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Statistical

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YE

DING

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REVIEW

Statistical

and

Bayesian

approaches

to

RNA

secondary

structure

prediction

YE

DING

Wadsworth

Center

New

York

State

Department

of

Health

Albany

NY

USA

ABSTRACT

Prediction

of

RNA

secondary

structure

is

a

fundamental

problem

in

computational

structural

biology

For

several

decades

free

energy

minimization

has

been

the

most

popular

method

for

prediction

from

a

single

sequence

In

recent

years

the

McCaskill

algorithm

for

computation

of

partition

function

and

base-pair

probabilities

has

become

increasingly

appreciated

This

paradigmshifting

work

has

inspired

the

developments

of

extended

partition

function

algorithms

statistical

sampling

and

clustering

and

application

of

Bayesian

statistical

inference

The

performance

of

thermodynamics-based

methods

is

limited

by

thermodynamic

rules

and

parameters

However

further

improvements

may

come

from

statistical

estimates

derived

from

structural

databases

for

thermodynamics

parameters

with

weak

or

little

experimental

data

The

Bayesian

inference

approach

appears

to

be

promising

in

this

context

Keywords

Boltzmann

ensemble

partition

function

sampling

Bayesian

inference

clustering

RNA

secondary

structure

INTRODUCTION

RNA

molecules

are

involved

in

some

of

the

cell

's

most

fundamental

processes

that

include

catalysis

pre-mRNA

splicing

and

RNA

editing

and

regulation

of

transcription

and

translation

To

a

large

degree

the

function

of

a

regulatory

RNA

molecule

is

determined

by

its

structure

Computational

methods

for

modeling

RNA

secondary

structure

provide

useful

initial

models

for

solving

the

tertiary

structure

by

crystallography

or

nuclear

magnetic

resonance

-LRB-

NMR

-RRB-

The

problem

of

computational

prediction

of

secondary

structure

for

a

single

RNA

sequence

dates

back

to

the

early

-LRB-

Tinoco

et

al.

-RRB-

Free

energy

minimization

has

been

the

most

popular

method

for

such

prediction

A

review

of

the

developments

of

this

paradigm

for

RNA

folding

can

be

found

elsewhere

-LRB-

Zuker

-RRB-

The

partition

function

approach

by

McCaskill

enables

rigorous

computation

of

base-pair

probabilities

and

heat

capacity

-LRB-

McCaskill

-RRB-

In

recent

years

there

has

been

increasing

interest

in

ensemble-base

approaches

that

extend

the

pioneering

work

of

McCaskill

This

review

is

focused

on

discussing

these

recent

developments

The

Bayesian

statistical

inference

approach

has

proven

to

be

highly

valuable

for

numerous

computational

biology

problems

A

Bayesian

framework

is

outlined

for

tackling

the

problem

of

statistical

estiReprint

requests

to

Ye

Ding

Wadsworth

Center

New

York

State

Department

of

Health

Center

for

Medical

Science

New

Scotland

Avenue

Albany

NY

USA

e-mail

yding@wadsworth.org

fax

-LRB-518-RRB- 402-4623

Article

and

publication

are

at

rna

mation

of

thermodynamic

parameters

using

RNA

structure

databases

Here

the

single

sequence

problem

is

the

primary

concern

Reviews

of

methods

based

on

covariation

analysis

of

homologous

RNAs

can

be

found

elsewhere

-LRB-

Zuker

Gardner

and

Giegerich

-RRB-

FREE

ENERGY

MINIMIZATION

PARADIGM

In

structural

computational

biology

free

energy

minimization

for

prediction

of

macromolecular

folding

is

a

longestablished

paradigm

It

assumes

that

at

equilibrium

the

solution

to

the

underlying

molecular

folding

problem

is

unique

and

that

the

molecule

folds

into

the

lowest

energy

state

Also

implicitly

assumed

is

that

the

free

energies

of

individual

structural

motifs

are

additive

This

paradigm

has

been

the

foundation

for

prediction

of

RNA

secondary

structure

for

over

three

decades

-LRB-

Tinoco

et

al.

Nussinov

and

Jacobson

Zuker

and

Stiegler

Mathews

et

al.

-RRB-

Other

applications

include

protein

folding

-LRB-

Anfinsen

Abagyan

-RRB-

and

transmembrane

helix

packing

-LRB-

Pappu

et

al.

-RRB-

For

RNA

secondary

structure

prediction

free

energy

parameters

for

basic

structural

motifs

are

estimated

or

extrapolated

from

chemical

melting

experiments

-LRB-

Xia

et

al.

Mathews

et

al.

-RRB-

The

discrete

optimization

problem

is

ill

conditioned

in

that

the

prediction

is

sensitive

to

small

changes

in

the

energy

parameters

-LRB-

Zuker

Layton

and

Bundschuh

-RRB-

Furthermore

there

is

substantial

uncertainty

in

the

energy

parameters

particularly

for

loops

For

these

reasons

efficient

algorithms

have

been

developed

for

not

only

RNA

-LRB-

-RRB-

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by

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Ding

TABLE

RNA

folding

programs

based

on

free

energy

minimization

Name

mfold

RNAstructure

Vienna

RNA

package

RNAshapes

pknotsRG

PKNOTS

RNALOSS

a

Functionsa

Folding

of

single

and

interacting

nucleic

acids

Window

software

for

RNA

folding

and

oligo

design

Optimal

and

complete

suboptimal

folding

Abstract

shape

representation

of

Vienna

suboptimal

foldings

Folding

including

a

class

of

simple

pseudoknots

Folding

including

a

class

of

pseudoknots

Locally

optimal

folding

URL

Only

directly

relevant

functions

are

listed

computing

the

minimum

free

energy

-LRB-

MFE

-RRB-

structure

but

also

for

generating

a

heuristic

set

of

suboptimal

structures

-LRB-

Zuker

and

Stiegler

Mathews

et

al.

-RRB-

An

alternative

approach

computes

all

suboptimal

foldings

within

an

energy

increment

above

the

MFE

-LRB-

Wuchty

et

al.

-RRB-

The

exponential

growth

in

the

number

of

these

foldings

motivated

recent

development

of

the

RNAshapes

method

for

the

efficient

representation

of

the

near-optimal

foldings

-LRB-

Giegerich

et

al.

-RRB-

The

complete

suboptimal

approach

addresses

the

low-energy

end

of

the

unweighted

energy

landscape

Neither

approach

guarantees

an

unbiased

representation

of

the

Boltzmann-weighted

ensemble

The

free

energy

minimization

algorithm

-LRB-

Zuker

and

Stiegler

-RRB-

and

the

algorithm

for

computing

suboptimal

structures

-LRB-

Wuchty

et

al.

-RRB-

have

been

extended

for

two

or

more

interacting

RNAs

-LRB-

Andronescu

et

al.

-RRB-

For

the

simple

Nussinov

Jacobson

energy

model

of

constant

basepair

energies

-LRB-

Nussinov

and

Jacobson

-RRB-

an

efficient

algorithm

has

been

developed

for

computation

of

the

number

of

structures

with

k

fewer

base

pairs

than

the

maximum

number

-LRB-

Clote

-RRB-

Free

energy

minimization

algorithms

have

also

been

developed

to

include

certain

types

of

pseudoknots

-LRB-

Rivas

and

Eddy

Reeder

and

Giegerich

-RRB-

but

applications

are

limited

to

short

or

moderate-length

sequences

depending

on

the

time

complexity

and

the

memory

requirement

of

the

particular

algorithm

A

list

of

the

programs

for

implementing

these

algorithms

is

presented

by

Table

PARTITION

FUNCTION

APPROACH

In

a

drastic

departure

from

free

energy

minimization

the

partition

function

approach

pioneered

by

McCaskill

-LRB-

-RRB-

laid

the

foundation

for

statistical

characterizations

of

the

equilibrium

ensemble

of

RNA

secondary

structures

In

particular

base-pair

probabilities

can

be

calculated

Similar

to

its

MFE

counterpart

the

algorithm

for

computing

partition

function

and

base-pair

probabilities

is

cubic

and

requires

quadratic

storage

The

significance

of

base-pair

probabilities

has

been

further

demonstrated

in

two

recent

studies

For

base

pairs

in

the

MFE

structure

those

with

higher

probabilities

have

higher

predictive

accuracy

measured

by

RNA

Vol

No.

positive

predictive

value

-LRB-

Mathews

-RRB-

The

positive

predictive

value

is

the

percentage

of

base

pairs

in

the

predicted

structure

that

are

in

the

structure

determined

by

comparative

sequence

analysis

Thus

base-pair

probabilities

provide

measures

of

confidence

for

MFE

predictions

That

study

was

based

on

a

new

partition

function

algorithm

that

accommodates

coaxial

stacking

and

more

recent

energy

parameters

Furthermore

base-pair

probabilities

are

found

to

be

less

affected

by

uncertainties

in

energy

parameters

than

is

the

MFE

structure

-LRB-

Layton

and

Bundschuh

-RRB-

The

McCaskill

algorithm

has

also

been

extended

to

include

a

class

of

pseudoknots

-LRB-

Dirks

and

Pierce

-RRB-

Like

the

partition

function

the

mean

and

variance

-LRB-

and

any

moments

in

general

-RRB-

of

the

Boltzmann-weighted

free

energy

distribution

can

be

calculated

and

these

ensemble

characteristics

are

reported

to

be

useful

for

distinguishing

biological

sequences

from

random

sequences

-LRB-

Miklos

et

al.

-RRB-

A

partition

function

algorithm

for

k-point

mutants

of

an

RNA

sequence

has

recently

been

described

-LRB-

Clote

et

al.

-RRB-

For

modeling

the

hybridization

of

two

nucleic

acid

molecules

the

Zuker

group

was

the

first

to

compute

partition

function

and

base-pair

probabilities

-LRB-

Dimitrov

and

Zuker

-RRB-

These

developments

are

indicative

of

the

recent

surge

in

interest

in

the

ensemble-based

approaches

Table

presents

a

list

of

programs

for

implementing

the

partition

function

algorithms

and

the

extensions

below

with

comprehensive

Turner

free

energy

parameters

STATISTICAL

SAMPLING

APPROACH

In

the

traceback

step

of

an

RNA

folding

algorithm

base

pairs

are

generated

one

at

a

time

according

a

chosen

principle

-LRB-

e.g.

energy

minimization

or

probabilistic

sampling

as

discussed

below

-RRB-

to

form

a

secondary

structure

The

longstanding

problem

of

a

statistical

representation

of

probable

foldings

can

be

addressed

by

a

sampling

extension

of

the

partition

function

approach

-LRB-

Ding

and

Lawrence

-RRB-

In

the

traceback

step

the

conditional

probabilities

computed

with

partition

functions

are

used

to

sample

a

new

base

pair

or

unpaired

base

-LRB-

s

-RRB-

given

partially

formed

structure

Thus

the

essence

of

the

sampling

algorithm

is

stochastic

traceback

This

algorithm

generates

a

sample

of

secondary

struc

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Statistical

methods

for

RNA

folding

TABLE

RNA

folding

programs

for

characterizing

Boltzmann

ensemble

of

RNA

secondary

structures

using

comprehensive

Turner

free

energy

parameters

Name

Sfold

RNAstructure

Vienna

RNA

package

NUPACK

a

Functionsa

Statistical

sampling

and

clustering

and

rational

design

of

nucleic

acids

Partition

function

and

base-pair

probabilities

Partition

function

and

base-pair

probabilities

Partition

function

and

base-pair

probabilities

including

a

class

of

pseudoknots

URL

Only

directly

relevant

functions

are

listed

tures

in

proportion

to

their

Boltzmann

weights

guaranteeing

a

statistical

representation

of

the

Boltzmann-weighted

ensemble

A

statistical

sample

of

the

ensemble

allows

sampling

estimates

of

the

probabilities

of

any

structural

motifs

from

the

simplest

elements

of

base

pair

and

unpaired

base

to

loops

of

various

types

to

more

complex

structures

consisting

of

stems

and

loops

that

may

be

of

special

interest

in

a

given

application

In

particular

probability

profiling

of

single-stranded

regions

in

RNA

secondary

structure

is

directly

applicable

to

the

rational

design

of

mRNA-targeting

nucleic

acids

-LRB-

Ding

and

Lawrence

Ding

Ding

et

al.

-RRB-

The

Boltzmann-weighted

density

of

states

-LRB-

BWDOS

-RRB-

-LRB-

Ding

and

Lawrence

-RRB-

characterizes

the

weighted

energy

landscape

whereas

a

densityof-states

algorithm

-LRB-

Cupal

et

al.

-RRB-

applicable

only

to

short

sequences

describes

the

unweighted

landscape

A

structure

sample

can

also

be

used

for

computation

of

other

characteristics

of

the

Boltzmann

ensemble

For

example

the

mean

and

the

variance

of

the

free

energy

distribution

can

be

estimated

by

a

sample

whereas

exact

calculations

require

laborious

algorithm

development

-LRB-

Miklos

et

al.

-RRB-

In

principle

a

sampling

extension

can

also

be

developed

for

a

partition

function

algorithm

including

pseudoknots

In

this

case

base-pair

probabilities

can

be

estimated

by

a

sample

and

the

estimates

should

closely

approximate

those

computed

by

a

high-order

algorithm

-LRB-

Dirks

and

Pierce

-RRB-

A

sample

of

moderate

size

drawn

from

the

ensemble

of

an

enormous

number

of

possible

structures

is

sufficient

to

guarantee

statistical

reproducibility

in

the

estimates

of

typical

sampling

statistics

The

reproducibility

is

best

demonstrated

when

two

independent

samples

do

not

have

a

single

structure

in

common

-LRB-

Ding

and

Lawrence

-RRB-

These

seemingly

surprising

observations

are

fully

expected

for

an

exact

sampling

algorithm

In

a

recent

study

on

both

structural

RNAs

and

mRNAs

-LRB-

D.

Mathews

pers

comm

-RRB-

base-pair

probabilities

estimated

by

the

following

three

methods

were

compared

to

those

computed

by

the

partition

function

approach

-LRB-

McCaskill

Mathews

-RRB-

-LRB-

-RRB-

the

heuristic

set

of

suboptimal

foldings

from

the

mfold

program

-LRB-

Zuker

-RRB-

-LRB-

-RRB-

the

complete

suboptimal

foldings

-LRB-

Wuchty

et

al.

-RRB-

and

-LRB-

-RRB-

statistical

sampling

-LRB-

Ding

and

Lawrence

-RRB-

The

same

thermodynamic

parameters

and

energy

functions

-LRB-

Mathews

et

al.

-RRB-

are

used

in

implementing

all

these

methods

These

three

methods

generate

different

sets

of

structures

while

the

partition

function

approach

does

not

generate

a

single

structure

However

the

partition

function

method

does

offer

exact

base-pair

probabilities

for

the

Boltzmann

ensemble

Thus

to

assess

how

well

a

set

of

structures

represents

the

Boltzmann

ensemble

the

estimates

of

the

base-pair

probabilities

using

this

set

can

be

compared

with

the

exact

probabilities

It

was

found

from

this

study

that

the

sampling

method

makes

far

better

estimates

than

do

the

other

two

methods

In

particular

a

small

sample

of

only

structures

performs

far

better

than

does

a

complete

set

of

suboptimal

structures

within

kT

of

the

lowest

free

energy

structure

The

improvement

by

sampling

is

over

an

order

of

magnitude

as

measured

by

the

square

root

of

the

mean

square

deviation

-LRB-

RMSD

-RRB-

The

major

reason

for

this

result

is

that

for

sequences

of

several

hundred

bases

or

longer

the

nearoptimal

foldings

constitute

only

a

small

portion

of

the

Boltzmann

ensemble

This

is

illustrated

in

Figure

for

an

RNase

P

sequence

there

is

little

overlap

between

the

BWDOS

for

a

sample

of

structures

and

the

unweighted

density

of

states

for

the

structures

with

the

lowest

energies

Another

reason

for

the

above

result

is

that

there

exist

entropic

clusters

in

the

ensemble

-LRB-

Ding

and

Lawrence

-RRB-

In

such

a

cluster

each

structure

has

a

much

higher

energy

-LRB-

lower

probability

-RRB-

than

does

the

MFE

however

the

cluster

will

be

represented

in

a

sample

simply

as

a

result

of

the

sheer

huge

number

of

structures

in

the

cluster

so

that

the

cluster

has

a

substantial

probability

-LRB-

sum

of

probabilities

of

individual

structures

in

the

cluster

-RRB-

CLUSTER

REPRESENTATION

OF

BOLTZMANN

ENSEMBLE

In

the

sampled

ensemble

distinct

structural

clusters

were

observed

-LRB-

Ding

and

Lawrence

-RRB-

This

observation

suggests

that

the

Boltzmann

ensemble

can

be

efficiently

represented

by

clusters

An

automated

clustering

procedure

and

tools

have

recently

been

developed

for

this

purpose

-LRB-

Chan

Downloaded

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by

Cold

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Laboratory

Press

Ding

ter

centroid

make

structural

predictions

that

are

substantially

improved

over

the

MFE

predictions

-LRB-

Ding

et

al.

-RRB-

a

result

that

further

validates

ensemblebased

approaches

In

a

recent

comparison

between

mRNAs

and

structural

RNAs

-LRB-

Y.

Ding

C.Y.

Chan

and

C.E.

Lawrence

in

prep

-RRB-

similarity

was

observed

for

the

number

of

clusters

and

the

energy

gap

between

the

MFE

structure

and

the

sampled

ensemble

However

for

structural

RNAs

there

are

more

high-frequency

base

pairs

in

both

the

Boltzmann

ensemble

and

the

clusters

and

the

clusters

are

more

compact

Thus

clustering

provides

a

new

method

for

detecting

differences

between

structural

and

nonstructural

RNAs

and

between

biological

RNA

sequences

and

random

sequences

-LRB-

Y.

Ding

C.Y.

Chan

and

C.E.

Lawrence

in

prep

-RRB-

FIGURE

The

energy

distribution

of

the

lowest

energy

structures

-LRB-

gray

bars

i.e.

the

partial

density

of

states

for

free

energy

below

kcal/mol

kcal/mol

above

the

MFE

-RRB-

and

the

energy

distribution

of

sampled

structures

-LRB-

black

bars

i.e.

the

density

of

states

of

the

weighted

ensemble

-RRB-

for

Heliobacterium

chlorum

RNase

P

RNA

of

nt

in

length

-LRB-

GenBank

accession

-RRB-

The

structure

determined

by

comparative

analysis

is

from

the

RNase

P

database

-LRB-

Brown

-RRB-

and

does

not

contain

pseudoknotted

base

pairs

For

a

comparison

based

on

the

same

implementation

of

the

thermodynamic

rules

all

other

structures

here

and

the

free

energies

of

all

the

structures

were

computed

with

the

Vienna

package

-LRB-

Hofacker

-RRB-

BAYESIAN

STATISTICAL

INFERENCE

APPROACH

et

al.

Ding

et

al.

Y.

Ding

C.Y.

Chan

and

C.E.

Lawrence

in

prep

-RRB-

The

procedure

returns

three

to

four

clusters

on

average

Another

advantage

of

clustering

is

that

the

centroid

structure

as

the

single

best

representative

of

the

cluster

can

be

easily

identified

with

little

computational

cost

The

centroid

of

any

set

of

structures

is

defined

as

the

Bayesian

inference

paradigm

structure

in

the

whole

ensemble

that

has

the

shortest

total

Bayesian

inference

methods

treat

all

quantities

in

a

problem

distance

to

structures

in

the

set

For

the

base-pair

distance

between

two

structures

the

centroid

is

simply

the

structure

as

random

variables

A

Bayesian

method

starts

with

the

specification

of

a

joint

distribution

of

all

quantities

of

formed

by

all

base

pairs

having

a

frequency

in

the

strucinterest

Basic

probability

rules

are

applied

to

derive

the

ture

set

-LRB-

Ding

et

al.

-RRB-

The

clusters

together

with

their

probabilities

-LRB-

estimated

by

frequencies

in

the

sample

-RRB-

and

their

centroids

present

a

complete

and

efficient

statistical

characterization

of

the

Boltzmann

ensemble

-LRB-

Fig.

-RRB-

Similar

to

the

reproducibility

of

ensemble-level

sampling

statistics

-LRB-

Ding

and

Lawrence

-RRB-

the

clusters

and

centroids

are

also

statistically

reproducible

from

one

sample

to

another

even

when

the

two

independent

samples

do

not

share

a

single

structure

-LRB-

Y.

Ding

C.Y.

Chan

and

C.E.

Lawrence

in

prep

-RRB-

It

FIGURE

Efficient

representation

of

the

Boltzmann

ensemble

of

secondary

structures

for

an

RNA

molecule

by

clusters

identified

from

a

statistically

representative

sample

of

structures

The

was

a

surprising

finding

that

the

centroid

size

or

probability

of

a

cluster

is

estimated

by

the

frequency

of

occurrence

of

the

cluster

in

the

of

the

sampled

ensemble

and

the

best

clus

sample

and

a

representative

of

the

cluster

is

its

centroid

structure

-LRB-

Ding

et

al.

-RRB-

RNA

Vol

No.

Since

the

early

Bayesian

methods

have

been

applied

to

a

wide

range

of

problems

in

the

burgeoning

fields

of

bioinformatics

and

genomics

The

Bayesian

Revolution

-LRB-

Beaumont

and

Rannala

Eddy

-RRB-

can

be

partly

credited

to

the

conceptual

simplicity

of

the

Bayesian

approach

-LRB-

as

opposed

to

the

difficulties

of

the

classic

frequentist

approach

Gelman

et

al.

-RRB-

and

partly

to

advances

in

computing

power

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Statistical

methods

for

RNA

folding

posterior

distributions

of

the

unknown

quantities

given

the

observed

data

Inferential

statements

are

made

rigorously

from

these

posterior

distributions

-LRB-

Gelman

et

al.

-RRB-

While

these

posterior

distributions

are

usually

easy

to

write

down

they

require

intensive

computations

that

have

become

feasible

only

with

advancing

computational

power

The

Bayesian

method

has

been

successfully

used

in

regulatory

motif

search

-LRB-

Lawrence

et

al.

Thompson

et

al.

-RRB-

sequence

alignment

-LRB-

Zhu

et

al.

-RRB-

and

sequence

segmentation

-LRB-

Liu

and

Lawrence

-RRB-

and

it

has

the

potential

to

provide

a

powerful

solution

to

the

general

problem

of

missing

data

in

the

analysis

of

biopolymer

sequences

-LRB-

Liu

and

Lawrence

-RRB-

Excellent

textbooks

on

Bayesian

analysis

are

available

-LRB-

Berger

Gelman

et

al.

Carlin

and

Louis

Liu

Jaynes

-RRB-

The

motivation

for

and

the

framework

of

the

Bayesian

methodology

are

outlined

below

The

focus

of

all

statistics

is

on

making

inferences

The

concept

of

statistical

inference

closely

follows

the

dictionary

definition

of

inference

the

process

of

deriving

a

conclusion

from

fact

and/or

premise

In

statistics

the

facts

are

the

observed

data

the

premise

is

represented

by

a

probabilistic

model

of

the

system

of

interest

and

the

conclusions

concern

unobserved

quantities

In

classical

statistics

inferences

are

made

by

finding

point

estimates

of

unknown

variables

with

maximum

likelihood

estimates

being

the

most

common

type

of

estimates

Uncertainty

is

addressed

by

setting

confidence

limits

on

these

estimates

Bayesian

statistics

has

a

more

ambitious

goal

to

find

the

probability

distribution

of

all

unknown

variables

after

considering

the

data

The

full

process

of

a

typical

Bayesian

analysis

can

be

described

as

consisting

of

three

main

steps

-LRB-

-RRB-

setting

up

a

full

probability

model

that

includes

all

of

the

variables

so

as

to

capture

the

relationships

among

these

variables

-LRB-

-RRB-

summarizing

the

findings

for

particular

quantities

of

interest

by

appropriate

posterior

distributions

and

-LRB-

-RRB-

evaluating

the

appropriateness

of

the

model

and

suggesting

improvements

-LRB-

Gelman

et

al.

-RRB-

A

standard

procedure

for

carrying

out

Step

is

to

first

write

down

the

likelihood

function

i.e.

the

probability

of

the

observed

data

given

the

unknowns

and

multiply

it

by

a

prior

distribution

i.e.

a

distribution

for

all

of

the

unobserved

variables

-LRB-

typically

unknown

parameters

-RRB-

Let

y

denote

the

observed

data

and

u

the

unobserved

parameter

The

joint

probability

distribution

is

represented

as

joint

likelihood

prior

i.e.

p

-LRB-

y

-RRB-

p

-LRB-

y

-RRB-

p

-LRB-

-RRB-

where

the

prior

distribution

on

u

p

-LRB-

u

-RRB-

reveals

what

is

known

about

the

parameter

without

the

knowledge

of

the

data

p

-LRB-

y

-RRB-

is

often

denoted

as

l

-LRB-

u

y

-RRB-

and

is

referred

to

as

the

likelihood

in

classical

statistics

that

is

based

on

a

model

of

the

underlying

process

Bayesian

inference

is

drawn

by

examination

of

the

probability

of

all

possible

values

of

the

parameter

after

consid

eration

of

the

data

Accordingly

Step

is

completed

by

obtaining

the

posterior

distribution

i.e.

the

conditional

distribution

of

the

parameter

given

the

data

through

the

application

of

Bayes

's

Theorem

to

the

joint

distribution

p

j

y

p

y

p

y

j

p

p

y

j

p

p

y

p

y

where

the

posterior

distribution

p

-LRB-

u

y

-RRB-

tells

us

what

is

known

about

u

given

knowledge

of

the

data

Thus

the

primary

tasks

of

Bayesian

analysis

are

to

develop

a

model

for

p

-LRB-

u

y

-RRB-

and

to

perform

computation

necessary

to

summarize

p

-LRB-

u

y

-RRB-

The

computation

of

the

marginal

likelihood

p

-LRB-

y

-RRB-

is

typically

the

most

challenging

part

of

the

Bayesian

analysis

The

popular

Gibbs

sampler

and

related

Monte

Carlo

methods

can

aid

Bayesian

computation

-LRB-

Liu

-RRB-

Bayesian

inference

can

not

only

be

readily

applied

to

any

probability

model

but

it

can

also

avoid

the

difficulties

of

asymptotic

inferences

for

interval

estimates

that

are

encountered

by

classical

statistics

It

provides

a

satisfactory

way

of

explicitly

introducing

and

keeping

track

of

assumptions

about

prior

knowledge

or

ignorance

When

no

or

minimal

prior

information

is

available

about

the

parameter

a

noninformative

prior

can

be

considered

in

the

application

of

the

Bayesian

machinery

for

making

inferences

Typically

a

uniform

density

giving

equal

weight

to

all

possible

values

of

the

parameter

is

used

for

this

purpose

For

a

continuous

parameter

defined

on

an

infinite

interval

the

uniform

density

is

referred

to

as

an

improper

prior

because

it

does

not

integrate

to

Use

of

an

improper

prior

can

lead

to

a

proper

posterior

distribution

but

caution

must

be

exercised

in

interpretation

of

the

results

-LRB-

Gelman

et

al.

-RRB-

In

this

case

the

posterior

mode

is

the

maximum

likelihood

estimate

because

the

posterior

distribution

is

proportional

to

the

likelihood

function

In

many

situations

noninformative

prior

Bayesian

inference

could

be

argued

to

be

the

single

most

powerful

method

of

statistical

analysis

-LRB-

Berger

-RRB-

Applications

to

RNA

structure

predictions

From

the

perspective

of

statistical

mechanics

the

secondary

structure

of

an

RNA

sequence

is

a

random

variable

that

follows

the

Boltzmann

equilibrium

distribution

From

a

full

Bayesian

viewpoint

the

Boltzmann

equilibrium

distribution

is

in

fact

a

conditional

probability

distribution

given

the

RNA

sequence

data

and

the

thermodynamic

parameters

-LRB-

Ding

and

Lawrence

-RRB-

The

Bayesian

inference

approach

was

first

applied

to

the

RNA

folding

problem

for

a

single

sequence

-LRB-

Ding

and

Lawrence

-RRB-

In

this

application

the

set

of

random

variables

includes

the

primary

sequence

data

-LRB-

observed

-RRB-

the

unknown

secondary

structure

and

number

of

destabilizing

loops

and

free

energy

parameters

Based

on

the

stacking

energy

model

and

noninformative

priors

the

Bayesian

algorithm

returns

posterior

distributions

for

the

number

of

destabilizing

loops

stacking

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Ding

energy

matrices

and

secondary

structures

In

particular

the

idea

of

generating

a

statistically

representative

sample

of

RNA

secondary

structures

after

partition

function

calculation

was

presented

for

the

first

time

The

Bayesian

approach

has

also

been

applied

to

the

prediction

of

the

common

secondary

structure

for

homologous

sequences

-LRB-

Knight

et

al.

-RRB-

This

procedure

works

on

a

reduced

structural

space

by

considering

a

set

of

foldings

based

on

thermodynamic

predictions

made

by

the

RNAsubopt

program

of

the

Vienna

package

-LRB-

Hofacker

-RRB-

Posterior

inference

is

performed

using

Bayes

's

Theorem

given

thermodynamic

predictions

alignment

-LRB-

mutual

information

-RRB-

and

chemical

mapping

information

and

under

the

assumptions

of

uniform

prior

for

the

structures

in

the

reduced

space

and

conditional

independence

for

data

from

different

sources

The

alternative

of

modeling

the

full

structure

space

as

considered

for

the

one-sequence

problem

-LRB-

Ding

and

Lawrence

-RRB-

appears

to

be

much

more

challenging

to

implement

for

the

multiple-sequence

problem

STATISTICAL

ESTIMATION

OF

THERMODYNAMIC

PARAMETERS

In

terms

of

free

energy

the

sampled

ensemble

can

be

much

closer

to

the

structure

determined

by

comparative

analysis

than

are

the

MFE

structure

and

the

near-optimal

structures

-LRB-

Fig.

-RRB-

nevertheless

there

is

much

room

for

improvement

If

the

free

energy

function

were

complete

and

if

all

thermodynamic

parameters

were

accurate

the

biological

structure

could

often

be

expected

to

have

the

lowest

energy

However

the

thermodynamics

parameters

for

RNA

for

which

thermodynamics

is

arguably

best

studied

among

macromolecules

are

incomplete

In

particular

extrapolations

for

large

loops

are

currently

necessary

Thus

any

free-energybased

algorithm

is

limited

by

the

parameters

used

Structure

predictions

might

be

improved

through

kinetic

modeling

of

RNA

folding

-LRB-

Doshi

et

al.

-RRB-

and

through

improvements

in

the

accuracy

of

the

free

energy

model

While

the

set

of

experimental

parameters

is

expected

to

continue

to

improve

an

alternative

is

to

statistically

estimate

the

parameters

by

taking

advantage

of

RNA

structure

databases

-LRB-

Larsen

and

Zwieb

Brown

Cannone

et

al.

Rosenblad

et

al.

Zwieb

et

al.

Sprinzl

and

Vassilenko

-RRB-

This

strategy

was

first

considered

by

Michael

Zuker

in

the

In

numerous

presentations

-LRB-

including

the

Computational

RNA

Workshop

in

Benasque

Spain

-RRB-

Zuker

described

pseudo

energy

rules

for

base-pair

stacking

and

small

structural

motifs

that

are

derived

by

computing

the

ratio

of

the

observed

frequencies

over

the

expected

frequencies

from

random

background

Essentially

based

on

the

same

framework

the

published

statistical

energies

for

base-pair

stacking

are

in

excellent

agreement

with

the

well-studied

experimental

energies

-LRB-

Dima

et

al.

-RRB-

suggesting

the

potential

of

such

RNA

Vol

No.

edge-based

approaches

to

improve

estimates

of

energies

for

loops

and

other

secondary

structural

motifs

The

idea

of

estimating

energy

parameters

with

known

structural

information

does

follow

the

way

a

Bayesian

thinks

about

a

problem

It

has

been

shown

that

the

primary

sequence

data

the

secondary

structure

and

the

free

energy

parameters

can

be

described

together

through

Bayesian

statistical

modeling

-LRB-

Ding

and

Lawrence

-RRB-

Like

the

relationships

among

pressure

volume

and

absolute

temperature

for

an

ideal

gas

of

a

fixed

number

of

moles

inference

on

any

variable

can

be

made

from

the

knowledge

of

the

other

two

variables

Analogously

databases

containing

secondary

structural

information

can

be

used

to

make

inferences

on

the

thermodynamic

parameters

The

accuracy

of

the

estimates

obviously

rests

on

the

assumption

that

the

structural

information

from

comparative

analysis

is

reliable

For

and

ribosomal

RNAs

comparative

analysis

identifies

nearly

all

of

the

base

pairs

present

in

the

crystal

structures

but

also

predicts

some

base

pairs

that

are

absent

in

the

crystal

structures

-LRB-

Cannone

et

al.

-RRB-

For

statistical

inference

independence

of

observations

simplifies

the

formulation

of

a

probability

model

in

particular

the

likelihood

function

Correlated

observations

complicate

the

analysis

For

the

problem

of

multiple

sequence

alignment

two

methods

have

been

proposed

to

address

the

issue

of

sequence

correlation

-LRB-

or

redundancy

-RRB-

due

to

similarity

among

related

sequences

One

approach

is

to

recruit

diversified

sequences

so

as

to

minimize

correlation

This

approach

is

not

objective

and

results

in

loss

of

information

and

automation

is

essential

as

data

sets

grow

-LRB-

Vingron

and

Sibbald

-RRB-

Another

approach

is

to

use

a

sequence

weighting

scheme

that

assigns

higher

weights

to

more

distantly

related

sequences

based

on

the

distance

between

a

sequence

and

an

ancestral

or

generalized

sequence

-LRB-

Luthy

et

al.

Thompson

et

al.

-RRB-

alternatively

it

can

assign

weights

based

on

the

diversity

observed

at

each

position

in

the

alignment

-LRB-

Henikoff

and

Henikoff

-RRB-

Conceptually

inferences

for

thermodynamic

parameters

can

be

performed

for

any

sequence

in

a

database

or

for

a

set

of

sequences

It

is

advantageous

to

pool

information

from

diverse

sequences

for

improved

accuracy

in

the

estimates

The

Turner

parameter

values

can

be

used

for

prior

specification

e.g.

uniform

priors

on

intervals

centering

at

Turner

values

The

prior

specification

is

the

first

step

of

Bayesian

statistical

inference

For

the

posterior

distribution

of

a

parameter

a

global

peak

is

very

interesting

If

the

peak

is

near

the

Tuner

parameter

value

this

indicates

that

the

Turner

parameter

is

supported

by

the

structural

data

used

If

the

peak

is

away

from

the

Turner

value

this

suggests

that

the

value

can

be

corrected

by

the

posterior

mode

If

the

posterior

is

flat

on

the

interval

this

may

mean

that

there

remains

substantial

uncertainty

about

the

parameter

so

that

a

much

larger

data

set

may

be

necessary

for

an

estimate

with

improved

accuracy

Alternatively

a

flat

posterior

may

sug

Downloaded

from

on

May

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by

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Spring

Harbor

Laboratory

Press

Statistical

methods

for

RNA

folding

gest

that

the

parameter

is

not

important

if

predictions

are

not

sensitive

to

changes

in

this

parameter

For

the

large

number

of

free

energy

parameters

the

high

dimensionality

of

the

estimation

problem

will

pose

computational

challenges

To

reduce

dimensionality

a

divide

and

conquer

strategy

can

be

considered

that

focuses

on

one

class

of

motifs

when

energies

of

other

motifs

are

considered

to

be

reliable

and

thus

are

assumed

to

be

known

For

example

for

estimating

the

parameters

for

multibranched

loops

tRNA

sequences

-LRB-

with

the

implications

of

modified

bases

taken

into

consideration

as

further

discussed

below

-RRB-

may

be

considered

and

it

can

be

assumed

that

the

energies

for

base-pair

stacking

and

small

loops

are

known

and

accurate

The

values

of

the

estimated

energies

for

prediction

improvement

can

be

assessed

by

the

positive

predictive

values

and

sensitivity

for

MFE

structure

and

centroids

computed

with

these

estimates

The

sensitivity

for

a

predicted

structure

is

the

percentage

of

base

pairs

in

the

structure

determined

by

comparative

sequence

analysis

that

are

also

present

in

the

predicted

structure

It

is

well

understood

that

modified

nucleotides

-LRB-

Rozenski

et

al.

-RRB-

present

problems

for

structure

predictions

for

tRNAs

Some

of

the

modified

nucleotides

either

can

not

form

the

correct

set

of

hydrogen

bonds

or

can

not

stack

-LRB-

D.

Mathews

pers

comm

-RRB-

These

nucleotides

can

be

handled

by

forcing

them

to

be

unpaired

for

folding

and

a

list

of

such

nucleotides

has

been

published

-LRB-

Mathews

et

al.

-RRB-

Other

modified

nucleotides

can

still

form

canonical

pairs

but

with

a

high

likelihood

of

altered

strength

for

every

nucleotide

-LRB-

D.

Mathews

pers

comm

-RRB-

For

these

nucleotides

their

energetic

contributions

may

also

be

estimated

by

applying

the

Bayesian

framework

with

an

appropriate

assembly

of

tRNA

sequences

that

contain

a

particular

modification

of

interest

The

Bayesian

inference

process

is

illustrated

in

Figure

for

the

simplest

case

of

one

sequence

one

structure

and

one

parameter

For

multiple

sequences

and

parameters

the

construction

of

the

likelihood

must

take

into

account

multiple

observations

-LRB-

e.g.

through

the

assumption

of

independence

-RRB-

and

posterior

inference

can

generally

be

facilitated

by

Gibbs

sampler

and

related

Monte

Carlo

methods

-LRB-

Liu

-RRB-

In

some

cases

e.g.

in

discrete

modeling

of

the

base-pair

stacking

parameters

-LRB-

Ding

and

Lawrence

-RRB-

the

marginal

likelihood

can

be

exactly

computed

CONCLUSIONS

The

paradigm-shifting

work

by

McCaskill

has

inspired

the

recent

developments

of

extended

partition

function

algorithms

for

modeling

single

molecular

folding

and

hybridization

of

two

nucleic

acid

molecules

sampling

extension

and

clustering

representation

of

sampled

ensemble

These

methods

enable

characterizations

of

the

equilibrium

structure

ensemble

that

are

not

possible

with

the

use

of

free

energy

minimization

FIGURE

The

Bayesian

inference

framework

for

estimating

a

thermodynamic

parameter

with

the

use

of

RNA

structure

databases

The

essence

of

the

Bayesian

viewpoint

is

that

both

the

data

and

the

parameter

are

random

variables

Here

the

primary

RNA

sequence

R

and

the

known

secondary

structure

I

are

data

available

from

RNA

structure

databases

and

J

is

the

unknown

thermodynamic

parameter

Based

on

a

likelihood

function

constructed

with

an

appropriate

model

and

a

prior

distribution

on

the

parameter

the

joint

probability

distribution

of

all

variables

will

allow

Bayesian

inference

to

be

drawn

about

the

thermodynamic

parameter

through

the

posterior

distribution

of

J

given

the

data

i.e.

P

-LRB-

J

R

I

-RRB-

Computational

solution

of

a

macromolecular

folding

problem

requires

two

components

sufficiently

complete

and

accurate

free

energies

and

a

procedure

to

find

the

minimum

of

the

energy

function

-LRB-

Abagyan

-RRB-

The

energy

model

is

incomplete

for

RNA

secondary

structure

The

ensemble

is

important

to

consider

even

in

the

ideal

case

of

the

complete

energy

model

because

macromolecular

folding

may

not

always

be

governed

by

the

Anfinsen

hypothesis

of

lowest

energy

state

-LRB-

Anfinsen

-RRB-

For

RNA

in

the

rugged

energy

landscape

it

might

be

possible

that

one

of

the

structural

clusters

approximately

represents

an

energy

well

that

is

utilized

in

folding

to

the

native

structure

while

the

others

could

be

associated

with

kinetic

traps

and

metastable

-LRB-

misfolded

-RRB-

states

that

can

be

involved

in

an

RNA

conformational

switch

In

the

post-genomic

era

the

Bayesian

approach

has

proven

to

be

highly

useful

to

address

uncertainties

in

large

noisy

biological

data

sets

-LRB-

Eddy

-RRB-

The

Bayesian

inference

approach

appears

to

be

well

suited

to

the

problem

of

estimating

thermodynamic

parameters

from

RNA

structure

databases

The

application

of

Bayesian

methods

to

computational

RNA

problems

lags

behind

the

applications

of

these

methods

to

sequence

analysis

and

other

genetic

problems

-LRB-

Beaumont

and

Rannala

-RRB-

The

potential

of

the

highly

Downloaded

from

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by

Cold

Spring

Harbor

Laboratory

Press

Ding

flexible

Bayesian

framework

is

worthy

of

full

exploration

in

the

era

of

RNAomics

ACKNOWLEDGMENTS

The

Computational

Molecular

Biology

and

Statistics

Core

at

the

Wadsworth

Center

is

acknowledged

for

providing

computing

resources

This

work

was

supported

in

part

by

National

Science

Foundation

grant

and

National

Institutes

of

Health

grant

to

Y.D.

The

author

thanks

Clarence

Chan

for

preparing

Figure

and

Eric

Westhof

Dave

Mathews

and

Steve

Carmack

for

helpful

suggestions

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Res

O O O O O O O O O O O O O O O O O O O O O B O B O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O B O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O B O B O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O 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O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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