Math 8803/4803, Spring 2008: Discrete Mathematical Biology

Math 8803/4803, Spring 2008:Discrete Mathematical Biology

Prof. Christine Heitsch

School of Mathematics

Georgia Institute of Technology

Lecture 2 – January 9, 2008

Overview

• Today: (an incomplete) mathematical primer

– Numbers in nature – integer sequences, recurrence relations,

mathematical induction, algorithms and computational

complexity, recursion, dynamic programming

– Shapes in nature – graphs and trees, Platonic solids, Euler

circuit / trail, Hamiltonian cycle / path

– Missing in action – strings, permutations, binomial

coefficients, pigeonhole principle, generating functions,. . .

• Friday:(an incomplete) biological primer

C. E. Heitsch, GA Tech 1

Phyllotaxis: optimizing seed dispersal

Cauliflower Pine cone Romanesco

Bellis perennis Coneflower Sunflower


Counting left and right spirals

Pine cone 8 right spirals 13 left spirals

Romanesco 13 right spirals 21 left spirals


Is there a pattern?

5 and 8 spirals 8 and 13 spirals 13 and 21 spirals

21 and 34 spirals 34 and 55 spirals 55 and 89 spirals


Enumerating Oryctolagus cuniculus

In 1202, Leonardo Fibonacci asked. . .

How many pairs of rabbits can be produced from a single pair inone year if it is assumed that every month each pair begets anew pair which from the second month on becomes productive?


The Fibonacci sequence fn

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 . . .

Defined as the recurrence relation fn = fn−1 + fn−2 for n ≥ 2with initial conditions f0 = 0, f1 = 1.

This (second order linear homogeneous recurrence relation with constant coefficients) can

be solved using the characteristic equation r2 − r − 1 = 0 to give

fn = 1√5

[(1+√

52

)n

−(

1−√

52

)n], n ≥ 0.

Even without this closed form solution, many properties of fn can

be proved using mathematical induction.


Some properties of fn

Claim: For all n ≥ 6, fn > (32)

n−1.

Two more “for the interested reader”. . .

Claim: Every integer n ≥ 1 can be expressed as the sum of

distinct Fibonacci numbers, no two of which are consecutive.

Claim: The representation from the previous claim is unique

if f1 is not allowed as a summand.


Proof by induction

Claim: For all n ≥ 6, fn > (32)

n−1.

Proof:

Let S(n) be the statement fn > (32)

n−1.

Base cases: For n = 6, fn = 8 and (32)

n−1 = 24332 . For n = 7, fn = 13 and

(32)

n−1 = 72964 . So, f6 > (3

2)5 and f7 > (3

2)6.

Assume that for all integers n with 7 ≤ n ≤ k that fn > (32)

n−1.

Consider n = k + 1. Then fk+1 = fk + fk−1 > (32)

k−1 + (32)

k−2 = (32)

k−2(52).

Since 52 > 9

4, we have fk+1 > (32)

k−2(32)

2 = (32)

k.

So, by the Principle of Mathematical Induction, S(n) is true for all n ≥ 6.


Algorithms and computational complexity

The amount of resources used by an algorithm (an unambiguously

specified procedure) to solve a problem is an important consideration.

Typical algorithmic resources are space (memory) and time.

Polynomial time/space algorithms are efficient (good);

exponential space/time algorithms are intractable (bad).


An obvious recursive algorithm

Fibonacci[n]

if (n = 0) then return (0)


else return(Fibonacci[n-1]+Fibonacci[n-2])

F3

F0

F2

F1

F1F0

F2

F1

F5

F3

F0

F2

F1

F1

F4

What is the complexity of this algorithm?


“Dynamic Programming”

Fibonacci[n]



else

F0 = 0

F1 = 1

For i = 1 to n, Fi = Fi−1 + Fi−2

What is the complexity of this algorithm?

In general, trading space for time to solve a recursion.


Folded RNA sequences form structures

Johnson et al. J Virol, 2004. Tihova et al. J Virol, 2004. Tang et al. Nat Str B, 2001.

How does the PaV RNA genome fold into a dodecahedral cage?


Platonic Solids


Some Graph Theory

A graph G is a set of vertices V , a set of edges E, and a mapping whichassociates each edge e ∈ E to an unordered pair of vertices {x, y} for x, y ∈ V .

A weighted graph G has a real number associated to each edge e.

A simple closed path is a walk in a graph G which begins and ends at the same vertex but is

otherwise a sequence of distinct vertices and distinct edges.

A connected graph that contains no nontrivial simple closed paths is a tree.

A rooted tree has a unique identified vertex (the root).

A child of a (parent) vertex is a vertex one edge farther away from the root.

A subtree of a vertex in a rooted tree is a child and all its descendents.


Bridges of Konigsberg

C

A

B

D

dc

a b

g

e

f

Is there an Euler circuit? Is there a Hamiltonian cycle?


Acknowledgements

• Plant Spirals Exhibit, Smith College Botanic Gardenhttp://maven.smith.edu/∼phyllo/EXPO/index.html

• Fibonacci Numbers and Naturehttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html


Documents

Math 8803/4803, Spring 2008: Discrete Mathematical Biology