Symbiosis

**Symbiosis I (
Semester 1)**. Biology and Statistics. A brief
description of the topics covered in the first semester.

Module 1
**Introduction**: The Scientific Method from the
biologist’s viewpoint and the role that statistics plays in
measuring variation by statistical hypotheses. The binomial
distribution is introduced to test hypotheses about population
proportions and the randomization test is introduced to test
hypotheses about the equality of means in tow populations. Case
studies of scientific hypotheses using von Helmont’s
experiments, Stanley Pruissner work on the prion hypothesis,
models of yellow fever as a mosquito vectored pathogen and then a
class project to ask if HIV can be transmitted by
mosquitoes.

Module 2
**The Cell** as the biological basis of life. Use of
statistical graphs and descriptive statistics including
correlation with data from cell size databases. The transcription
from descriptive to inference through randomization tests and
bootstrapping. Biological consequences of surface area/volume
rations and mitosis as a consequence of growth.

Module 3.
**Size and scale**. Introduction to the concepts of
scaling and allometry. Differences between isometric and
allometric scaling. Fractal branching for surface area/volume
problems. Slope as a rate of change. Log-Log plots and the Power
Law. The exponential function and the normal distribution. Linear
regression and transformations.

Module 4.
**Mendelian genetics**. The use of Mendel’s
original data and how to draw conclusions based on probability.
Goodness of Fit tests and independence tests are done using
Mendel’s data. The use of the Punnett’s Square and
probability trees to demonstrate how the probability of
combination of alleles is a representation of the meiosis
process. Conditional probability, Bayes rule, Poisson and normal
approximation to the binomial distribution and an introduction to
sampling methods are the main statistical topics of the
module.

Module 5.
**DNA Molecular Biology and the Genome**. DNA
replication and sequence analysis. Revisiting mitosis at the
molecular level. Goodness of Fit test and independence tests to
study nucleotide and di-nucleotide frequencies in sequence data.
Matrices and graphs in the context of transitional (conditional)
probabilities. Repeats and palindromes (related to restriction
enzyme sites) using probability, random walks in the context of
sequence comparisons. Examination of genomes and genome sequences
for defined elements.

Module 6.
**Evolution**. Using probability functions to
demonstrate changes in gene frequency over generational times.
Evolution concepts including a Walk through DEEP TIME, protein
structure, genetic code and mutations. Changes in gene frequency
in populations and an introduction to the Wright-Fisher model.
Hardy-Weinberg equilibrium, steady state conditions and the
t-test.

**Symbiosis II (
Semester 2)**: Biology and Calculus. A
brief description of the topics covered in the second semester.
One of the goals of the SYMBIOSIS project is to cover the topics
in a first semester calculus course by the end of Symbiosis II.
Thus, the calculus in Symbiosis II is actually a continuation of
calculus concepts begun in Symbiosis I.

Module 7.
**Population Ecology**. Introduction to population
ecology that uses an increasingly complex sequence of models to
explore the annual salmon smolt migration from the headwaters of
the Columbia River to the Pacific Ocean. The module begins with a
simple emigration/immigration difference equation model, thus
giving an accessible context within which to introduce ecological
models and rates of change. The inclusion of mortality and
reproduction then motivates the development of instantaneous
rates of change and the derivative, during which the rules of
differentiation are also developed.

Module 8.
**Ecological Interactions**. Focuses on
species-species interactions with an emphasis on competition,
predation, symbiosis, and parasitism. This allows several
different models to be introduced and the continued development
of the calculus. In particular, these models serve as both a
pretext and a context for the development of the chain rule, of
the properties of the derivative, and of the use of derivatives
in qualitative exploration of mathematical models. This includes
topics such as differentiability implying continuity,
monotonicity and concavity, and the Mean Value Theorem, among
others. This also allows us to talk about equilibria in a
meaningful way that includes specifics and quantitative
results.

Module 9.
**Ecological Models**. The theme of ecology
continues via a different set of mathematical concepts –
namely, optimization, limiting processes, structural ecology and
simple matrix algebra, and game theoretic concepts. The goal of
this module is to show how ecological processes tend toward
optimal strategies, as well as being a means of concluding the
conceptual development of differential calculus.

Module 10.
**Chronobiology**. Circadian rhythms and other
periodic biological phenomena that allows for the introduction
and development of the trigonometric functions. The discussion of
the trigonometric functions is delayed not only because students
often struggle with trigonometry, but also because periodic
processes occur frequently enough in biology so as to allow them
to be the subject of an entire module. The development of
the trigonometric functions in this context includes the coverage
of harmonic oscillations, derivatives of trigonometric functions,
and the statistical concept of a periodogram.

Module 11.
**Plant Physiology/Integrals** Introduction of the
concept of the definite integral using integration in the study
of plant physiology. The emphasis is on cumulative processes, as
such accumulations are modeled by mathematical integrals. In
particular, the concept of a relative rate of change, which is
important both mathematically and biologically, is introduced and
developed. Such development motivates and benefits from the
concept of the definite integral as a tool for measuring areas,
volumes, and masses. The fundamental theorem of calculus is
developed and illustrated via several examples, as are some basic
techniques for working with integrals.

Module 12.
**Enzymes and Energy**. Differential equation models
are used to study enzymes and energy. These topics includes the
Michealis-Menten equations, the use of integrals in solving
differential equations, and the use of integrals in the study of
cumulative processes other than those which produce changes in
area or volume. This module not only wraps up the coverage of the
integral calculus with arguably its most important application
– the solution and study of differential equations –
but it also broadens students’ perspectives to the
importance of fields such as physics and chemistry to the pursuit
of biology.

The mathematics component is also assessed since the students have to take the Math Department’s Gateway exam that demonstrates competency in calculus concepts. Successful passage of this exam is required to receive credit from calculus and allows students to take Calculus II.

**Symbiosis III (
Semester 3)**: Biology, Math and
Bioinformatics. A brief description of the topics covered in the
first semester. In Symbiosis III the quantitative
components include calculus, matrices, graph theory, advanced
statistical topics such as non-linear estimation, multivariate
methods and an introduction to bioinformatics. The biological
topics are Neurons, Membranes, Developmental Biology and
BioInformatics.

Module 13.
**Transport across the membrane**. Fick’s Law,
Danielli-Davson equation and integration to find concentrations.
CFTR activity on ion transport. Diffusion, facilitative diffusion
and active transport. Trans-membrane proteins. Consequences of
the sodium potassium pump on these processes is
developed.

Module 14.
**Neuroscience**. Excitable cells and membrane
potential in nerve transmission and muscle contractions. Action
potential and signal propagation. Synapses. Ladder circuit
approximation and Ohm’s Law. Cable equation, Hodgkin Huxley
Equations. Artificial neural network.

Module 15.
**Photosynthesis**. Cyclic and noncyclic electron
transport, ATP generation using membrane potential. Integrals and
differential equations in Bear-Lambert Law. Estimating parameters
of an enzymatic pathway.

Module 16.
**Developmental Biology**. A broad description of
basic concepts in developmental biology. Topics include
morphogenesis, determination and differentiation, morphogens and
polarity gradients to establish fields for gene activation. Other
topics include fertilization, cleavage, gastrulation;
differential gene expression and developmental gene interactions.
The approach examines gene expression in relation to
development.

Module 17.
**Genomics and molecular biology**. A short module
to introduce basic molecular tools such as cloning, vectors,
nucleic acid hybridization (southerns and northerns), genomic
sequencing.

Module 18.
**Gene expression and BioInformatics**. A longer
module that uses the previous modules to further examine
bioinformatics using gene expression and microarrays.
Introduction to NCBI datasets, specifically microarrays from
various studies. Datasets are downloaded and “cleaned
up”. Log2 values of differential gene expression is
introduced and used in analysis. Microarray analysis project. Use
of Minitab statistic analysis program. Normalization of values in
dataset. Calculation of MA plots and regression analysis of the
graphs.