Math for Biology - An Introduction Terri A. Grosso Outline Math for Biology - An Introduction Differential Equations - An Overview The Law of Terri A. Grosso Mass Action Enzyme Kinetics CMACS Workshop 2012 January 6, 2011 Terri A. Grosso Math for Biology - An Introduction Math for Biology - An Introduction Terri A. Grosso Outline 1 Differential Equations - An Overview Differential Equations - An Overview The Law of 2 The Law of Mass Action Mass Action Enzyme Kinetics 3 Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - Our Goal Math for Biology - An Introduction Terri A. Grosso Outline We will NOT be solving differential equations Differential Equations - An Overview The tools - Rule Bender and BioNetGen - will do that for The Law of us Mass Action This lecture is designed to give some background about Enzyme Kinetics what the programs are doing Terri A. Grosso Math for Biology - An Introduction Differential Equations - An Overview Math for Biology - An Introduction Terri A. Grosso Differential Equations contain the derivatives of (possibly) unknown functions. Outline Differential Represent how a function is changing. Equations - An Overview We work with first-order differential equations - only The Law of Mass Action include first derivatives Enzyme Generally real-world differential equations are not directly Kinetics solvable. Often we use numerical approximations to get an idea of the unknown function's shape. Terri A. Grosso Math for Biology - An Introduction A few solutions. Figure: Some solutions to f 0(x) = 2 Differential Equations - Starting from the solution Math for 0 Biology - An A differential equation: f (x) = C Introduction Terri A. Grosso Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - Starting from the solution Math for 0 Biology - An A differential equation: f (x) = C Introduction A few solutions. Terri A. Grosso Figure: Some solutions to f 0(x) = 2 Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction A few solutions. Figure: Some solutions to f 0(x) = 2x Differential Equations - Starting from the solution Math for 0 Biology - An A differential equation: f (x) = Cx Introduction Terri A. Grosso Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - Starting from the solution Math for 0 Biology - An A differential equation: f (x) = Cx Introduction A few solutions. Terri A. Grosso Figure: Some solutions to f 0(x) = 2x Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction 0 Exercise: Given f (x) = 2x and (x0; f (x0)) = (4; 22), what is the solution? f (x) = x2 + 6. Differential Equations - Initial Conditions Math for Biology - An Introduction Terri A. Grosso How do we know which is the correct solution? Outline Differential Need to know the value for a point - the initial conditions. Equations - An Overview Only one necessary for these types of problems. Need an The Law of initial condition for each variable in the equation. Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction f (x) = x2 + 6. Differential Equations - Initial Conditions Math for Biology - An Introduction Terri A. Grosso How do we know which is the correct solution? Outline Differential Need to know the value for a point - the initial conditions. Equations - An Overview Only one necessary for these types of problems. Need an The Law of initial condition for each variable in the equation. Mass Action 0 Enzyme Exercise: Given f (x) = 2x and (x0; f (x0)) = (4; 22), what Kinetics is the solution? Terri A. Grosso Math for Biology - An Introduction Differential Equations - Initial Conditions Math for Biology - An Introduction Terri A. Grosso How do we know which is the correct solution? Outline Differential Need to know the value for a point - the initial conditions. Equations - An Overview Only one necessary for these types of problems. Need an The Law of initial condition for each variable in the equation. Mass Action 0 Enzyme Exercise: Given f (x) = 2x and (x0; f (x0)) = (4; 22), what Kinetics is the solution? f (x) = x2 + 6. Terri A. Grosso Math for Biology - An Introduction Differential Equations - A Slightly More Complex Example Math for Biology - An Introduction Terri A. Grosso The Logistic Curve Outline Models population growth Differential Equations - Differential equation: An Overview d The Law of P(t) = P(t)(1 − P(t)) Mass Action dt Enzyme When does P(t) not change? In other words, when is the Kinetics derivative equal to 0? Under what conditions is the derivative positive? Negative? Terri A. Grosso Math for Biology - An Introduction P(0) = :5 1 P(t) = 1 + e−t Differential Equations - A Slightly More Complex Example Math for Biology - An Introduction The Logistic Curve - Solution Terri A. What more do we need before we find a solution? Grosso Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction 1 P(t) = 1 + e−t Differential Equations - A Slightly More Complex Example Math for Biology - An Introduction The Logistic Curve - Solution Terri A. What more do we need before we find a solution? Grosso P(0) = :5 Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - A Slightly More Complex Example Math for Biology - An Introduction The Logistic Curve - Solution Terri A. What more do we need before we find a solution? Grosso P(0) = :5 Outline 1 Differential P(t) = Equations - −t An Overview 1 + e The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - How about this one? Math for Biology - An Introduction Terri A. Grosso Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Biochemical Reactions - An Application of Differential Equations Math for Biology - An Introduction Terri A. Grosso Outline How can we represent the concentrations of molecules in Differential solution? Equations - An Overview We can represent how much the concentrations change The Law of Mass Action over time as differential equations. Enzyme A set of differential equations that closely describe how a Kinetics system develops is a model of the system. Terri A. Grosso Math for Biology - An Introduction Biochemical Reactions - Terminology Review Math for Biology - An Introduction Terri A. Grosso Chemical Reaction A process that changes a set of Outline chemical species into another Differential Reactants The initial set of chemical species Equations - An Overview Products The new set of chemical species The Law of Mass Action A basic synthesis reaction A + B ! C Enzyme Kinetics An equilibrium reaction A + B )−*− C Conservation of Mass The mass of the products has to equal that of the reactants (in a closed system) Terri A. Grosso Math for Biology - An Introduction Biochemical Reactions - Some Basic Questions Math for Biology - An Introduction Terri A. Grosso Outline How quickly does a biochemical reaction take place? Differential Equations - An Overview How will different concentrations of the reactants affect The Law of the reaction rate? Mass Action What will be the concentrations of the reactants and Enzyme Kinetics products at equilibrium? Terri A. Grosso Math for Biology - An Introduction The Law of Mass Action Math for Biology - An Introduction Terri A. Describes the rate at which chemicals collide and form Grosso new compounds Outline It's a model that describes molecular interactions Differential Equations - Example: A + B ! C An Overview The Law of Concentration is represented as [A], [B] and [C]. Mass Action The rate can be expressed as the change in the amount of Enzyme d[C] Kinetics compound C: dt This rate is determined by the number of collisions between A and B and the probability that a collision will lead to the combination of the molecules. Terri A. Grosso Math for Biology - An Introduction The Law of Mass Action Math for Biology - An Introduction Terri A. Grosso d[C] Outline = k[A][B] Differential dt Equations - An Overview Called the Law of Mass Action The Law of k is the rate constant. Takes into account shapes, Mass Action Enzyme attraction and temperature. Kinetics k is different for every reaction. Terri A. Grosso Math for Biology - An Introduction Equilibrium Constant Math for Biology - An Introduction k+ Terri A. −* Grosso A + B )− C k− Outline A is consumed by forward reaction and produced by the Differential reverse reaction, so Equations - An Overview d[A] The Law of = k−[C] − k+[A][B] Mass Action dt At equilibrium, the reactions cancel each other out and Enzyme Kinetics k− [A]eq[B]eq ≡ Keq = k+ [C]eq Exercise: Show that this equation follows from the previous one Terri A. Grosso Math for Biology - An Introduction Equilibrium Constant - Exercise Math for Biology - An Introduction Terri A. Grosso k− [A]eq[B]eq Outline ≡ Keq = Differential k+ [C]eq Equations - An Overview What is the relationship between the equilibrium The Law of concentrations of A, B and C if Keq is greater than 1? Mass Action Less than 1? Enzyme Kinetics Almost equal to 1? Terri A. Grosso Math for Biology - An Introduction Enzyme Basics Math for Biology - An Introduction Terri A. Grosso Enzymes help to convert substrates into products Outline Differential Catalysts - affect the rate of the reaction but are not Equations - An Overview changed by it The Law of Speed up biological reactions by up to 10 million times Mass Action Enzyme Very specific - usually one enzyme catalyzes one reaction Kinetics Regulated by feedback loops - like those found in signalling pathways Terri A.