Mathematical Modeling and Biology Bo Deng Introduction Examples of - PowerPoint PPT Presentation

Mathematical Modeling and Biology Mathematical Modeling and Biology Bo Deng Introduction Examples of Models Bo Deng Consistency Model Test Department of Mathematics Mathematical University of Nebraska – Lincoln Biology Conclusion March 10, 2016 www.math.unl.edu/ ∼ bdeng1 1 / 24

What is modeling? Mathematical Modeling and Biology Bo Deng Mathematical modeling is Introduction Examples of to translate nature into mathematics Models Consistency Model Test Mathematical Biology Conclusion 2 / 24

What is modeling? Mathematical Modeling and Biology Bo Deng Mathematical modeling is Introduction Examples of to translate nature into mathematics Models Consistency to be logically consistent Model Test Mathematical Biology Conclusion 2 / 24

What is modeling? Mathematical Modeling and Biology Bo Deng Mathematical modeling is Introduction Examples of to translate nature into mathematics Models Consistency to be logically consistent Model Test Mathematical to fit the past and to predict future Biology Conclusion 2 / 24

What is modeling? Mathematical Modeling and Biology Bo Deng Mathematical modeling is Introduction Examples of to translate nature into mathematics Models Consistency to be logically consistent Model Test Mathematical to fit the past and to predict future Biology Conclusion to fail against the test of time, i.e. to give way to better models 2 / 24

Human history has two periods – before and after calculus (1686/1687) Mathematical Modeling and Issac Newton (1642-1727) is the founding father of Biology mathematical modeling Bo Deng Introduction Examples of Models Consistency Model Test Mathematical Biology Conclusion 3 / 24

Human history has two periods – before and after calculus (1686/1687) Mathematical Modeling and Issac Newton (1642-1727) is the founding father of Biology mathematical modeling Bo Deng Introduction James Clerk Maxwell (1831-1879), Albert Einstein Examples of Models (1879-1955), Erwin Schr¨ odinger (1887-1961), Claude Consistency Shannon (1916-2001) are some of the luminary disciples Model Test Mathematical Biology Conclusion 3 / 24

Human history has two periods – before and after calculus (1686/1687) Mathematical Modeling and Issac Newton (1642-1727) is the founding father of Biology mathematical modeling Bo Deng Introduction James Clerk Maxwell (1831-1879), Albert Einstein Examples of Models (1879-1955), Erwin Schr¨ odinger (1887-1961), Claude Consistency Shannon (1916-2001) are some of the luminary disciples Model Test Mathematical Biology Calculus is the principle language of nature Conclusion 3 / 24

Human history has two periods – before and after calculus (1686/1687) Mathematical Modeling and Issac Newton (1642-1727) is the founding father of Biology mathematical modeling Bo Deng Introduction James Clerk Maxwell (1831-1879), Albert Einstein Examples of Models (1879-1955), Erwin Schr¨ odinger (1887-1961), Claude Consistency Shannon (1916-2001) are some of the luminary disciples Model Test Mathematical Biology Calculus is the principle language of nature Conclusion This century is the century of mathematical biology, which is to translate Charles Darwin’s (1809-1882) theory into mathematics 3 / 24

Model as approximation – Newton’s planetary motion Mathematical Modeling and Biology Planet r � Sun Bo Deng r 2 � Introduction r 1 � Examples of  � r 1 − � r 2 Models m 1 ¨ r 1 � = − Gm 1 m 2   r 2 � 3 Consistency � � r 1 − �    � r 2 − � r 1 Model Test m 2 ¨ � r 2 = − Gm 1 m 2 Mathematical r 1 � 3 � � r 2 − �   Biology   � r = � r 1 − � r 2  Conclusion 4 / 24

Model as approximation – Newton’s planetary motion Mathematical Modeling and Biology Planet � r Sun Bo Deng � r 2 Introduction � r 1 Examples of  � r 1 − � r 2 Models m 1 ¨ � r 1 = − Gm 1 m 2   r 2 � 3 Consistency � � r 1 − �    � r 2 − � r 1 Model Test m 2 ¨ r 2 � = − Gm 1 m 2 Mathematical r 1 � 3 � � r 2 − �   Biology   � r = � r 1 − � r 2  Conclusion A few calculus maneuvers lead to ρ r ( θ ) = 1 + ǫ cos θ with the eccentricity 0 ≤ ǫ < 1 for elliptic orbits 4 / 24

Special Relativity – Einstein’s model of space and time Mathematical Modeling and Biology One Assumption: Bo Deng The speed of light is constant for every stationary observer Introduction Examples of y y ¯ Models Consistency ¯ v K K Model Test Mathematical Biology Conclusion x x ¯ 0 0 5 / 24

Special Relativity – Einstein’s model of space and time Mathematical Modeling and Biology One Assumption: Bo Deng The speed of light is constant for every stationary observer Introduction Examples of y y ¯ Models Consistency ¯ v K K Model Test Mathematical Biology Conclusion x x ¯ 0 0 A few calculus maneuvers lead to E = mc 2 , and more 5 / 24

Special Relativity — Einstein’s model of space and time Mathematical One Assumption: Modeling and Biology The speed of light is constant for every stationary observer Bo Deng y y ¯ Introduction Examples of Models ¯ K L K √ ct Consistency c 2 − v 2 ] t = c ¯ [ t Model Test Mathematical Biology x x ¯ 0 vt 0 Conclusion 6 / 24

Special Relativity — Einstein’s model of space and time Mathematical One Assumption: Modeling and Biology The speed of light is constant for every stationary observer Bo Deng y ¯ y Introduction Examples of Models ¯ K L K √ ct Consistency c 2 − v 2 ] t = c ¯ [ t Model Test Mathematical Biology x x ¯ 0 vt 0 Conclusion Prediction: Time dilation for K -frame observer 1 − ( v/c ) 2 > L L c = ¯ t = t � c 6 / 24

General Relativity — Model of space and time in acceleration Mathematical y y ¯ Modeling and Biology Bo Deng v 1 = a ∆ t + v 0 Introduction c ∆ t Examples of Models Consistency v 1 ∆ t Model Test c ∆ t Mathematical Biology Conclusion x ¯ x v 0 ∆ t 7 / 24

General Relativity — Model of space and time in acceleration Mathematical y y ¯ Modeling and Biology Bo Deng v 1 = a ∆ t + v 0 Introduction c ∆ t Examples of Models Consistency v 1 ∆ t Model Test c ∆ t Mathematical Biology Conclusion x x ¯ v 0 ∆ t Prediction: Light beam bends under acceleration or near massive bodies 7 / 24

Mathematical model need not be mathematical Mathematical Modeling and Gregor Johann Mendel (1822-1884) found the first Biology mathematical model in biology, leading to the discovery of Bo Deng gene Introduction Examples of Parent Genotype Models m rD × f DD or Consistency m rr × f DD or m rr × f rD or m DD × f rD m DD × f rr m DD × f DD m rD × f rr Model Test m rD × f rD m rr × f rr Mathematical Biology Conclusion Offspring Genotype z ′ 1 1/4 0 1/2 0 0 rr z ′ 0 1/2 0 1/2 1 1/2 rD z ′ 0 1/4 1 0 0 1/2 DD 8 / 24

One More Example: Structure of DNA by modeling Mathematical Modeling and Biology Bo Deng Introduction Examples of Models Consistency Model Test Mathematical Biology Conclusion Rosalind Franklin and Maurice Wilkins had the data, but James D. Watson and Francis Crick had the frame of mind to model the data (1953) 9 / 24

Another More – Predation in Ecology Mathematical The mathematical model was discovered by Crawford Stanley Modeling and Biology (Buzz) Holling (1930- ) in 1959 Bo Deng T d — average time a predator takes to discover a prey Introduction Examples of Models Consistency Model Test Mathematical Biology Conclusion 10 / 24

Another More – Predation in Ecology Mathematical The mathematical model was discovered by Crawford Stanley Modeling and Biology (Buzz) Holling (1930- ) in 1959 Bo Deng T d — average time a predator takes to discover a prey Introduction Examples of T k — average time a predator takes to kill a prey Models Consistency Model Test Mathematical Biology Conclusion 10 / 24

Another More – Predation in Ecology Mathematical The mathematical model was discovered by Crawford Stanley Modeling and Biology (Buzz) Holling (1930- ) in 1959 Bo Deng T d — average time a predator takes to discover a prey Introduction Examples of T k — average time a predator takes to kill a prey Models T d,k = T d + T k — average time a predator takes to Consistency discovery and kill a prey Model Test Mathematical Biology Conclusion 10 / 24

Another More – Predation in Ecology Mathematical The mathematical model was discovered by Crawford Stanley Modeling and Biology (Buzz) Holling (1930- ) in 1959 Bo Deng T d — average time a predator takes to discover a prey Introduction Examples of T k — average time a predator takes to kill a prey Models T d,k = T d + T k — average time a predator takes to Consistency discovery and kill a prey Model Test R d = 1 Mathematical — rate of discovery, i.e. number of preys a Biology T d Conclusion predator would find in a unit time R k = 1 — rate of killing, i.e. number of preys a predator T k would kill in a unit time 1 1 R d,k = = — rate of discovery and killing T d,k T d + T k 10 / 24