**Unit 3: Modeling Biological Processes**
*Nearly any biological process can be reduced to a mathematical formula
and modeled on a computer. The large, multivariable datasets involved
in the study of complex processes can in fact be understood much more
quickly and easily with the help of computer modeling. An animal
behaviorist can use mathematical prediction models to instantaneously
predict the reaction that a subject will have to a stimulus, though the
spectrum of possible outcomes and the complexity of the mathematical
model might appear quite daunting to someone attempting to model it
manually. Alternately, using computational modeling, a biochemist could
quickly predict the variety of outcomes that a cellular cascade reaction
might have based upon the amount of substrate introduced to the cell
and, in this way, make a better estimate as to the amount that should be
used in a particular experiment or in order to achieve a particular
desired outcome. In this unit, we will look at examples of
computational modeling in biochemistry, cell biology, neuroscience,
population biology, evolution, and behavior. You will learn the basic
procedures for computational modeling in each of these fields and will
gain a fuller understanding of the flexibility and universal
applicability of mathematical modeling.*

**Unit 3 Time Advisory**

This unit will take approximately 27.5 hours to complete.

☐ Subunit 3.1: 2.5 hours

☐ Subunit 3.2: 8 hours

☐ Subunit 3.3: 5.5 hours

☐ Subunit 3.4: 1 hour

☐ Subunit 3.5: 7 hours

☐ Subunit 3.6: 3.5 hours

**Unit3 Learning Outcomes**

Upon completion of this unit, students will be able to:
- Describe mathematical approaches to biochemistry and cellular
pathways in computational biology.
- Explain the process of neuronal signaling along with the
mathematical components.
- Describe how the evolutionary process can be modeled using various
statistics.
- Describe the mathematical treatment of population growth.

**3.1 Biochemistry Techniques**
*Note: Biochemistry is the study of the chemical reactions that occur in
biological settings. In this section, we will learn how to model the
behavior of biologically-relevant molecules and the reaction rates of
chemical reactions catalyzed by enzymes.*

**3.1.1 Mathematical Treatment of Biochemistry**
**3.1.1.1 Molecular Dynamics**
- **Reading: Wikipedia’s “Molecular Dynamics”**
Link: Wikipedia’s “Molecular
Dynamics” (HTML)

```
Instructions: The page defines molecular dynamics and presents many
of the details of the simulations and algorithms involved. How can
molecular dynamics be used to understand the movements of atoms or
even proteins?
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.1.1.2 Enzymes and Kinetics**
- **Reading: Wikipedia’s “Enzyme Kinetics”**
Link: Wikipedia’s “Enzyme
Kinetics” (HTML)

```
Instructions: This entry introduces you to enzyme kinetics,
including single and multiple substrate assays and Michaelis-Menten
and non-Michaelis-Menten kinetics. You should be able to describe
the factors that influence enzyme kinetics and use the
Michaelis-Menten equation.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2 Cellular Pathways**
*Note: Chemical reactions within cells often depend upon intricate
cascades of reactions, where an interruption of any step in the sequence
can completely skew the final product by either preventing the
production of an important protein or inhibiting the breakdown and
removal of a detrimental compound from tissue. In this subunit, we will
learn how cellular processes can be modeled and predicted using the
computer.*

**3.2.1 Mathematical Treatment of Cellular Pathways**
- **Reading: Developmental Cell: Alex Mogilner, Roy Wollman, and
Wallace F. Marshall’s “Quantitative Modeling in Cell Biology”**
Link: Developmental Cell: Alex Mogilner, Roy Wollman, and Wallace F.
Marshall’s “Quantitative Modeling in Cell
Biology”
(PDF)

```
Instructions: Click on the link entitled “Quantitative Modeling in
Cell Biology” to download the PDF. Read the article and focus on
why and how models are used in cellular pathways and what
mathematical approaches are involved.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2.1.1 Cellular Pathways**
- **Reading: Nature Cell Biology: Bree B. Aldridge, John M. Burke,
Douglas A. Lauffenburger, and Peter K. Sorger’s “Physiochemical
Modeling of Cell Signaling Pathways”**
The Saylor Foundation does not yet have materials for this portion
of the course. If you are interested in contributing your content to
fill this gap or aware of a resource that could be used here, please
submit it here.

```
[Submit Materials](/contribute/)
```

**3.2.2 Prediction Modeling**
**3.2.2.1 Ordinary Differential Equation Modeling**
- **Reading: George Washington University: Michael J. Coleman’s
“Population Models with Ordinary Differential Equations”**
Link: George Washington University: Michael J. Coleman’s
“Population Models with Ordinary Differential
Equations”
(PDF)

```
Instructions: Scroll down and click on the link entitled “Slides
here” to download the PDF. This is a technical reading that
provides examples and includes mathematical derivations using
partial differential equations. The goal here is to understand
differential equations in general then apply them to single species
models of population growth followed by multiple species models.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2.2.2 Partial Differential Equations**
- **Reading: Nature Cell Biology: Bree B. Aldridge, John M. Burke,
Douglas A. Lauffenburger and Peter K. Sorger’s “Physiochemical
Modeling of Cell Signal Pathways”**
Link: Nature Cell Biology: Bree B. Aldridge, John M. Burke, Douglas
A. Lauffenburger and Peter K. Sorger’s “Physiochemical Modeling of
Cell Signal Pathways”
(PDF)

```
Instructions: Click on the link entitled “Physiochemical Modeling
of Cell Signal Pathways” to download the PDF. This is a review
article that should give you a good introduction to model design and
mathematical modeling of biochemical pathways.
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2.2.3 Flux-Balance Analysis**
- **Reading: Nature.com’s “Flux Balance Analysis Primer”**
Link: Nature.com’s “Flux Balance Analysis
Primer”
(HTML)

```
Instructions: This link will introduce the basic concepts of
Flux-Balance Analysis and the complete mathematical background
behind it. It includes information about metabolic pathway
construction as well. You may want to search for articles listed in
the literature cited section to gain more insight into the method of
flux-balance analysis.
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2.2.4 Analysis of Extreme Pathways**
- **Reading: University of California, San Diego: Systems Biology
Research Group’s “Extreme Pathway Analysis”**
Link: University of California, San Diego: Systems Biology Research
Group’s “Extreme Pathway
Analysis”
(HTML)

```
Instructions: Scroll down the page to find several published
articles on extreme pathway analysis. The goal here is to become
familiar with extreme pathway analysis and how it is being used.
You should be able to achieve this by reading a couple of the
articles listed under related publications on the web page.
Studying this material should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.2.4 Statistical Testing for Accuracy**
**3.3 Neuronal Signaling**
*Note: The process of thinking and doing is so complex, it is hard to
understand how it could be duplicated in any way. However, it all comes
down to individual action potentials, the firing of individual neurons
within a network, and the ways in which those firings affect other
neutrons within the network. Here, we will learn about the mathematical
modeling of action potentials (and changes in membrane potential that
may or may not create an action potential) and discover how we to
predict the behavior of a single neuron or a simple neural network based
on these models.*

**3.3.1 Mathematical Treatment of Neuronal Signaling**
**3.3.1.1 The Action Potential**
- **Reading: Bryn Mawr College: Rebecca Vandiver’s “Hodgkin-Huxley
Model”**
Link: Bryn Mawr College: Rebecca Vandiver’s “Hodgkin-Huxley
Model”
(PDF)

```
Instructions: Click on the link entitled “Hodgkin-Huxley Model” to
download the PDF. This presentation covers the Hodgkin-Huxley model
of action potentials and includes a mathematical treatment. You
should be able to qualitatively describe the model and how it is
applied to action potentials.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.1.2 Changes in Membrane Potential**
- **Reading: Journal of Physiology: A. V. Hill’s “A New Mathematical
Treatment of Changes of Ionic Concentration in Muscle and Nerve
Under the Action of Electric Currents, with a Theory as to Their
Mode of Excitation”**
Link: Journal of Physiology: A. V. Hill’s “A New Mathematical
Treatment of Changes of Ionic Concentration in Muscle and Nerve
Under the Action of Electric Currents, with a Theory as to Their
Mode of
Excitation”
(PDF)

```
Instructions: Read this article on the mathematics associated with
membrane potentials. What is the Nernst Theory and how does the
information in this reading contribute to our understanding of ion
concentrations associated with nerve conduction?
Reading this material should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.2 Prediction Modeling**
**3.3.2.1 Single-Neuron Modeling**
- **Reading: University of Sussex: Andrew Phillipides’ “Neuronal
Signaling in Real Neurons”**
Link: University of Sussex: Andrew Phillipides’ “Neuronal Signaling
in Real
Neurons”
(PPT or HTML)

```
Instructions: Scroll down the page and click on the Lecture 2 link.
This presentation covers resistance, capacitances, and models
associated with electrical impulse conduction. Pay particular
attention to the Hodgkin-Huxley model.
Studying this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.2.2 Artificial Neural Networks**
- **Reading: learnartificialneuralnetworks.com’s “Artificial Neural
Networks”**
Link: learnartificialneuralnetworks.com’s “Artificial Neural
Networks” (HTML)

```
Instructions: This comprehensive website provides everything you
need to know about artificial networks, from an introduction to
artificial neural networks to an explanation of how to train
networks.
Studying this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.4 Population Biology**
*Note: Predicting population viability and growth based upon
multivariable data sets that contain information on the age and sex of
individuals, the predation rate, and limiting resource factors is done
much more efficiently on a computer than it is manually. Here we will
learn to predict population growth in a simple population using computer
modeling.*

**3.4.1 Mathematical Treatment of Population Biology**
- **Reading: Michigan State University: Steven Vieira’s “Population
Growth Models”**
Link: Michigan State University: Steven Vieira’s “Population Growth
Models” (HTML)

```
Instructions: This page discusses several population growth models
and the math associated with them. Pay attention to key terms and
the differences between exponential and logistic growth.
Reading this material should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.4.2 Predicting Population Growth**
- **Reading: Michigan State University: Steven Vieira’s “Population
Growth Models”**
Link: Michigan State University: Steven Vieira’s “Population Growth
Models” (HTML)

```
Instructions: This page discusses several population growth models
and the math associated with them. This is the same reading as in
3.4.1.1. You should focus here on mathematical formulas that allow
you to predict future population size based on current conditions.
Reading this material should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.5 Evolution**
*Note: Computer modeling comes in handy when studying the evolution of a
population, species, or group of species. Using allelic frequency (the
frequency with which a particular gene sequence appears at a given
locus) and a mathematical calculation of the Hardy-Weinberg Equilibrium,
we can quickly see whether a population is currently evolving or not.
By comparing the physical or molecular characteristics of different
species and mathematically sorting out the species with highest numbers
of similarities, we can create phylogenies, or “family trees” of
species, which estimate the process of evolution and divergence among
those species over millions of years. In this section, we will learn
how to predict evolutionary change within a population and evolutionary
connection between populations using computer modeling methods.*

**3.5.1 Mathematical Treatment of Evolution**
**3.5.1.1 Hardy-Weinberg Equilibrium**
- **Reading: Genetics.org: Jennifer Shoemaker, Ian Painter, and B. S.
Weir’s “A Bayesian Characterization of Hardy-Weinberg
Disequilibrium”**
Link: Genetics.org: Jennifer Shoemaker, Ian Painter, and B. S.
Weir’s “A Bayesian Characterization of Hardy-Weinberg
Disequilibrium”
(HTML)

```
Instructions: This article provides a little background on
estimating Hardy-Weinberg equilibrium and demonstrates how a
Bayesian approach can replace other methods used to estimate
Hardy-Weinberg equilibrium. In addition to understanding the
Bayesian approach mentioned in the paper, you should also have an
understanding of Hardy-Weinberg equilibrium.
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**Reading: Jon Wakefield’s “Bayesian Methods for Examining Hardy-Weinberg Equilibrium”**Link: University of Washington: Jon Wakefield’s “Bayesian Methods for Examining Hardy-Weinberg Equilibrium” (PDF)Instructions: This paper describes various aspects of establishing Hardy-Weinberg equilibrium and demonstrates how a Bayesian approach can be used.

Reading this material should take approximately 1.3 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.5.1.2 Creating Phylogenies**
- **Reading: Indiana University: ENSI/SENSI’s “Making Cladograms”**
Link: Indiana University: ENSI/SENSI’s “Making
Cladograms”
(HTML and PDF)

```
Instructions: This is a well-developed lesson plan that guides you
through the creation of a cladogram. After reading the page, scroll
to the bottom and view the 6 page PDF on creating a cladogram. Work
through this as it is the best way to learn how cladograms are
developed.
Reading this material should take approximately 2 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.5.2 Modeling Evolutionary Change**
- **Reading: University of Utah: David M. Hillis, John P. Huelsenbeck,
and Clifford W. Cunningham’s “Application and Accuracy of Molecular
Phylogenies”**
Link: University of Utah: David M. Hillis, John P. Huelsenbeck, and
Clifford W. Cunningham’s “Application and Accuracy of Molecular
Phylogenies”
(PDF)

```
Instructions: Click on the the link entitled
“Hillis\_etal\_Science\_94.pdf” to download the PDF. The paper
discusses various models of evolution and different methods of
creating phylogenies. You should be able to compare the performance
of parsimony, neighbor-joining, and UPGMA.
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.5.3 Modeling Evolutionary Connections**
- **Reading: Proceedings of the National Academy of Sciences: Mónica
Medina’s “Genomes, Phylogeny, and Evolutionary Systems Biology”**
Link: Proceedings of the National Academy of Sciences: Mónica
Medina’s “Genomes, Phylogeny, and Evolutionary Systems
Biology”
(HTML)

```
Instructions: This article explains how genomic data has improved
our understanding of phylogenies and also links genomic data to the
rapidly developing field of systems biology. You should be able to
describe advances in biology using genomic data, systems biology,
and transcriptional networks.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.6 Behavior**
*Note: To think that behavior can be mathematically predicted seems
counter-intuitive, yet a number of mathematical formulas and matrices
have been created and successfully deployed in order to predict the
outcomes of conflicts between animal behavior and psychology. The study
of behavior through mathematics is famously known as “Game Theory.” In
this section, we will learn about a few standard methods of predicting
behavior through Game Theory and discover how to model them on the
computer.*

**3.6.1 Mathematical Treatment of Behavior**
**3.6.1.1 Game Theory**
- **Reading: University of California, Los Angeles: David K. Levine’s
“What Is Game Theory?”**
Link: University of California, Los Angeles: David K. Levine’s
“What Is Game
Theory?”
(HTML)

```
Instructions: This is a good introductory reading on the subject of
game theory. It includes an instructive example and some simple
math behind the prisoners dilemma.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.6.1.2 Prisoner's Dilemma**
- **Reading: University of Iowa Math Club: Erin Pearse’s “The
Prisoner’s Dilemma”**
Link: University of Iowa Math Club: Erin Pearse’s “The Prisoner’s
Dilemma” (PDF)

```
Instructions: Click on the link entitled “The Prisoner's Dilemma.
Invited speaker. April 23, 2009. Grinnell College.” to download the
PDF. This is a presentation that covers the game, the mathematics
behind the game, and variations on the theme. Pay attention to
definitions and to real world application of the Prisoner’s
Dilemma.
Reading this material should take approximately 1.3 hours.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.6.2 Prediction Modeling**
**3.6.2.1 Collective Behavior**
- **Reading: Trends in Cognitive Science: Robert L. Goldstone and
Marco A. Janssen’s “Computational Models of Collective Behavior”**
Link: Arizona State University: R. Goldstone and M. Janssen’s
“Computational Models of Collective
Behavior” (PDF)

```
Instructions: Scroll down to the line “70. Goldstone, R.L. and M.A.
Janssen (2005) Computational models of collective behaviour, Trends
in Cognitive Science 9(9): 424-430” and click the link to download
the PDF. You should be able to compare and contrast the different
Agent-Based models from the reading.
Reading this material should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```