Math for Biology - An Introduction Terri A. Grosso Math for Biology - An Introduction Outline Differential Equations - An Overview Terri A. Grosso The Law of Mass Action Enzyme CMACS Workshop 2012 Kinetics January 6, 2011 Terri A. Grosso Math for Biology - An Introduction
Math for Biology - An Introduction Terri A. Grosso 1 Differential Equations - An Overview Outline Differential Equations - An Overview 2 The Law of Mass Action The Law of 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 The tools - Rule Bender and BioNetGen - will do that for Equations - An Overview us The Law of Mass Action This lecture is designed to give some background about Enzyme what the programs are doing Kinetics Terri A. Grosso Math for Biology - An Introduction
Differential Equations - An Overview Math for Biology - An Introduction Terri A. Differential Equations contain the derivatives of (possibly) Grosso unknown functions. Outline Represent how a function is changing. Differential Equations - We work with first-order differential equations - only An Overview The Law of include first derivatives Mass Action Generally real-world differential equations are not directly Enzyme 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
Differential Equations - Starting from the solution Math for A differential equation: f ′ ( x ) = C 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
Differential Equations - Starting from the solution Math for A differential equation: f ′ ( x ) = C Biology - An Introduction A few solutions. Terri A. Grosso Figure: Some solutions to f ′ ( x ) = 2 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 A differential equation: f ′ ( x ) = Cx 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
Differential Equations - Starting from the solution Math for A differential equation: f ′ ( x ) = Cx Biology - An Introduction A few solutions. Terri A. Grosso Figure: Some solutions to f ′ ( x ) = 2 x Outline Differential Equations - An Overview The Law of Mass Action Enzyme Kinetics 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 Need to know the value for a point - the initial conditions. Differential Equations - Only one necessary for these types of problems. Need an An Overview initial condition for each variable in the equation. The Law of Mass Action Enzyme Kinetics 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 Need to know the value for a point - the initial conditions. Differential Equations - Only one necessary for these types of problems. Need an An Overview initial condition for each variable in the equation. The Law of Mass Action Exercise: Given f ′ ( x ) = 2 x and ( x 0 , f ( x 0 )) = (4 , 22), what Enzyme 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 Need to know the value for a point - the initial conditions. Differential Equations - Only one necessary for these types of problems. Need an An Overview initial condition for each variable in the equation. The Law of Mass Action Exercise: Given f ′ ( x ) = 2 x and ( x 0 , f ( x 0 )) = (4 , 22), what Enzyme Kinetics is the solution? f ( x ) = x 2 + 6. Terri A. Grosso Math for Biology - An Introduction
Differential Equations - A Slightly More Complex Example Math for Biology - An Introduction Terri A. The Logistic Curve Grosso Models population growth Outline Differential Differential equation: Equations - An Overview d dt P ( t ) = P ( t )(1 − P ( t )) The Law of Mass Action When does P ( t ) not change? In other words, when is the Enzyme Kinetics derivative equal to 0? Under what conditions is the derivative positive? Negative? Terri A. Grosso Math for Biology - An Introduction
Differential Equations - A Slightly More Complex Example Math for Biology - An The Logistic Curve - Solution Introduction What more do we need before we find a solution? 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 - A Slightly More Complex Example Math for Biology - An The Logistic Curve - Solution Introduction What more do we need before we find a solution? Terri A. 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 The Logistic Curve - Solution Introduction What more do we need before we find a solution? Terri A. Grosso P (0) = . 5 Outline 1 P ( t ) = Differential 1 + e − t Equations - An Overview 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 How can we represent the concentrations of molecules in Outline solution? Differential Equations - An Overview We can represent how much the concentrations change The Law of over time as differential equations. Mass Action A set of differential equations that closely describe how a Enzyme 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. Chemical Reaction A process that changes a set of Grosso chemical species into another Outline Reactants The initial set of chemical species Differential Equations - Products The new set of chemical species An Overview The Law of A basic synthesis reaction A + B → C Mass Action Enzyme An equilibrium reaction A + B − ⇀ − C Kinetics ↽ 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 How will different concentrations of the reactants affect Equations - An Overview the reaction rate? The Law of Mass Action What will be the concentrations of the reactants and Enzyme products at equilibrium? Kinetics Terri A. Grosso Math for Biology - An Introduction
The Law of Mass Action Math for Biology - An Introduction Describes the rate at which chemicals collide and form Terri A. Grosso new compounds Outline It’s a model that describes molecular interactions Differential Example: A + B → C Equations - An Overview Concentration is represented as [ A ], [ B ] and [ C ]. The Law of Mass Action The rate can be expressed as the change in the amount of Enzyme compound C: d [ C ] Kinetics 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 ] dt Differential Equations - Called the Law of Mass Action An Overview k is the rate constant. Takes into account shapes, The Law of Mass Action attraction and temperature. Enzyme 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. A + B − ⇀ C ↽ − Grosso k − A is consumed by forward reaction and produced by the Outline reverse reaction, so Differential Equations - d [ A ] An Overview = k − [ C ] − k + [ A ][ B ] The Law of dt Mass Action At equilibrium, the reactions cancel each other out and Enzyme ≡ K eq = [ A ] eq [ B ] eq Kinetics k − [ C ] eq k + Exercise: Show that this equation follows from the previous one Terri A. Grosso Math for Biology - An Introduction
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