Counting position weight matrices in a sequence & an application to discriminative motif finding Saurabh Sinha Computer Science University of Illinois, Urbana-Champaign

Transcriptional Regulation GENE ACAGTGA TRANSCRIPTION FACTOR PROTEIN

GENE ACAGTGA TRANSCRIPTION FACTOR PROTEIN Transcriptional Regulation

Binding sites and motifs Transcription factor binding sites in a gene’s neighborhood are the fundamental units of the regulatory network Transcription factor binding is specific, hence binding sites are similar to each other, but variability is often seen A motif is the common sequence pattern among binding sites of transcription factor

Motif models Consensus string, e.g., ACGWGT Position Weight Matrix (PWM)

Position Weight Matrix A C G T ACCCGTT ACCGGTT ACAGGAT ACCGGTT ACATGAT Binding sites PWM

Databases of PWMs Transfac has ~100s of PWMs for human Jaspar: a smaller, perhaps better curated database of PWMs Organism specific databases coming up frequenctly PWMs in databases often derived from experimentally validated binding sites

Bioinformatics of PWMs Popular motif model i.e., several motif finding algorithms that attempt to find PWMs from sequences Gibbs sampling: one of the earliest; tries to sample PWMs with high “relative entropy” MEME: another early algorithm; uses expectation maximization to find PWMs that best “model the sequences” Many more algorithms to find PWMs from a set of sequences

Problem: counting motifs Given DNA sequence, and a consensus motif (say “ACGWGT”), count the motif in the sequence Trivial solution What if the motif is a Position Weight Matrix (PWM) ? Why hasn’t this problem been looked at? Because previous algorithms used different scores of PWMs: how “sharp” they are, how well they explain data, etc.

Counting matches to a PWM: A possibility For each site s in sequence, compute If Pr(s | W) > some threshold, call s a site Count number of sites in sequence No distinction between strong and weak sites, as long as they are above threshold binary scheme, not realistic

A wish-list (for the score) Score should consider both strong and weak occurrences of motif Score should assign appropriate weights to strong and weak occurrences Score should be aware that there may also be sites of other known motifs in the sequence The list goes on : score should be efficiently computable, score should be differentiable, score should …

The “w-score” Defined by a probabilistic model of sequence generation Given one or more motifs, and a background distribution, defines a probability space on sequences A simple (zeroth order) Hidden Markov model (HMM)

Probabilistic Model: toy example Given two motifs W 1,W 2, a “background” motif W b, and a sequence length L Pr(W i  W j ) = p j transition probability When in state W i, emit a substring s chosen with probability Pr(s | W i ) emission probability Stop when length of emitted sequence is L W1W1 W2W2 WbWb A stochastic process generating sequences of length L

A “path” through the HMM One possible path T 1 W1W1 W1W1 W2W2 WbWb WbWb WbWb W2W2 WbWb WbWb W2W2 Another possible path T 2

Likelihood of sequence & paths A path of the HMM defines the locations of motif matches For a sequence S & a path T, the joint probability Pr(S,T) is easy to compute Conditional probability of a path T, given the data S, is: Strong matches make the probability higher Paths with weak matches have lower conditional probabilities W1W1 W1W1 W2W2 WbWb WbWb WbWb W2W2 WbWb WbWb W2W2

Let the number of occurrences of a motif (say W 1 ) in path T be Compute: In words: An average of the motif count, with weights equal to the probability of T given S The “w-score”

The “w-score” (Cont’d) Score depends both on number and quality of matches to motif. Every substring is a potential binding site, and paths placing the motif there will contribute to the count Pr(T | S) depends on the match strength of all motifs, not just the one being counted

The wish-list (again) Score should give consider both strong and weak occurrences of motif Score should assign appropriate weights to strong and weak occurrences Score should be aware that there may also be sites of other known motifs in the sequence    An exciting new feature of this motif score

Computational pros and cons The w-score computation takes time, where L is sequence length, and l m is the motif length. This is relatively expensive The w-score can be differentiated with respect to all of the PWM parameters in time Important feature for search algorithms

Using the “w-score” in discriminative motif finding

Discriminative motif finding Suppose we have a set of co-regulated genes, i.e., we believe they have binding sites of the same transcription factor (in their regulatory control regions) Traditionally, motif finding tries to find these binding sites, based on over-representation, conservation etc. Often we also know a set of genes that should NOT have binding sites of that transcription factor Examples: ChIP-on-chip, In situ hybridization pictures of Drosophila embryo, etc.

Problem formulation Given two sets of sequences S + and S - Find a motif that has many occurrences in S + and few occurrences in S - Maximize the difference in the average counts of the motif in the two sets Let  W (S) = count of a motif W in sequence S Maximize:

Optimization problem Find motif W that maximizes

Derivatives of objective function Let W k  be the PWM entry for base  in column k We can efficiently compute We can efficiently differentiate our objective function

Algorithm Search space: Set of n = 20 substrings of sequences in S + (called “site set”) Objective function: Construct PWM W from site-set, compute score Length of sites is user-defined

Algorithm S+S+ Current site-set C

Algorithm S+S+ Replace one site with any site from sequence Pick a replacement that improves objective function

Algorithm Current solution (site-set): C Candidate new solution: C Many possibilities for C (every substring of every sequence in S + is a possible replacement) Evaluate objective function on each candidate C Too slow ! Use derivative information !

Algorithm Estimate the objective function value for each candidate C using partial derivatives and first order approximation Examine each candidate in decreasing order of estimated score If a candidate C found with greater score than C, choose it.

Algorithm illustration Estimated scores 11 Accurate score 10 Accurate score 13 Accurate score Current score = 12

Algorithm Properties Objective function has many desirable properties, but is an expensive operation Derivative computation has the same time complexity, and is used to guide search Avoids local optima by searching in a discretized PWM space Performs significantly better and/or faster than Gibbs sampling and Conjugate Gradients, for this particular score

Discriminative PWM Search (DIPS) Software available Can easily handle data sets of ~100 sequences Can find multiple motifs iteratively, but without masking: Find a PWM, then include it in the model as a known PWM, find another PWM, and so on

Performance tests Tested on synthetic data Compared to traditional motif finder as well as two discriminative motif finders Superior performance in the presence of “distractor” motifs it really helps to be able to count a motif in the presence of other known motifs

Tests on Drosophila Enhancers HEAD TAIL Protein Concentration BICOID (ACTIVATOR)

Tests on Drosophila Enhancers HEAD TAIL Protein Concentration CAUDAL (ACTIVATOR)

Tests on Drosophila Enhancers HEAD TAIL Protein Concentration KRUPPEL (REPRESSOR)

DIPS runs S + = promoters of genes expressed in anterior half of embryo S - = promoters of genes expressed in posterior half of embryo Top motif: Bicoid ! HEAD TAIL Protein Concentration BICOID (ACTIVATOR)

DIPS runs S + = promoters of genes expressed in posterior half of embryo S - = promoters of genes expressed in anterior half of embryo Top motif: Caudal ! HEAD TAIL Protein Concentration CAUDAL (ACTIVATOR)

DIPS runs S + = promoters of genes expressed around the middle 20% of embryo S - = promoters of genes expressed in middle 20% of embryo Top motif: Kruppel ! HEAD TAIL Protein Concentration

Summary of results

Social regulation in honey bee Transition from nursing in the hive to foraging for food is age related, but also regulated by the needs of the colony 32 genes demonstrated to be significantly differentially expressed in brains of nurses and foragers (21 active in foragers only, 11 active in nurses only) DIPS run on 2Kbp promoters of these social behavior-related genes

Results on honey bee genes

Conclusion Discriminative motif finding increasingly becoming a necessary analysis Motif finding in the presence of other known motifs also becoming relevant A search algorithm that maximizes any objective function of the motif counts in the sequences (as long as its differentiable) Several extensions and variations possible

Acknowledgements Eric Siggia, Eran Segal Yoseph Barash (“LearnPSSM”) Andrew Smith (“DME”)

Reference ISMB 2006 (Brazil); Bioinformatics journal.