JM - http://folding.chmcc.org 1

Introduction to Bioinformatics: Lecture XVIGlobal Optimization and Monte Carlo

Jarek MellerJarek Meller

Division of Biomedical Informatics, Division of Biomedical Informatics, Children’s Hospital Research Foundation Children’s Hospital Research Foundation & Department of Biomedical Engineering, UC& Department of Biomedical Engineering, UC

JM - http://folding.chmcc.org 2

Outline of the lecture

Global optimization and local minima problem Physical map assembly, ab initio protein

folding and likelihood maximization as examples of global optimization problems

Biased random search heuristics Monte Carlo approach Biological motivations and genetic algorithms

JM - http://folding.chmcc.org 3

Optimization, steepest descent and local minima

Optimization is a procedure in which an extremum of a function is sought. When the relevant extremum is the minimum of a function the optimization procedure is called minimization.

f(x)

Global minimumLocal minimum

Local minimumLocal minimum

JM - http://folding.chmcc.org 4

Rugged landscapes and local minima (maxima) problem

JM - http://folding.chmcc.org 5

Algorithmic complexity of global optimization

Polynomial vs. exponential complexity, e.g., n2 vs. 2n

steps to obtain the optimal solution where n denotes the overall “size of the input”

Global optimization term is used to refer to optimization problems for which no polynomial time algorithm that guarantees optimal solution is known

In general global optimization implies that there might be multiple local minima and thus one is likely to find a local rather than the global optimum

Let us revisit some of the global optimization problems that we stumbled on so far …

6

The problem of ordering clone libraries with STS markers in the presence of errors

5 4 1 3 2

1 0 1 0 1 0

2 0 1 1 0 1

3 0 1 0 1 1

4 1 0 0 1 0

DNA clone 1 clone 2 clone 3 clone 4

STS: 1 2 3 4 5

1 2 3 4 5

1 0 0 1 1 0

2 1 1 0 1 0

3 0 1 1 1 0

4 0 0 1 0 1

In the presence of experimental errors the problem leads to globaloptimization problem (see Pevzner, Chapter 3).

STS

Clone

JM - http://folding.chmcc.org 7

Heuristic solutions may still provide good probe ordering

The number of “gaps” (blocks of zeros in rows) in the hybridization matrixmay be used as a cost function, since hybridization errors typically splitblocks of ones (false negatives) or split a gap into two gaps (false positive).

The problem of finding a permutation that minimizes the number of gapscan be cast as a Traveling Salesman Problem (TSP), in which cities are the columns of the hybridization matrix (plus an additional column of zeros)and the distance between two cities is the number of positions in which the two columns differ (Hamming dist.)

Thus, an efficient algorithm is unlikely in general case (unless P=NP) andheuristic solutions are being sought that provide good probe ordering, atleast for most cases (e.g. Alizadeh et. al., 1995)

JM - http://folding.chmcc.org 8

Profile HMMs and likelihood optimization when states (optimal multiple alignments) are not known

JM - http://folding.chmcc.org 9

Random biased search: ideas and heuristics

GA, MC, SA (MC with a smoothing)

Fitness lanscapes

Biological and physical systems solve these “unsolvable” problems:From optimization to biology and back to optimization

JM - http://folding.chmcc.org 10

Literature watch: 10 years of DNA computing

Adleman LM, Molecular computation of solutions to combinatorial problems,Science 266:1021-4 (1994)

RS Braich, N Chelyapov, C Johnson, PWK Rothemund, and L Adleman,Solution of a 20-Variable 3-SAT Problem on a DNA Computer, Science 296: 499-502 (2002)

JM - http://folding.chmcc.org 11

Monte Carlo random search

A simulation technique for conformational sampling and optimization based on a random search for energetically favourable conformations.

Finding global (or at least “good” local) minimum by biasedrandom walk may take some luck …

JM - http://folding.chmcc.org 12

Monte Carlo algorithm

The core of MC algorithm is a heuristic prescription for a plausible

pattern of changes in the configurations assumed by the system. Such an elementary “move” depends on the type of the problem.

In the realm of protein structure it may be for instance a rotation around a randomly chosen backbone bond. A long series of random moves is generated with only some of them considered as “good” moves.

The advantage of MC method is its generality and a relatively weak dependence on the dimensionality of the system. However, finding a “move” which would ensure efficient sampling may be a highly non-trivial problem.

JM - http://folding.chmcc.org 13

Monte Carlo algorithm

In the standard Metropolis MC a move is accepted

unconditionally if the new configuration results in a better (lower) potential energy. Otherwise it is accepted with a probability given by the Boltzmann factor:

)exp(Tk

UP

B

rr

)()( rr UUU denotes the change in the potential energy associated with a move

rr

JM - http://folding.chmcc.org 14

Climbing mountains easier: simulated annealing

Increasing the effective “temperature” means higher

probability of accepting moves that increase the energy

Thus, the likelihood of escaping from a local minimum may be tuned

Heating and cooling cycles, in analogy to physical systems

In the limit of infinitely slow cooling simulated annealing is guaranteed to provide the global minimum

JM - http://folding.chmcc.org 15

From biology to optimization: genetic algorithms

Genetic algorithm (GA). A class of algorithms inspired by the mechanisms of genetics, which has been applied to global optimization (especially combinatorial optimization problems). It requires the specification of three operations (each is typically probabilistic) on objects, called "strings" (these could be real-valued vectors)

0. Initialize population 1. Select parents for reproduction and “evolutionary” operators (e.g. mutation and crossover) 2. Perform operations to generate intermediate population and evaluate their fitness (value of the objective function to be optimized) 3. Select a subpopulation for next generation (survival of the fittest)4. Repeat 1-3 until some stopping rule is reached

JM - http://folding.chmcc.org 16

Genetic algorithm: operators and adaptation

Reproduction - combining strings in the population to create a new string (offspring); Example: Taking 1st character from 1st parent + rest of string from 2nd parent: [001001] + [111111] ===> [011111]

Mutation - spontaneous alteration of characters in a string; Example: [001001] ===> [101001]

Crossover - combining strings to exchange values, creating new strings in their place. Example: With crossover location at 2: [001001] & [111111] ===> [001111], [111001]

JM - http://folding.chmcc.org 17

Genetic algorithms for global optimization

The original GA was proposed by John Holland and used crossover and total population replacement. This means a population with 2N objects (called chromosomes) form N pairings of parents that produce 2N offsprings. The offsprings comprise the new generation, and they become the total population, replacing their parents. More generally, a population of size N produces an intermediate population of N+M, from which Ñis kept to form the new population. One way to choose which Ñsurvive is by those with the greatest fitness values –survival of the fittest.

JM - http://folding.chmcc.org 18

Random biased search: ideas and heuristics

GA, MC, SA (MC with a smoothing)

Fitness lanscapes