Mathematical Modeling And Biology
MathematicalModeling andBiologyBo DengMathematical Modeling and BiologyIntroductionExamples ofModelsBo DengConsistencyModel TestMathematicalBiologyDepartment of MathematicsUniversity of Nebraska – LincolnConclusionMarch 10, 2016www.math.unl.edu/ bdeng11 / 24
What is modeling?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion2 / 24Mathematical modeling isto translate nature into mathematics
What is modeling?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsMathematical modeling isto translate nature into mathematicsConsistencyModel TestMathematicalBiologyConclusion2 / 24to be logically consistent
What is modeling?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsMathematical modeling isto translate nature into mathematicsConsistencyModel Testto be logically consistentMathematicalBiologyto fit the past and to predict futureConclusion2 / 24
What is modeling?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsMathematical modeling isto translate nature into mathematicsConsistencyModel Testto be logically consistentMathematicalBiologyto fit the past and to predict futureConclusionto fail against the test of time, i.e. to give way to bettermodels2 / 24
Human history has two periods – before and aftercalculus (1686/1687)MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion3 / 24Issac Newton (1642-1727) is the founding father ofmathematical modeling
Human history has two periods – before and aftercalculus (1686/1687)MathematicalModeling andBiologyBo DengIssac Newton (1642-1727) is the founding father ofmathematical modelingIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion3 / 24James Clerk Maxwell (1831-1879), Albert Einstein(1879-1955), Erwin Schrödinger (1887-1961), ClaudeShannon (1916-2001) are some of the luminary disciples
Human history has two periods – before and aftercalculus (1686/1687)MathematicalModeling andBiologyBo DengIssac Newton (1642-1727) is the founding father ofmathematical modelingIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion3 / 24James Clerk Maxwell (1831-1879), Albert Einstein(1879-1955), Erwin Schrödinger (1887-1961), ClaudeShannon (1916-2001) are some of the luminary disciplesCalculus is the principle language of nature
Human history has two periods – before and aftercalculus (1686/1687)MathematicalModeling andBiologyBo DengIssac Newton (1642-1727) is the founding father ofmathematical modelingIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionJames Clerk Maxwell (1831-1879), Albert Einstein(1879-1955), Erwin Schrödinger (1887-1961), ClaudeShannon (1916-2001) are some of the luminary disciplesCalculus is the principle language of natureThis century is the century of mathematical biology, whichis to translate Charles Darwin’s (1809-1882) theory intomathematics3 / 24
Model as approximation – Newton’s planetarymotionMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion4 / 24Planet rSun r2 r1 r1 r2 m1 r 1 Gm1 m2 k r1 r2 k3 r2 r1m2 r 2 Gm1 m2 k r2 r1 k3 r r1 r2
Model as approximation – Newton’s planetarymotionMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionPlanet rSun r2 r1 r1 r2 m1 r 1 Gm1 m2 k r1 r2 k3 r2 r1m2 r 2 Gm1 m2 k r2 r1 k3 r r1 r2A few calculus maneuvers lead tor(θ) ρ1 cos θwith the eccentricity 0 1 for elliptic orbits4 / 24
Special Relativity – Einstein’s model of space andtimeMathematicalModeling andBiologyBo DengIntroductionOne Assumption:The speed of light is constant for every stationary observerExamples ofModelsȳyConsistencyModel TestvKK̄MathematicalBiologyConclusion05 / 240xx̄
Special Relativity – Einstein’s model of space andtimeMathematicalModeling andBiologyBo DengIntroductionOne Assumption:The speed of light is constant for every stationary observerExamples ofModelsȳyConsistencyModel TestvKK̄MathematicalBiologyConclusion00xx̄A few calculus maneuvers lead to E mc2 , and more5 / 24
Special Relativity — Einstein’s model of space andtimeMathematicalModeling andBiologyBo DengOne Assumption:The speed of light is constant for every stationary observerExamples ofModelsConsistencyȳyIntroductionKctModel TestLK̄ 22[ c v ]t ct̄MathematicalBiologyConclusion6 / 240vt 0xx̄
Special Relativity — Einstein’s model of space andtimeMathematicalModeling andBiologyBo DengOne Assumption:The speed of light is constant for every stationary observerExamples ofModelsConsistencyȳyIntroductionKctModel TestLK̄ 22[ c v ]t ct̄MathematicalBiologyConclusion0vt 0xx̄Prediction: Time dilation for K-frame observerLLt p t̄2cc 1 (v/c)6 / 24
General Relativity — Model of space and time inaccelerationMathematicalModeling andBiologyȳyBo Dengv1 a t v0Introductionc tExamples ofModelsConsistencyModel TestMathematicalBiologyc tv1 tConclusionv0 t7 / 24xx̄
General Relativity — Model of space and time inaccelerationMathematicalModeling andBiologyȳyBo Dengv1 a t v0Introductionc tExamples ofModelsConsistencyModel TestMathematicalBiologyc tv1 tConclusionv0 txx̄Prediction: Light beam bendsunder acceleration or near massivebodies7 / 24
Mathematical model need not be mathematicalMathematicalModeling andBiologyBo DengGregor Johann Mendel (1822-1884) found the firstmathematical model in biology, leading to the discovery ofgeneIntroductionParent Genotypemrr frrmrD frDmDD fDDmrr frD ormrD frrmrr fDD ormDD frrmrD fDD ormDD frDExamples yModel 0zrr0zrD0zDD8 / 24
One More Example: Structure of DNA by modelingMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionRosalind Franklin and Maurice Wilkins had the data, butJames D. Watson and Francis Crick had the frame ofmind to model the data (1953)9 / 24
Another More – Predation in EcologyMathematicalModeling andBiologyThe mathematical model was discovered by Crawford Stanley(Buzz) Holling (1930- ) in 1959Bo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion10 / 24Td — average time a predator takes to discover a prey
Another More – Predation in EcologyMathematicalModeling andBiologyThe mathematical model was discovered by Crawford Stanley(Buzz) Holling (1930- ) in 1959Bo DengIntroductionTd — average time a predator takes to discover a preyExamples ofModelsTk — average time a predator takes to kill a preyConsistencyModel TestMathematicalBiologyConclusion10 / 24
Another More – Predation in EcologyMathematicalModeling andBiologyThe mathematical model was discovered by Crawford Stanley(Buzz) Holling (1930- ) in 1959Bo DengIntroductionTd — average time a predator takes to discover a preyExamples ofModelsTk — average time a predator takes to kill a preyConsistencyModel TestMathematicalBiologyConclusion10 / 24Td,k Td Tk — average time a predator takes todiscovery and kill a prey
Another More – Predation in EcologyMathematicalModeling andBiologyThe mathematical model was discovered by Crawford Stanley(Buzz) Holling (1930- ) in 1959Bo DengIntroductionTd — average time a predator takes to discover a preyExamples ofModelsTk — average time a predator takes to kill a preyConsistencyModel TestMathematicalBiologyConclusion10 / 24Td,k Td Tk — average time a predator takes todiscovery and kill a prey1— rate of discovery, i.e. number of preys aRd Tdpredator would find in a unit time1Rk — rate of killing, i.e. number of preys a predatorTkwould kill in a unit time11Rd,k — rate of discovery and killingTd,kTd Tk
Model of Predation in EcologyMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion11 / 24And Holling’s predation function form:Rd,k 1/TdRd1 Td Tk1 Tk (1/Td )1 Tk Rd
Model of Predation in EcologyMathematicalModeling andBiologyAnd Holling’s predation function form:Bo DengRd,k IntroductionExamples ofModelsConsistencyModel Test1/TdRd1 Td Tk1 Tk (1/Td )1 Tk RdPrediction: Assume the discovery rate is proportional tothe prey population X, Rd aX. Then the Holling TypeII predation rate must saturate as X MathematicalBiologyaX1 X 1 Tk aXTklim Rd,k limX ConclusionRd,k1Tk011 / 24X
ConsistencyMathematicalModeling andBiologyBo DengNot every piece of mathematics can be a physical law ormodel. Logical consistency is the first and necessaryconstraintIntroductionExamples ofModelsTime Invariance Principle (TIP)ConsistencyA model must has the same functional form for every timeindependent observationModel TestMathematicalBiologyConclusion12 / 24
ConsistencyMathematicalModeling andBiologyBo DengNot every piece of mathematics can be a physical law ormodel. Logical consistency is the first and necessaryconstraintIntroductionExamples ofModelsTime Invariance Principle (TIP)ConsistencyA model must has the same functional form for every timeindependent observationModel TestMathematicalBiologyConclusionNewtonian mechanics is TIP-consistent:stx(s, x0 )x012 / 24x(t s, x0 ) x(t, x(s, x0 ))
Special Relativity is self-consistentMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion13 / 24Let P be a point, having K (x, y, z, t) coordinate in theK-frame and K̄ (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.Then they are exchangeable via a linear transformationdepending the speed v:K̄ KL(v)
Special Relativity is self-consistentMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsLet P be a point, having K (x, y, z, t) coordinate in theK-frame and K̄ (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.Then they are exchangeable via a linear transformationdepending the speed v:K̄ KL(v)ConsistencyModel TestMathematicalBiologyConclusionLet K̃ (x̃, ỹ, z̃, t̃) be the coordinate of the same point ina K̃-frame moving at speed u with respect to theK̄-frame. Then we haveK̃ K̄L(u) KL(v)L(u) KL(w) with w 13 / 24u v1 uvc2
Special Relativity is self-consistentMathematicalModeling andBiologyBo DengIntroductionLet P be a point, having K (x, y, z, t) coordinate in theK-frame and K̄ (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.Then they are exchangeable via a linear transformationdepending the speed v:Examples ofModelsK̄ KL(v)ConsistencyModel TestMathematicalBiologyConclusionLet K̃ (x̃, ỹ, z̃, t̃) be the coordinate of the same point ina K̃-frame moving at speed u with respect to theK̄-frame. Then we haveK̃ K̄L(u) KL(v)L(u) KL(w) with w u v1 uvc2u vfor elements u, v ( c, c)1 uvc2defines a commutative groupThe operation u v 13 / 24
Holling’s predation model is consistentMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion14 / 24Tc — average time to consume a prey
Holling’s predation model is consistentMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion14 / 24Tc — average time to consume a preyTd,k,c Td Tk Tc — average time to discover, kill,and consume a prey
Holling’s predation model is consistentMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyTc — average time to consume a preyTd,k,c Td Tk Tc — average time to discover, kill,and consume a preyThen the rate of predation is self-consistent:Rd,k,c Conclusion 14 / 241Td,k,c 1Td Tk TcRd,kRd 1 Tc Rd,k1 (Tk Tc )Rd
Pay the TIP, or elseMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion15 / 24All differential equation models are TIP-consistent
Pay the TIP, or elseMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion15 / 24All differential equation models are TIP-consistentMost mapping models in ecology are TIP-inconsistent
Pay the TIP, or elseMathematicalModeling andBiologyBo DengAll differential equation models are TIP-consistentMost mapping models in ecology are TIP-inconsistentIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion15 / 24Example: Logistic mapxn 1 Qλ (xn ) λxn (1 xn )cannot be a model for which n represents time
Pay the TIP, or elseMathematicalModeling andBiologyBo DengAll differential equation models are TIP-consistentMost mapping models in ecology are TIP-inconsistentIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyExample: Logistic mapxn 1 Qλ (xn ) λxn (1 xn )cannot be a model for which n represents timeConclusionThe time n 2 observation yields a different functionalform:xn 2 Qλ (xn 1 ) Qλ (Qλ (xn )) 6 Qµ (xn )for any value µ. Strike one on the logistic map15 / 24
Model Test – Finding the Best FitMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion16 / 24x̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for anatural process which are modeled by competing modelsy(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,and initial state y0 , z0
Model Test – Finding the Best FitMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyx̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for anatural process which are modeled by competing modelsy(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,and initial state y0 , z0Model selection criterion: All else being equal whicheverhas a smaller error is the benchmark model by default:Model TestMathematicalBiologyConclusionEy min(y0 ,p)Ez min(z0 ,q)16 / 24nX[y(ti ; y0 , p) x̄i ]2i 1nXi 1[z(ti ; z0 , q) x̄i ]2
Model Test – Finding the Best FitMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyx̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for anatural process which are modeled by competing modelsy(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,and initial state y0 , z0Model selection criterion: All else being equal whicheverhas a smaller error is the benchmark model by default:Model TestMathematicalBiologyConclusionEy min(y0 ,p)Ez min(z0 ,q)16 / 24nX[y(ti ; y0 , p) x̄i ]2i 1nX[z(ti ; z0 , q) x̄i ]2i 1A parameter value is only meaningful to its model, and itcan only be derived by best-fitting the observed data tothe model
Model Test – Fit the past, predict the futureMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion17 / 24Edmond Halley (1656-1742) used Newtonian mechanics topredict the 1758 return of Halley’s Comet, giving thecomet its name
Model Test – Fit the past, predict the futureMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion17 / 24Edmond Halley (1656-1742) used Newtonian mechanics topredict the 1758 return of Halley’s Comet, giving thecomet its nameArthur Eddington (1882-1944) used the total solar eclipseof May 29, 1919 to confirm general relativity’s predictionfor the bending of starlight by the Sun, making Einstein aninstant world celebrity
Model Test – Fit the past, predict the futureMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion17 / 24Edmond Halley (1656-1742) used Newtonian mechanics topredict the 1758 return of Halley’s Comet, giving thecomet its nameArthur Eddington (1882-1944) used the total solar eclipseof May 29, 1919 to confirm general relativity’s predictionfor the bending of starlight by the Sun, making Einstein aninstant world celebrityGregor Mendel’s Laws of Inheritance (1866) wasrediscovered in 1900, ushering in the science of moderngenetics
Model Test – Fit the past, predict the futureMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionEdmond Halley (1656-1742) used Newtonian mechanics topredict the 1758 return of Halley’s Comet, giving thecomet its nameArthur Eddington (1882-1944) used the total solar eclipseof May 29, 1919 to confirm general relativity’s predictionfor the bending of starlight by the Sun, making Einstein aninstant world celebrityGregor Mendel’s Laws of Inheritance (1866) wasrediscovered in 1900, ushering in the science of moderngeneticsHolling’s model of predation is ubiquitous in theoreticalecology17 / 24
Mathematical Biology — To Translate Evolution toMathematicsMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion18 / 24Example: One Life RuleEvery organism lives only once and must die in any finite timein the presence of infinite population density
Mathematical Biology — To Translate Evolution toMathematicsMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion18 / 24Example: One Life RuleEvery organism lives only once and must die in any finite timein the presence of infinite population densityIn math translation: Let xt be the population at time t.Then the per-capita change must satisfyxt x0xt 1 1x0x0
Mathematical Biology — To Translate Evolution toMathematicsMathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionExample: One Life RuleEvery organism lives only once and must die in any finite timein the presence of infinite population densityIn math translation: Let xt be the population at time t.Then the per-capita change must satisfyxt x0xt 1 1x0x0Lead toxt x0 1One Life Rule limx0 x0and to the logistic equationẋ(t) rx(t)[1 x(t)/K]18 / 24with x(t) xt , r the max per-capita growth rate, and Kthe carrying capacity
Footnote: model or no model, generalization orrelativism is often the problemMathematicalModeling andBiologyBo DengStrike two on the logistic map x1 λx0 (1 x0 ):IntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion19 / 24limx0 x1 x0 lim [λ(1 x0 ) 1] 6 1x0 x0
Footnote: model or no model, generalization orrelativism is often the problemMathematicalModeling andBiologyBo DengStrike two on the logistic map x1 λx0 (1 x0 ):IntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion19 / 24limx0 x1 x0 lim [λ(1 x0 ) 1] 6 1x0 x0While the logistic equation, x0 (t) rx(t)(1 x(t)/K),dogged another consistency bullet x(t; x0 ) x0Klim lim 1 1x0 x0 (K x0 )e rtx0 x0
Footnote: model or no model, generalization orrelativism is often the problemMathematicalModeling andBiologyBo DengStrike two on the logistic map x1 λx0 (1 x0 ):IntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionlimx0 x1 x0 lim [λ(1 x0 ) 1] 6 1x0 x0While the logistic equation, x0 (t) rx(t)(1 x(t)/K),dogged another consistency bullet x(t; x0 ) x0Klim lim 1 1x0 x0 (K x0 )e rtx0 x0There should be no different versions of the same reality,but refined approximations19 / 24
One More Example: Why DNA is coded in 4 bases?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusion20 / 24
One More Example: Why DNA is coded in 4 bases?MathematicalModeling andBiologyBo DengIntroductionExamples ofModelsConsistencyModel TestMathematicalBiologyConclusionThe AT pair has one weak O-H bond but
This century is the century of mathematical biology, which is to translate Charles Darwin’s (1809-1882) theory into mathematics 3/24. Mathematical Modeling and Biology Bo Deng Introduction Examples of Models Consistency Model Test Mathematical Biology Conclusion Model as approxim
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