Introduction to Mathematical BiologyMATH CS 120, Professor: Paul J. Atzberger

• Advances in experimental techniques are yielding an unprecedented amount of quantitative data for biological systems.

• Mathematical approaches are playing an important role in the modeling and analysis of biological systems.

• Subset of topics to be discussed in the class:– Population Modeling, Infectious Disease Modeling– Introduction to Molecular Biology, Gene Regulatory Networks – Microorganism Motility and Molecular Motor Proteins– Neuronal Signaling : Action Potentials – Developmental Biology : Pattern Formation

Fibonacci’s Rabbits • Historical Background:

• Fibonacci (Leonardo of Pisa) was born in 1170 AD.

• Educated in North Africa (father was a diplomat)

• Published liber abaci in 1202 introducing Arab-Hindu numerals into general use in Europe. (Rules of Arithmetic with Decimal Numbers)

• Posed the following problem:

• Suppose rabbits multiply in pairs every month, requiring at least one year of maturing before being able to reproduce.

• How many pairs of rabbits will there be after one year?

• For simplicity he assumed the rabbits were protected from predators and bad weather so that none of them ever died.

Fibonacci’s Rabbits Answer:

• Let N_k = number of pairs of rabbits at month k.

• At the beginning we have N_0 = 1 (mature pair).

• After one month the pair produces one pair of rabbits N_1 = N_0 + 1 = 2.

• After one additional month there is now only N_0 pairs which are mature which produce N_0 new rabbits giving N_2 = N_1 + N_0 = 3.

• Now after an additional month there are now N_1 pairs which are mature which produce N_1 new rabbits giving N_3 = N_2 + N_1 = 5.

• The pattern in general is that the number of rabbits in month i is given by N_i = N_{i – 1} + N_{i – 2}.

• This is the Fibonacci series!

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

• After one year there are 377 pairs of rabbits.

Preditor-Prey Populations

• What effect would predators have on the population of rabbits?

• Above is data for actual population of Canadian lynx and snowshoe hares.

• The populations of the lynx and hares appear to have fairly regular oscillations over time which are slightly out of phase.

• Spikes in the hare population are followed by increases in the lynx population and vice versa (see 1915 and 1925).

• How might this be explained?

• How might a quantitative mathematical model be formulated for these populations? Can models reproduce the observed oscillations?

Locust Populations

• Brown Locusts of South Africa are found to exhibit a 17 year cycle.

• The size of the population is found to correlate with climate observations.

• Locusts cause major damage to crops.• Prediction of population and size is an interesting

problem.• Prediction is important to mobilizing resources to

respond to protect crops.• Models of the population are important for

understanding effective pest control programs and their ecological impact.

Ecological Systems

• We see that even relatively simple systems consisting of even only two populations can exhibit surprising counter-intuitive behaviors.

• How might we try to understand more complicated populations with many different interacting species?

• How might we understand the interrelationships between species populations, geographical land features, climate, and human populations?

• Relying on intuition alone will probably not be sufficient to understand these issues.

Ecological Systems

• Technological advances such as geographic information systems, animal tagging, and shared ecological data bases are providing unprecedented information for this purpose.

• Mathematical modeling, analysis, and computational simulations are playing an important role in deepening our understanding of these interrelationships.

• Findings from this analysis could have important impact on public policies: such as hunting and fishing regulations, environmental pollution legislation, green house gas emissions control.

Human Populations

• What is the maximum stability sustainable population for a geographic region given food supply, water supply, and other natural and economic resources?

• How do infectious disease spread in a population and what policies might work best in controlling an outbreak?

• Mathematical approaches can help in answer these questions and analyzing possible scenarios.

Worldwide Human Populations

• dd

6.6 billion people worldwide (estimate as of September 2007).

Human Populations

Early Modeling of Human PopulationsThomas Malthus

• Thomas Malthus (1766-1834) was a political economist and cleric in England.

• He posited that human populations ought to grow by some proportion over a fixed period of time in the absence of environmental or social constraints (geometric growth).

• He stated that human populations double about every 25 years.

• He also posited that agricultural production grows only as a proportion of time (arithmetic growth).

Early Modeling of Human PopulationsThomas Malthus

•This led him to the conclusion that if populations were unchecked this would lead inevitably to severe shortages of food and horrible social problems.

• His predictions did not in the end hold for the populations of Europe (greater gains in the capacity to food production occurred through industrialization and agricultural science to keep pace with population).

• However, his ideas were very influential in Charles Darwin’s conception of the principle of Natural Selection and evolution.

Human Populations and Disease

HIV Virus Salmonella Typhimurium Bacterial Infection

• How are diseases spread in a population? By what mechanisms (airborne, physical contact, food & water supply)? Are social factors important?

• What features should be taken into account? Is only a subset of the population susceptible? What is the outcome of having the disease (death, being infectious, immunity after recovery, etc…)?

• What key factors might be important in controlling the disease and helpful in setting public policies?

• Mathematical approaches have played an important role in gaining insights into these issues.

Mechanisms of Disease Infection (Cellular Level)

HIV Virus

• More microscopically, we can consider diseases at the level of individual cells.

• How do diseases operate at the level of individual cells?

• What pathways of the cell might we design compounds to impact to help disrupt the virus life-cycle?

• This requires insights into the biochemistry of cells and the general mechanisms by with cellular processes are regulated.

Cell Regulation

• Many biochemical experiments have been performed and yield information about how various proteins interact. What kinds of overall behaviors can arise from networks of interacting proteins?

• Given this information can be determine key mechanisms by which the cell processes information about its internal state and environment?

• Mathematical modeling and analysis is playing a key role in determining generic features of biochemical systems which allow for networks to have behaviors similar, for example, to switches, toggles, and clocks.

Electrical Signaling in Neural Tissues

cortical neuronsneuron

dendritic tree

human brain

• How are signals propagated in neuronal tissues?

• How do large networks of interacting neurons process information?

• For instance, how does our nervous system first encode into an electrical signal stimuli and then process this information? sounds, images, smells?

• The giant squid axon is one model system used to study the first question about the basic mechanism of signal propagation.

Action Potentials

• Electrical signals are received from the cell dendrites.

• These signals are processed by the nerve cell and a “decision” is made whether or not to fire an action potential down the axon.

• The axon carries the electrical signal to the synaptic knobs were neurotransmitters are released.

• Neurotransmitters trigger local electrical signals down the dendrites of neighboring nerve cells again starting the cycle.

Giant Squid (Logilo Pealeii)

• Can grow 60 to 70 feet long.

• Predatory (thought to attack whales and other large sea creatures).

• Very powerful suckers on the tentacles are used to grab prey.

• Historic accounts of squid attacking ships.

Giant Squid

Imagined Predatory Giant Squid Attacking a Whale

Giant Squid Axons

Nerve Axons can have diameter up to 1mm (convenient for making measurements)

Action Potentials

• Basic mechanism involves currents of ions: Na, Ca, K, Cl transverse to the axon which opens neighboring ion channels.

• This cascade along the axon propagate these transverse currents away from the cell body (soma).

• Hodgkin-Huxley Model.

Belousov-Zhabotinsky Reaction

Pattern Formation

brussalator

• Biochemical systems are found to exhibit intriguing patterns (sometime oscillating in time)

• How do these patterns arise from initially homogenous mixtures of chemical species?.

• How might biological systems use such chemically generated patterns?

Pattern Formation

• Pattern forming chemical reactions and related mechanisms are thought to play an important role in developmental biology.

• We shall discuss models of pattern formulation through the mechanisms of “Turing Instabilities.”

Conclusions

• Biological systems present many interesting and challenging problems.

• In this class we shall address the following questions:

– How can mathematical approaches be used to model biological systems?

– How can mathematical analysis be used to identify generic mechanisms by which biological system potentially operate?

– How can mathematical analysis and computational simulation be used to explain quantitatively/qualitatively experimental observations?