Symbiosis

**IBMS 1100**
is the first course in a 3 course sequence that integrates biology, statistics, and mathematics, As a result, the mathematics and statistics is introduced, explored, and developed in biological contexts, including surface area to volume ratios, isometric and allometric scaling, fractals in biology, and difference equations and discrete systems in genetics, evolution, and the study of DNA. Pre-calculus concepts and limits are also introduced and developed in IBMS 1100, both due to the natural contexts which arise for doing so (such as log-log plots) and because a major goal of the Symbiosis project is to spread the coverage of calculus I across 2 semesters as a way of promoting greater student success in both calculus comprehension and skill development.

**Teaching method**
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:
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Lectures were prepared mainly in power-point. Hands-on class activities and data analysis in the computer lab were used when appropriate on addition to the wet/dry lab component.

**Textbook**
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Complete class notes, on addition to power point presentations, were written for this course by the instructors under a grant from HHMI, they are available from the D2L platform.

**Statistical software**
: Minitab, R, Maple, Java Applications, Image J, Web-based Applets and Activities

The study of **
Biology
** is introduced. Aspects of what hypotheses are and how they are tested leads into statistical inference. Examples of hypothesis testing such as von Helmont’s plant growth test and Stanley Prussiner’s Prion Hypothesis are discussed. An introduction of Arbovirus infection of Yellow Fever leads to a discussion of viruses and definition of life. The hypothesis of whether AIDS can be transmitted by mosquitoes is used as an example of the use of quantitative biology. The five themes of biology are introduced as the thread of further modules.

What is **
Statistics
**? Role of statistics in the scientific method. An introduction to the role of mathematics and statistics in science in general. Randomization test to test the hypothesis of equal means (medians, variances) of two populations based on experimental data. Why do we study probability? Basic definitions: random experiment, sample space, event. Definitions of probability: classical, relative frequency, axiomatic definition and its consequences; independent events; replicates of a random experiment; Pascal triangle and basic combinatorics.
Types of random variables, mass or probability function and density functions
.
Discrete probability distributions
; b
inomial distribution; applying the binomial distribution to do test of hypothesis about a population proportion. First glance at the limit concept (probability as limit of a relative frequency, along with difficulties in using such a definition). First glance at mathematical models.

Introduction to the cell. What is the cell and why are they small? What is the concept of multicellularity? The organization of the cell and what are the consequences of the components functions. TANSTAAFL (There ain’t no such thing as a free lunch), a more wide ranging discussion of consequences starting from the more formally known Second Law of Thermodynamics. The transmission of information into and out of the cell. The cell cycle and mitosis as a consequence of cell growth, repair and quiescence. Data production: observational studies and experiments. Basic definitions: population, sample, individual, variables (categorical & quantitative).

Displaying and summarizing data for categorical variables, tables and graphs, relative risk, odds ratio, measuring agreement in matched-pairs situations. Displaying and summarizing data for quantitative variables, tables and graphs for one, two and several variables at the time. What are the data telling us? How to decide between the different statistical graphs? Location (mean vs. median, five number summaries) and variability statistics. Sources of variability. Looking at paired data. Correlation. What is statistical inference? Introducing the idea of sampling variability and sampling distribution. Exact sampling distribution of a sample proportion (based on the Binomial distribution) and its application to hypothesis testing and estimating with confidence. Bootstrapping to do inference about a population mean. Randomization or permutation test to test hypotheses about a parameter (mean, median or variance) in two populations.

What happens to an organism as it grows bigger? Can ants really toss locomotives off the tracks? Can King Kong jump off the Empire State Building? Can Tyrannosaurus rex really run at 80 kph? This module examines the functions that describe what happens when organisms grow (or shrink). Included are organism size as a determining factor in shape, the differences between isometry and allometry, problems with isometric scaling in biology, bacteria size, shape, organization, cell wall structure, and other characteristics. Exponential growth of bacterial populations. Biological models with mass as the independent variable. Area, volume, and surface area to volume ratio. Isometric scaling, slope, equations of lines, allometry and power laws. Limits as tools for approximation. The exponential function. Logarithms. Linear regression and transformed variables. Normal distribution, Fractal Geometry as it relates to biological organisms and the surface area to volume ratio.

Why was Gregor Mendel able to elucidate the laws that determine how organisms pass genetic information from one generation to the next? This crucial process was discovered and then ignored for almost 40 years and yet was the key that Darwin was missing to explain Evolution. The data and processes that Mendel used to determine these principles are examined. In this context, Meiosis is described as the cellular equivalent of Mendelian Laws.

A coin model to understand genotypes and phenotypes for all combinations of homozygous and heterozygous parents. Punnet squares and probability trees, ‘back- testing’. Comparing experimental results with the expected results under an assumed model: Chi-square test of goodness of fit. Review of probability basics. Chi-square test of independence. Fisher’s exact test. Test of homogeneity. Describing dependence with relative risk and odds ratio. Conditional probability and Bayes rule. Discrete distributions, expected value and variance, discrete uniform, Bernoulli, Binomial and its use to test hypotheses about a population proportion. Power of a test. Determining sample size based on the desired power for a test. Poisson distribution, binomial and normal approximations to the Poisson distribution. Introduction to sampling: population, sampling frame, sampling size, sampling methods (simple, systematic, cluster, two-stage, stratified), transect sampling, sampling and non-sampling error, capture/recapture and distance sampling.

Mendelian Laws describe how information is passed from generation to generation, but the molecular processes were not determined until the nature and structure of DNA was described. The structure of this molecule and the consequences of replication are covered. Is DNA the same in different organisms? Quantitative tools to look at the composition of the information are developed. DNA as nucleotide sequences, nucleotide frequency, GC content. Independence and conditional probability in the DNA environment. Transition matrix, graph to represent transition matrices. Probability of a given sequence of nucleotides, repeats of a single nucleotide, length of the repeat, geometric distribution. Palindromes, probability of any palindrome and of specific palindromes, space in between palindromes. Comparing two sequences of nucleotides. Similarities that happen just by chance. Random walks (and their use in testing for similarities). Sampling distribution of the sample mean and its use in confidence interval estimation and hypotheses testing. Approximated distribution (normal) of the sample proportion and its use in confidence interval estimation and hypotheses testing. Necessary sample size calculation in the case of estimation based on desired precision and confidence and the case of testing hypothesis based on the desired power. The t-student distribution and its application to inference for the sample mean.

*
“Nothing in Biology makes sense, except in the light of Evolution”
* by Theodosius Dobzhansky is the quote that sums up the importance of Evolution to Biology. The genetic basis of Evolution has been described and the applications of these principles to examples are covered, i.e. applications of probability and statistics to populations. Evolution as it relates to population size and density. The Wright-fisher model with the Hardy-Weinberg equations as a special case: rigorous development of the limit concept; continuity; discrete dynamical systems; effect of sample size in the Chi-square test; an introduction to graphs and their use in genetics.