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. Grosso Math for Biology - An Introduction How Enzymes Work - An example
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 Should go up linearly That’s not what happens The rate only increases to a maximum value
Enzyme Kinetics - A Law Breaker
Math for Biology - An Introduction
Terri A. Grosso Assume a model of an enzyme catalyzed reaction: Outline S + E → P + E Differential Equations - An Overview If we increase the concentration of the substrate, what
The Law of happens to the reaction rate? Mass Action
Enzyme Kinetics
Terri A. Grosso Math for Biology - An Introduction That’s not what happens The rate only increases to a maximum value
Enzyme Kinetics - A Law Breaker
Math for Biology - An Introduction
Terri A. Grosso Assume a model of an enzyme catalyzed reaction: Outline S + E → P + E Differential Equations - An Overview If we increase the concentration of the substrate, what
The Law of happens to the reaction rate? Mass Action
Enzyme Should go up linearly Kinetics
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - A Law Breaker
Math for Biology - An Introduction
Terri A. Grosso Assume a model of an enzyme catalyzed reaction: Outline S + E → P + E Differential Equations - An Overview If we increase the concentration of the substrate, what
The Law of happens to the reaction rate? Mass Action
Enzyme Should go up linearly Kinetics That’s not what happens The rate only increases to a maximum value
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - A Better Model
Math for Biology - An Introduction k1 k2 Terri A. S + E )−*− C )−*− P + E Grosso k−1 k−2
Outline Substrate combines with Enzyme to form Complex Differential Complex breaks down into Product and Enzyme Equations - An Overview But the Product is mostly removed, so that reverse The Law of Mass Action reaction doesn’t really occur Enzyme Can assume that reaction doesn’t happen. The Kinetics conventional form: k1 k S + E )−*− C −→2 P + E k−1 Called the Michaelis-Menten Model of enzyme kinetics
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - Rates of Change
Math for Biology - An k1 k Introduction S + E )−*− C −→2 P + E Terri A. k−1 Grosso For ease of writing, let s = [S], c = [C], e = [E], and Outline p = [P]. Differential Equations - Using Law of Mass Action, can write four differential An Overview equations: The Law of Mass Action ds = k−1c − k1se Enzyme dt Kinetics de = (k + k )c − k se dt −1 2 1 dc = k se − (k + k )c dt 1 2 −1 dp = k c dt 2
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - Michaelis-Menten Equation
Math for Biology - An Introduction Given the differential equations and some assumptions, it Terri A. Grosso is possible to approximate the rate of product formation Definitions: Outline
Differential v the rate at which the product is formed Equations - k2 the rate constant for dissociation of the enzyme-product An Overview complex The Law of Mass Action [E]0 the enzyme concentration Enzyme [S] the substrate concentration Kinetics Km the Michaelis constant which measures the affinity of the substrate for the enzyme. The Michaelis-Menten equation: [S] v = k2[E]0 Km + [S]
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - Application to the Frog Cell Cycle
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 Enzyme Kinetics - Exercise 1
Math for Biology - An Introduction
Terri A. Grosso
Outline
Differential Equations - An Overview
The Law of Mass Action Enzyme Identify, substrate, enzyme and product Kinetics Ignoring ATP, write the forward (phosphorylating) reaction following the Michaelis-Menten model What is the differential equation for the change in concentration of Wee1? Wee1-P? Use the Michaelis-Menten reaction to write a formula for the rate of product formation.
Terri A. Grosso Math for Biology - An Introduction Enzyme Kinetics - Exercise 2
Math for Biology - An Introduction
Terri A. Grosso
Outline
Differential Equations - An Overview
The Law of Mass Action With the people near you, choose a reaction from the cycle Enzyme Kinetics Identify, substrate, enzyme and product Ignoring ATP, write the reaction following the Michaelis-Menten model Use the Michaelis-Menten reaction to write a formula for the rate of product formation. Be ready to present to the group
Terri A. Grosso Math for Biology - An Introduction