Math 5198 Mathematics for Bioscientists
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1 Math 5198 Mathematics for Bioscientists Lecture 1: Course Conduct/Overview Stephen Billups University of Colorado at Denver Math 5198Mathematics for Bioscientists p.1/22
2 Housekeeping Syllabus CCB MERC Lab (Science 130) for DERIVE and MATLAB Homework 1 Read Sections Problems (Due Tuesday, Aug. 31, 2004 at the beginning of class) Sec 1.1: 21,22,23,24,25,26 Sec 1.2: 1,3,4,6,8,10, 13-18, 20,24,25,26,30,36,38 Sec 1.3: 2,4,6,12,16,20,22,30,34,44,53 Math 5198Mathematics for Bioscientists p.2/22
3 Introduction Three main components of the course: Differential Equations Linear Algebra Graph Theory If time permits, we may also look at optimization. Math 5198Mathematics for Bioscientists p.3/22
4 Ex. 1: Pharmacokinetics Application of Differential Equations To decide the right dosing of a drug, we need to know what the concentration of the drug in the bloodstream will be at any given time. The concentration of a drug in the body decreases at a rate proportional to the concentration. c (t) = kc(t). Question: What would you expect to be true about k? Answer: k should be negative for concentration to decrease. Math 5198Mathematics for Bioscientists p.4/22
5 Ex. 1, continued (from Math 5198Mathematics for Bioscientists p.5/22
6 Ex. 2: Stochiometric Equations Application of Linear Algebra Reactions must satisfy stochiometric equations 2NO 2 + F 2 2NO 2 F Biological systems involve many such reactions, resulting in large systems of linear equations. Math 5198Mathematics for Bioscientists p.6/22
7 Example 3: Metabolic Networks Application of Graph Theory (from KEGG database: Math 5198Mathematics for Bioscientists p.7/22
8 Course Goals: Become comfortable with mathematical modeling as a tool for understanding biological systems. Learn fundamental mathematical tools, notation, and terminology that you are likely to encounter in the new era of biology. Prepare for advanced study in computational biology. Math 5198Mathematics for Bioscientists p.8/22
9 Motivation Biology is becoming a computational science. Human Genome Project High Throughput Technologies Mathematical and computational tools are needed to analyze massive amounts of data. Systems Biology: Advances in 21st century will involve building mathematical models of biological systems and experimenting in silico. Math 5198Mathematics for Bioscientists p.9/22
10 Introduction to Differential Equations Biological systems are dynamic they are constantly moving and changing 1. Populations grow and shrink 2. Hormone levels fluctuate 3. Neurons fire 4. Blood flows Differential equations are the main mathematical tool for understanding how systems change. Math 5198Mathematics for Bioscientists p.10/22
11 What is a Differential Equation? An equation relating functions to their derivatives: Examples: dy dx = y + x d 2 x dt 2 + sin x = 0 y 2y + 5y + y = e x u u x + u t = 0 uu x + u t = 0. Math 5198Mathematics for Bioscientists p.11/22
12 Elements of Differential Equations independent variables dependent variables derivatives of dependent variables Math 5198Mathematics for Bioscientists p.12/22
13 Types of Differential Equations Ordinary vs. Partial Ordinary: contains only ordinary derivatives (e.g. dy dx ). Has only one independent variable. Partial: contains partial derivatives, (e.g. u ). Has > 1 t independent variables. Order: order of highest derivative. Linear vs. nonlinear A linear ordinary diff. eq. has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). Math 5198Mathematics for Bioscientists p.13/22
14 xercise: Classify these Differential Equatio dy dx = y + x d 2 x dt + sin x = 0 y 2y + 5y + y = e x u u x + u t = 0 uu x + u t = 0. Math 5198Mathematics for Bioscientists p.14/22
15 General Form of ODE We can write an Ordinary Differential Equation (ODE) in the form F (x, y, y,..., y (n) ) = 0. Example: dy = 3ydx = dy dx 3y = 0. Math 5198Mathematics for Bioscientists p.15/22
16 Solutions A solution to an nth order ODE F (x, y,..., y (n) ) = 0 on the interval a < x < b is any function φ(x) satisfying on a < x < b. Example 1: Check: y (x) = 2e 2x = 2y. F (x, φ, φ,..., φ (n) ) = 0 dy dx = 2y = y(x) = e2x. Question: What is the interval? Math 5198Mathematics for Bioscientists p.16/22
17 Solutions, (cont) Example 2: d 2 u + 16u = 0 = u(x) = sin 4x. dx2 Check: u (x) = 4 cos 4x, u (x) = 16 sin 4x, so d 2 u + 16u = 16 sin 4x + 16 sin 4x = 0. dx2 Math 5198Mathematics for Bioscientists p.17/22
18 Exercise Is y = 4 x 2 a solution to the differential equation dy dx = x y? Math 5198Mathematics for Bioscientists p.18/22
19 Solution Plug y = 4 x 2 into the ODE: dy dx = 1 2 x ( 2x) = 4 x2 y. Math 5198Mathematics for Bioscientists p.19/22
20 But Wait! The solution only makes sense for 2 < x < 2 Why? Answer: 4 x 2 is undefined for x < 2 and x > 2, and its derivative is undefined for x = ±2. Math 5198Mathematics for Bioscientists p.20/22
21 Comment In general, ODEs have solutions only for open intervals. We must be careful that any solution we compute makes sense everywhere we care about. Math 5198Mathematics for Bioscientists p.21/22
22 Homework Read Sections Problems (Due Tuesday, Aug. 31, 2004 at the beginning of class) Sec 1.1: 21,22,23,24,25,26 Sec 1.2: 1,3,4,6,8,10, 13-18, 20,24,25,26,30,36,38 Sec 1.3: 2,4,6,12,16,20,22,30,34,44,53 Math 5198Mathematics for Bioscientists p.22/22
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