Math For Biology - An Introduction

Math forBiology - AnIntroductionTerri A.GrossoOutlineMath for Biology - An IntroductionDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoCMACS Workshop 2012January 6, 2011Terri A. GrossoMath for Biology - An Introduction

Math forBiology - AnIntroductionTerri A.GrossoOutline1 Differential Equations - An OverviewDifferentialEquations An OverviewThe Law ofMass Action2 The Law of Mass ActionEnzymeKinetics3 Enzyme KineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Our GoalMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsWe will NOT be solving differential equationsThe tools - Rule Bender and BioNetGen - will do that forusThis lecture is designed to give some background aboutwhat the programs are doingTerri A. GrossoMath for Biology - An Introduction

Differential Equations - An OverviewMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsDifferential Equations contain the derivatives of (possibly)unknown functions.Represent how a function is changing.We work with first-order differential equations - onlyinclude first derivativesGenerally real-world differential equations are not directlysolvable.Often we use numerical approximations to get an idea ofthe unknown function’s shape.Terri A. GrossoMath for Biology - An Introduction

Differential Equations - Starting from the solutionMath forBiology - AnIntroductionA differential equation: f 0 (x) CTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Starting from the solutionMath forBiology - AnIntroductionTerri A.GrossoA differential equation: f 0 (x) CA few solutions.Figure: Some solutions to f 0 (x) 2OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Starting from the solutionMath forBiology - AnIntroductionA differential equation: f 0 (x) CxTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Starting from the solutionMath forBiology - AnIntroductionTerri A.GrossoA differential equation: f 0 (x) CxA few solutions.Figure: Some solutions to f 0 (x) 2xOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Initial ConditionsMath forBiology - AnIntroductionTerri A.GrossoHow do we know which is the correct solution?OutlineDifferentialEquations An OverviewThe Law ofMass ActionNeed to know the value for a point - the initial conditions.Only one necessary for these types of problems. Need aninitial condition for each variable in the equation.EnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - Initial ConditionsMath forBiology - AnIntroductionTerri A.GrossoHow do we know which is the correct solution?OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsNeed to know the value for a point - the initial conditions.Only one necessary for these types of problems. Need aninitial condition for each variable in the equation.Exercise: Given f 0 (x) 2x and (x0 , f (x0 )) (4, 22), whatis the solution?Terri A. GrossoMath for Biology - An Introduction

Differential Equations - Initial ConditionsMath forBiology - AnIntroductionTerri A.GrossoHow do we know which is the correct solution?OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsNeed to know the value for a point - the initial conditions.Only one necessary for these types of problems. Need aninitial condition for each variable in the equation.Exercise: Given f 0 (x) 2x and (x0 , f (x0 )) (4, 22), whatis the solution?f (x) x 2 6.Terri A. GrossoMath for Biology - An Introduction

Differential Equations - A Slightly More ComplexExampleMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsThe Logistic CurveModels population growthDifferential equation:dP(t) P(t)(1 P(t))dtWhen does P(t) not change? In other words, when is thederivative equal to 0?Under what conditions is the derivative positive?Negative?Terri A. GrossoMath for Biology - An Introduction

Differential Equations - A Slightly More ComplexExampleMath forBiology - AnIntroductionTerri A.GrossoThe Logistic Curve - SolutionWhat more do we need before we find a solution?OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - A Slightly More ComplexExampleMath forBiology - AnIntroductionTerri A.GrossoOutlineThe Logistic Curve - SolutionWhat more do we need before we find a solution?P(0) .5DifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - A Slightly More ComplexExampleMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Logistic Curve - SolutionWhat more do we need before we find a solution?P(0) .5P(t) 11 e tThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Differential Equations - How about this one?Math forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Biochemical Reactions - An Application ofDifferential EquationsMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsHow can we represent the concentrations of molecules insolution?We can represent how much the concentrations changeover time as differential equations.A set of differential equations that closely describe how asystem develops is a model of the system.Terri A. GrossoMath for Biology - An Introduction

Biochemical Reactions - Terminology ReviewMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsChemical Reaction A process that changes a set ofchemical species into anotherReactants The initial set of chemical speciesProducts The new set of chemical speciesA basic synthesis reaction A B CAn equilibrium reaction A B *) CConservation of Mass The mass of the products has toequal that of the reactants (in a closed system)Terri A. GrossoMath for Biology - An Introduction

Biochemical Reactions - Some Basic QuestionsMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsHow quickly does a biochemical reaction take place?How will different concentrations of the reactants affectthe reaction rate?What will be the concentrations of the reactants andproducts at equilibrium?Terri A. GrossoMath for Biology - An Introduction

The Law of Mass ActionMath forBiology - AnIntroductionTerri A.GrossoDescribes the rate at which chemicals collide and formnew compoundsOutlineIt’s a model that describes molecular interactionsDifferentialEquations An OverviewExample: A B CThe Law ofMass ActionEnzymeKineticsConcentration is represented as [A], [B] and [C ].The rate can be expressed as the change in the amount ofd[C ]compound C:dtThis rate is determined by the number of collisionsbetween A and B and the probability that a collision willlead to the combination of the molecules.Terri A. GrossoMath for Biology - An Introduction

The Law of Mass ActionMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsd[C ] k[A][B]dtCalled the Law of Mass Actionk is the rate constant. Takes into account shapes,attraction and temperature.k is different for every reaction.Terri A. GrossoMath for Biology - An Introduction

Equilibrium ConstantMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsk A B *) Ck A is consumed by forward reaction and produced by thereverse reaction, sod[A] k [C ] k [A][B]dtAt equilibrium, the reactions cancel each other out and[A]eq [B]eqk Keq k [C ]eqExercise: Show that this equation follows from theprevious oneTerri A. GrossoMath for Biology - An Introduction

Equilibrium Constant - ExerciseMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKinetics[A]eq [B]eqk Keq k [C ]eqWhat is the relationship between the equilibriumconcentrations of A, B and C if Keq is greater than 1?Less than 1?Almost equal to 1?Terri A. GrossoMath for Biology - An Introduction

Enzyme BasicsMath forBiology - AnIntroductionTerri A.GrossoEnzymes help to convert substrates into productsOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsCatalysts - affect the rate of the reaction but are notchanged by itSpeed up biological reactions by up to 10 million timesVery specific - usually one enzyme catalyzes one reactionRegulated by feedback loops - like those found insignalling pathwaysTerri A. GrossoMath for Biology - An Introduction

How Enzymes Work - An exampleMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Enzyme Kinetics - A Law BreakerMath forBiology - AnIntroductionTerri A.GrossoAssume a model of an enzyme catalyzed reaction:OutlineDifferentialEquations An OverviewThe Law ofMass ActionS E P EIf we increase the concentration of the substrate, whathappens to the reaction rate?EnzymeKineticsTerri A. GrossoMath for Biology - An Introduction

Enzyme Kinetics - A Law BreakerMath forBiology - AnIntroductionTerri A.GrossoAssume a model of an enzyme catalyzed reaction:OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsS E P EIf we increase the concentration of the substrate, whathappens to the reaction rate?Should go up linearlyTerri A. GrossoMath for Biology - An Introduction

Enzyme Kinetics - A Law BreakerMath forBiology - AnIntroductionTerri A.GrossoAssume a model of an enzyme catalyzed reaction:OutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsS E P EIf we increase the concentration of the substrate, whathappens to the reaction rate?Should go up linearlyThat’s not what happensThe rate only increases to a maximum valueTerri A. GrossoMath for Biology - An Introduction

Enzyme Kinetics - A Better ModelMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsk1k2k 1k 2S E *) C *) P ESubstrate combines with Enzyme to form ComplexComplex breaks down into Product and EnzymeBut the Product is mostly removed, so that reversereaction doesn’t really occurCan assume that reaction doesn’t happen. Theconventional form:k1S E *) Ck2 k 1P ECalled the Michaelis-Menten Model of enzyme kineticsTerri A. GrossoMath for Biology - An Introduction

Enzyme Kinetics - Rates of ChangeMath forBiology - AnIntroductionTerri A.GrossoOutlineDifferentialEquations An OverviewThe Law ofMass ActionEnzymeKineticsk1S E *) Ck2 k 1P EFor ease of writing, let s [S], c [C], e [E], andp [P].Using Law of Mass Action, can write four differentialequations:dsdtdedtdcdtdpdt k 1 c k1 se (k 1 k2 )c k1 se k1 se (k2 k 1 )c k2 cTerri A. GrossoMath for Biology - An Introduction