Mathematically Modeling Biochemistry

Transcription

1 Mathematically Modeling Biochemistry Nathan Addy Scientific Programmer October 23, 2008 Overview What are the systems we study? What approaches to do we use to study them? Why these approaches? How does calculus fit into this picture? 1

2 The Cell A container of complicated chemistry Cells move information around using Signal Transduction Pathways. Any time the body has to communicate with itself, it uses signal transduction. 2

3 Cells move information around. Any time the body has to communicate with itself, it uses signal transduction. Cells move information around. Any time the body has to communicate with itself, it uses signal transduction. Brain releases adrenaline into the blood 3

4 Cells move information around. Any time the body has to communicate with itself, it uses signal transduction. Different cells have receptors (locks) for adrenaline (the key) Cells move information around. Any time the body has to communicate with itself, it uses signal transduction. Some cells turn off Digestive cells start digesting Non-essential blood vessles constrict Some cells turn on Lungs use more energy to pick up more oxygen Essential blood vessles dialate Cells burn go into overdrive energy production 4

5 Specificity is based on cellular response to second messengers The alpha pathway is a specific signal transduction pathway in yeast. 5

6 Our goal is to understand this pathway Make predictions Given an input, can we predict will the output will be? Solve the System If we desire a particular output, can we determine what inputs to feed into the system? Control the system If a particular input causes some output, can we determine how to fix the system so that it causes some other output? To understand systems, we need models A model is an framework(abstraction) that represents our understanding of what is really going on. There are many types of models Conceptual models Mathematical models Other kinds of models 6

7 Conceptual Model Very Simple model for cholesterol levels in the body. Exercise Eating foods high in cholesterol Cholesterol levels Conceptual Model Exercise Eating foods high in cholesterol Can be very useful, but are limited in how well they help you understand the system. Cholesterol levels 7

8 Mathematical Model: Use language of math to describe the system L = Cholesterol Levels C = Calories burned while exercising E = Cholesterol intake a = rate of cholesterol removed per calorie burned b = rate of cholesteral absorbed per cholesterol eaten a >> b dl/dt = b * (de/dt) - a * (dc/dt) Mathematical Model L = Cholesterol Levels C = Calories burned while exercising E = Cholesterol intake a = rate of cholesterol removed per calorie burned b = rate of cholesteral absorbed per cholesterol eaten a >> b dl/dt = b * de/dt - a * dc/dt Why might a mathematical model be more useful than other types of models for understanding and controlling the system? 8

9 Mathematical Model L = Cholesterol Levels C = Calories burned while exercising E = Cholesterol intake a = rate of cholesterol removed per calorie burned b = rate of cholesteral absorbed per cholesterol eaten a >> b dl/dt = b * de/dt - a * dc/dt It is more detailed, reflecting better understanding Gives you things to measure (suggests experiments) We have lots of tools for evaluating and working with these models. Modeling biochemistry: enzymes 9

10 What are enzymes? An uphill chemical reaction. Enzymes are helper molecules + Enzymes make uphill chemical reactions go downhill. 10

11 Enzymes are helper molecules They catalyze chemical reactions Developing a mathematical model for what is happening here 11

12 Law of Mass-action For a chemical reaction A + B -> C, the rate at which This reaction occurs depends on a fundamental constant, a reaction rate k, as well as the amount of A and the amount of B present. D[C]/dt = k * [A] * [B] Developing a mathematical model for what is happening here What can we say about d[p]/dt? Assumptions: The rate of change of ES is much lower than the rates of change of S and of P -- this happens when E << S 12

13 So with these assumptions, the reaction Will proceed at a rate: These equations imply how to experimentally learn about this system: They tell us the important details: Vmax and Km. By doing experiments where we fix [E0] and vary [S], we can determine both K_m and [S]. 13

14 The mathematical tools we use aren t always calculus based. Models don t have to use the tools of calculus (continuous functions, deterministic) Many of the models we use are probabilistic. If individual reactions can be modeled using calculus, why use probability? Probability represents what is happening more accurately (the kinetics). Models can be represented more accurately (the variables). 14

15 Lots of tools in the mathematical modeler s toolbox Calculus thinking/reasoning is used all over the place. Calculus tools are used often but not everywhere. Conclusions Understanding biology can often be best done with mathematical models. Mathematical models support prediction, solving, and fixing to the highest degrees of accuracy. Specific tools of calculus are very useful in this process. Calculus thinking is absolutely essential. 15