Thanksgiving!

For more links to resources for mathematical modeling of gene expression, select 'Read More'. Did I miss an important resource, or do you have a favorite method for modeling transcription and translation? Please share your thoughts below as a comment!

More Life Rafts for Life Scientists

The links below provide more help to the naive life scientist that is drowning in attempts to understanding mathematical models of gene expression and cell behavior. They include tools for representing mathematical formulas, tools for solving them, and some helpful reference.

Math Magic Lite
The authors of this software claims it is the ultimate equation editor on the planet. It certainly is the best option I have found for representing mathematical formulas without spending any money (freeware). It is also very easy to use. If you begin to model gene expression, you can now write your differential equations in style! (Or other equations, for that matter). I highly recommend this for any scenario in which you need to express any formulas (even chemical reactions) in writing. I used this very program to make the front piece image for this article.

Mathematical modeling of gene expression: a guide for the perplexed biologist:
This review article surveys several different methods currently used to model gene expression. Although it is written in a way to be accessible to the biologist, the article is concise and might not be as illuminating as the Aleem thesis linked to above. Still, I recommend this paper, especially as a springboard into more advanced topics and primary literature. 

Paul's Online Math Notes
Euler's Method for approximating differential equations. This method does not deliver the solution, but can be useful for getting an idea of how a system behaves. The steady state levels reached are often accurate, but the exact dynamics will not be precise using Euler's Method by hand.

Khan Academy's Playlist for Differential Equations
These videos are a great introduction to differential equations. If you lack the prerequisite knowledge but are interested in modeling with differential equations, or at least understanding models built upon this branch of mathematics, then I recommend checking out the first few videos in this series.

Wolfram Alpha Calculator
This calculator purports to have a functionality akin to the computer on 'Star Trek': queries on math and science, from any field, can be answered. The feature most relevant for this article is the ability of the calculator to solve some differential equations. This tool makes solving these equations (if they can be solved, that is) easy: no extra software is required (read: no need to spend money), and no knowledge of R or other programming languages is required.

The Deep End

Caution: the remaining links below may contain advanced mathematics. Although I can't profess to fully understand some of them, it is somewhat enlightening to at least note the broad organization of their model, and to see the advantages of one system over another.

Modeling Gene Expression with Differential Equations
While this paper is over a decade old, it is an interesting read. The math featured here is complex but deals primarily with differential equations, with models of both protein and RNA levels for a auto-repressor system.

Modeling the stochastic dynamics of gene expression in single cells: A Birth and Death Markov Chain Analysis
A stochastic model for gene expression. A detailed model (although some considerations left out, such as number of ribosomes and free amino acids).

Backwards SDE approach to modelling of gene expression
This paper describes the use of a stochastic differential equation model, introducing noise while measuring continuous change

Ranking of Gene Regulators Through Differential Equations and Gaussian Processes
This paper leverages differential equation models in a slightly different way: by fitting known expression data to a model, the research hope to be able to identify the regulatory factors that influence the expression. (Sort of a reverse modeling approach, at least compared to the main focus of this post).

Differential Equation Models for Gene Regulatory Network

This poster provides a concise reference for the difference between ordinary, partial, and stochastic differential equations and their use in modeling gene expression. Unfortunately, the explanations are curt if not lacking altogether. 


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