The Ensemble of RNA Structures Example: best structures of the RNA - PowerPoint PPT Presentation

The Ensemble of RNA Structures Example: best structures of the RNA sequence GGGGGUAUAGCUCAGGGGUAGAGCAUUUGACUGCAGAUCAAGAGGUCCCUGGUUCAAAUCCAGGUGCCCCCU free energy in kcal/mol (((((((..((((.......))))...........((((....))))(((((.......)))))))))))). -28.10 (((((((..((((.......))))....((((.(.......).))))(((((.......)))))))))))). -27.90 ((((((((.((((.......))))(((((((((..((((....))))..)))).)))))....)))))))). -27.80 ((((((((.((((.......))))(((((((((..((((....))))..))).))))))....)))))))). -27.80 (((((((..((((.......))))....((((...........))))(((((.......)))))))))))). -27.60 (((((((..((((.......))))....(((..(.......)..)))(((((.......)))))))))))). -27.50 ((((((((.((((.......)))).((((((((..((((....))))..)))).)))).....)))))))). -27.20 ((((((((.((((.......)))).((((((((..((((....))))..))).))))).....)))))))). -27.20 ((((((((.((((.......))))...........((((....)))).((((.......)))))))))))). -27.20 ((((((...((((.......))))...........((((....))))(((((.......))))).)))))). -27.20 (((((((...(((...(((...(((......)))..)))..)))...(((((.......)))))))))))). -27.10 ((((((((.((((.......))))((((((((...((((....))))...))).)))))....)))))))). -27.00 ((((((((.((((.......))))((((((((...((((....))))...)).))))))....)))))))). -27.00 ((((((((.((((.......))))....((((.(.......).)))).((((.......)))))))))))). -27.00 (((((((..((((.......)))).((((((....).))))).....(((((.......)))))))))))). -27.00 (((((((..((((.......))))...........(((......)))(((((.......)))))))))))). -27.00 ((((((...((((.......))))....((((.(.......).))))(((((.......))))).)))))). -27.00 ((((((((.((((.......))))(((((((((..(((......)))..)))).)))))....)))))))). -26.70 ((((((((.((((.......))))(((((((((..(((......)))..))).))))))....)))))))). -26.70 ((((((((.((((.......))))....((((...........)))).((((.......)))))))))))). -26.70 (((((((..((((.......)))).(((((.......))))).....(((((.......)))))))))))). -26.70 S.Will, 18.417, Fall 2011 ((((((...((((.......))))....((((...........))))(((((.......))))).)))))). -26.70 The set of all non-crossing RNA structures of an RNA sequence S is called (structure) ensemble P of S .

Is Minimal Free Energy Structure Prediction Useful? • BIG PLUS: loop-based energy model quite realistic • Still mfe structure may be “wrong”: Why? • Lesson: be careful, be sceptical! (as always, but in particular when biology is involved) • What would you improve? S.Will, 18.417, Fall 2011

Probability of a Structure How probable is an RNA structure P for a RNA sequence S ? GOAL: define probability Pr[ P | S ]. IDEA: Think of RNA folding as a dynamic system of structures (=states of the system). Given much time, a sequence S will form every possible structure P . For each structure there is a probability for observing it at a given time. This means: we look for a probability distribution! Requirements: probability depends on energy — the lower the more probable. No additional assumptions! S.Will, 18.417, Fall 2011

Distribution of States in a System Definition (Boltzmann distribution) Let X = { X 1 , . . . , X N } denote a system of states, where state X i has energy E i . The system is Boltzmann distributed with temperature T iff Pr[ X i ] = exp ( − β E i ) / Z for Z := � i exp ( − β E i ), where β = ( k B T ) − 1 . Remarks • broadly used in physics to describe systems of whatever • Boltzmann distribution is usually assumed for the thermodynamic equilibrium (i.e. after sufficiently much time) • transfer to RNA easy to see: structures=states, energies • why temperature? • very high temperature: all states equally probable S.Will, 18.417, Fall 2011 • very low temperature: only best states occur • k B ≈ 1 . 38 × 10 − 23 J / K is known as Boltzmann constant ; β is called inverse temperature . • call exp ( − β E i ) Boltzmann weight of X i .

What next? We assume that the structure ensemble of an RNA sequence is Boltzmann distributed. • What are the benefits? (More than just probabilities of structures . . . ) • Why is it reasonable to assume Boltzmann distribution? (Well, a physicist told me . . . ) • How to calculate probabilities efficiently? (McCaskill’s algorithm) S.Will, 18.417, Fall 2011

Benefits of Assuming Boltzmann Definition Probability of a structure P for S: Pr[ P | S ] := exp ( − β E ( P )) / Z . Allows more profound weighting of structures in the ensemble. We need efficient computation of partition function Z ! Even more interesting: probability of structural elements Definition Probability of a base pair ( i , j ) for S: � Pr[( i , j ) | S ] := Pr[ P | S ] P ∋ ( i , j ) Again, we need Z (and some more). Base pair probabilities enable a new S.Will, 18.417, Fall 2011 view at the structure ensemble (visually but also algorithmically!). Remark: For RNA, we have “real” temperature, e.g. T = 37 ◦ C , which determines β = ( k B T ) − 1 . For calculations pay attention to physical units!

An Immediate Use of Base Pair Probabilities MFE structure and base pair probability dot plot 1 of a tRNA dot.ps GGGGGUAUAGCUCAGGGGUAGAGCAUUUGACUGCAGAUCAAGAGGUCCCUGGUUCAAAUCCAGGUGCCCCCU U G G G G G U A U A G C U C A G G G G U A G A G C A U U U G A C U G C A G A U C A A G A G G U C C C U G G U U C A A A U C C A G G U G C C C C C U G G G G G U A U A G C U C A G G G G U A G A G C A U U U G A C U G C A G A U C A A G A G G U C C C U G G U U C A A A U C C A G G U G C C C C C U G G G G G U A U A G C U C A G G G G U A G A G C A U U U G A C U G C A G A U C A A G A G G U C C C U G G U U C A A A U C C A G G U G C C C C C U G C G C G C G C G C U G A U G A G U G G A C A U G C G C G U AG A G C C U A A U A U A U C G U A U G C G U U C G C C A G C S.Will, 18.417, Fall 2011 U A G U G C A A A G G G G G G U A U A G C U C A G G G G U A G A G C A U U U G A C U G C A G A U C A A G A G G U C C C U G G U U C A A A U C C A G G U G C C C C C U 1 computed by “RNAfold -p”

Why Do We Assume Boltzmann We will give an argument from information theory. We will show: The Boltzmann distribution makes the least number of assumptions. Formally, the B.d. is the distribution with the lowest information content/maximal (Shannon) entropy. As a consequence: without further information about our system, Boltzmann is our best choice. [ What could “further information” mean in a biological context? ] S.Will, 18.417, Fall 2011

Shannon Entropy (by Example) We toss a coin. For our coin, heads and tails show up with respective probabilities p and q (not necessarily fair). How uncertain are we about the result? p = 0 . 5 , q = 0 . 5 ⇒ Answer: expected 1.0 H = 1 — maximal p * log2(1/p) + q * log2(1/q) 0.8 information uncertainty 0.6 p = 1 , q = 0 ⇒ 1 1 0.4 H = p log b p + q log b q . H = 0 — no uncer- 0.2 tainty 0.0 0.2 0.4 0.6 0.8 1.0 p This is Shannon entropy — a measure of uncertainty. In general, define the Shannon entropy 2 as N S.Will, 18.417, Fall 2011 � H ( � p ) := − p i log b p i . i =1 2 of a probability distribution � p over N states X 1 . . . X N

Shannon Entropy (by Example) We toss a coin. For our coin, heads and tails show up with respective probabilities p and q (not necessarily fair). How uncertain are we about the result? p = 0 . 5 , q = 0 . 5 ⇒ Answer: expected 1.0 H = 1 — maximal p * log2(1/p) + q * log2(1/q) 0.8 information uncertainty 0.6 p = 1 , q = 0 ⇒ 1 1 0.4 H = p log b p + q log b q . H = 0 — no uncer- 0.2 tainty 0.0 0.2 0.4 0.6 0.8 1.0 p This is Shannon entropy — a measure of uncertainty. In general, define the Shannon entropy 2 as N S.Will, 18.417, Fall 2011 � H ( � p ) := − p i log b p i . i =1 2 of a probability distribution � p over N states X 1 . . . X N

Shannon Entropy (by Example) We toss a coin. For our coin, heads and tails show up with respective probabilities p and q (not necessarily fair). How uncertain are we about the result? p = 0 . 5 , q = 0 . 5 ⇒ Answer: expected 1.0 H = 1 — maximal p * log2(1/p) + q * log2(1/q) 0.8 information uncertainty 0.6 p = 1 , q = 0 ⇒ 1 1 0.4 H = p log b p + q log b q . H = 0 — no uncer- 0.2 tainty 0.0 0.2 0.4 0.6 0.8 1.0 p This is Shannon entropy — a measure of uncertainty. In general, define the Shannon entropy 2 as N S.Will, 18.417, Fall 2011 � H ( � p ) := − p i log b p i . i =1 2 of a probability distribution � p over N states X 1 . . . X N

Formalizing “Least number of assumptions” Example: Assume: we have N events. Without further assumptions, we will naturally assume the uniform distribution p i = 1 N . This is the uniquely defined distribution maximizing the entropy H ( � p ) = − � i p i log b p i . It is found by solving the following optimization problem: maximize the function � H ( � p ) = − p i log b p i S.Will, 18.417, Fall 2011 i under the side condition � i p i = 1 .

Formalizing “Least number of assumptions” Theorem: Given a system of states X 1 . . . X N and energies E i for X i . The Boltzmann distribution is the probability distribution � p that maximizes Shannon entropy N � H ( � p ) = − p i log b p i i =1 under the assumption of known average energy of the system N � < E > = p i E i . i =1 S.Will, 18.417, Fall 2011

Proof We show that the Boltzmann distribution is uniquely obtained by solving N � 3 maximize function H ( � p ) = − p i ln p i i =1 under the side conditions • C 1 ( � p ) = � i p i − 1 = 0 and • C 2 ( � p ) = � i p i E i − < E > = 0 by using the method of Lagrange multipliers. S.Will, 18.417, Fall 2011 3 whether using ln or log b is equivalent for maximization