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RNA Design Optimization: A Survey and Recent

Advances

Denny Chen Dai

∗

Senior Supervisor: Kay C. Wiese

School of Computing Science

Simon Fraser University

cda18@cs.sfu.ca

Abstract

RNA design problem is a recently emerging research topic motivated by applica-

tions such as customized drug design and the self-assembly of RNA nano-objects.

This paper gives a survey of the recent advances in RNA design. We discuss the

empirical hardness of solving the problem as well as the combinatorial properties

of its underlying sequence-structure map. A literature review on existing algo-

rithmic solutions is given and comparisons are made among them. An algorithm

performance prediction model is introduced and its relevance to RNA design is

addressed. We conclude by proposing that RNA design could be extended into

a multi-objective optimization problem and this research topic is worth further

exploring.

1 Introduction

RNA is a single stranded sequence and this strand can fold back onto itself. Consider an RNA

molecule as a strand over four types of bases: Adenine (A), Cytosine (C), Guanine (G), and Uracil

(U). Intra-molecular base pairs can form bonds between different nucleotides (nt). AU and GC

are called the Watson-Crick base pairs which are most commonly found in RNAs. However, non-

Watson-Crick base pairing can also occur, for example, the GU and AC wobble pairs [22] [24]. In

summary, the most stable and commonly seen are GC, AU, and GU, and their mirrors, CG, UA, and

UG. These are called the canonical base pairs.

The RNA secondary structure is formed through base pairing between nucleotide bases at different

positions of the primary sequence. In nature, these pairing relations represent the hydrogen bonds

between nitrogens and free energy exists among them. Therefore, different secondary structures

imply different free energy levels. RNA secondary structure becomes stable under a particular en-

ergy level called the ground state, where the structure achieves a minimum free energy (MFE) level

among all possible secondary structure conformations.

In the classical RNA secondary structure prediction problem, we seek for a given RNA primary

sequence its corresponding MFE structure. Efﬁcient dynamic programming algorithms exist [16]

that ﬁnd the MFE structure in O(n

3

). However, this is achieved under a simpliﬁed free energy as-

sumption. In a complete energy model, the problem appears to be NP-hard. The reverse problem,

namely RNA Design, also appears to be NP hard [15] [26]. In RNA design, we seek for a given

structure conﬁguration, the target RNA primary sequence that would fold into this structure as its

MFE ground state. The ﬁnding of such sequence(s) involves exploring the exponentially large se-

quence space whose size far surpasses that of the structure space [27]. Recent advances in solving

∗

This work issubmitted as a supporting document for the Phd depth examination at the School of Computing

Science, Simon Fraser University.

1

the design problem are motivated by a number of promising applications such as customized drug

design and the self-assembly of RNA nano-objects.

This survey provides a brief overviewof the recent advances in RNA design. The rest of the paper is

organized as follows: in section 2, we introduce the RNA design problem and discuss issues related

to the empirical hardness of solving it; computationalmethods for estimating this empirical hardness

is also discussed; in section 3, we give a brief literature review on existing algorithmic solutions for

RNA design and presented our recently developed local search method, rnaDesign; in section 4,

we introduce the topic of algorithm performance prediction and presented a regression model for

RNA design prediction; in section 5, we discuss the feasibility of formalizing RNA design into a

multi-objective optimization problem and conclusions are drawn in section 6.

2 RNA Design Problem

RNA primary sequence is deﬁned as a string of length n over alphabet set Σ = {A, C, G, U },

representing the four nucleotide units. It is experimentally known that RNA sequences will fold into

a particular spatial structure in order to achieve certain biological functions. The secondary structure

of an RNA molecule is a coarse-grained simpliﬁcation of the more complex three dimensional RNA

tertiary structure. Formally, given a single stranded RNA sequence r of length N, where r =

(r

1

, r

2

, . . . , r

N

) and r

i

∈ {A, C, G, U } for which i ∈ [1, N ], a secondary structure is a set of

ordered base pairs (i, j), with 1 ≤ i < j ≤ N satisfying the following constraints:

1. j − i > 3, i.e. adjacent bases cannot be paired, and

2. {i, j} ∩ {i

0

, j

0

} = Ø, i.e. the base pairs do not conﬂict with each other.

In RNA design, we search for RNA primary sequences folding into a predeﬁnedsecondary structure.

In this problem, a target structure S

∗

is given which deﬁnes a set of pairing relationships among

nucleotides:

{(i, j)|i, j ∈ [1, N]} (1)

where N is the size of the structure, and i, j correspond to nucleotide locations where a pairing bond

exists. For a given RNA primary sequence r, a minimum free energy structure exists representing

the ground-state secondary structure for r:

S

o

r

= min{e(S

r

)|S

r

∈ Ω} (2)

Here we deﬁne Ω as the set of all possible structures over r and e(S

r

) as the thermodynamic energy

on S

r

. In the design problem, given the target structure S

∗

, the objective is to ﬁnd a sequence r,

whose MFE structure S

o

r

conforms to S

∗

:

d(S

o

r

, S

∗

) = 0 (3)

where d is the structure distance between two RNA secondary structures.

2.1 Combinatorial Properties and Empirical Hardness

Solving the RNA design problem requires searching the large combinatorial space of all possible

sequences for the one that folds into the predeﬁned structure. For any RNA secondary structure

consisting of n nucleotides, the underlying sequence space contains 4

n

unique candidates. It is also

empirically known that the total number of unique RNA secondary structures is much smaller, with

an estimated upper bound [15] of

(0.7137 ∗ n

3/2

∗ 2.2888

n

) (4)

Therefore, there exists a many-to-one mapping relationship between the sequence space and the

secondary structure space. A mapping between sequence r and structure s exists if and only if s is

the MFE structure of r. Here we do not consider the possibility of one sequence folding into two

2

MFE structures (RNA switches). Two sequences r

1

and r

2

that map to the same structure are called

neutral counterparts, indicating that both sequences share the same native MFE structure. For any

given structure s, the set of all sequences that map to s constitutes a combinatorial set called the

neutral network. The earlier work of [15] studies the combinatorial properties of neutral network

and [7] addresses its biological importance in terms of evolutionary transition as well as genotype

& phenotype correlation.

The sequence-structure mapping problem was experimentally investigated in [27]. It is found that

neutral sequences are in fact percolated throughout the whole sequence space; on the other hand,

there exists a high degree of connectivity among neutral sequences: here we consider an undirected

connection between neutral sequence r

1

and r

2

exists if and only if the hamming distance between

r

1

and r

2

is 1. In evolutionary dynamics, such connections represent a one-point mutation from

one RNA primary sequence towards another. A high degree of connectivity in the neutral network

indicates that the RNA secondary structure tends to be conserved during sequence mutations. The

mutational landscape of RNA sequences were studied in [28] and its biological relevance was dis-

cussed.

Recent work [26] demonstrated the NP-completenessof RNA design by ﬁrst transformingthe design

problem into an inverse HMM. In inverse HMM, the hidden path way is equivalent to the RNA

secondary structure and the emitting sequence produced by such a path way is equivalent to the

RNA primary sequence. A polynomial reduction from 3-SAT is then presented to show that there

exists no polynomial runtime algorithm capable of solving RNA design unless P equals NP.

2.2 Sequence Neighborhood Boundary Estimation

The presence of a many-to-one mapping correlation states that, in order to solve the RNA design

problem, one only needs to search for a corresponding neutral network for the target secondary

structure. Once such network is found, a series of one-point mutation sequence solutions will be ob-

tained. However,the distribution of these neutral networksin the sequence space is largely unknown.

This is due to the limitation of current computing power as one need to exhaustively enumerate the

sequence space whose capacity is exponential in the sequence length.

Schuster et. al empirically studied the structure density surface of arbitrary RNA sequences, and

showed that there exists a small-radius hamming neighborhoodball around arbitrary sequences [27];

within such neighborhood ball, the structure coverage is high such that most common secondary

structures could be mapped from at least one sequence within the ball. Therefore, an estimation of

the hamming boundary of such neighborhood ball will give an empirical upper bound for ﬁnding

arbitrary neutral networks in the sequence space.

To estimate the neighborhood ball boundary, Schuster et. al proposeda closest approachingdistance

measurement that ﬁnds an upper bound for a minimum distance between sequences folded into two

different structures. However, such method is computationally costly therefore becomes infeasible

on large RNA sequences.

We recently proposed an alternative method, the Incremental Redundancy Estimation (IRE) ap-

proach for determining the sequence neighborhood boundary [8]. We conﬁrmed through empirical

experiments the existence of small-radius neighborhood balls centered at arbitrary sequences; it is

shown that the neighborhood ball boundary is much smaller than the corresponding sequence size.

The method also scales well with sequence size and is capable of estimating the hamming boundary

on large sequence with good accuracy. The result of this work is that the hamming boundary of the

neighborhood ball gives an empirical constraint for algorithms that explore the sequence space for

RNA design solving; utilizing the combinatorial properties of the neighborhood structure may also

provide insight towards novel algorithm design for RNA design problem.

3 RNA Design Algorithms

In the literature, heuristic search algorithm is established as the standard technique for tackling the

RNA design problem, as it enables an effective and efﬁcient exploration of the high-dimensional

sequence-structure space. RNA design is the inverse of the classical RNA secondary structure pre-

diction problem, and the study of this problem provides complementary insights into the theory

3

concerning evolutionary dynamics [7]. The general design issues are addressed in [12]. Typically, a

good RNA design shall exhibit both high sequence afﬁnity and structure speciﬁcity:

• Sequence afﬁnity is deﬁned such that the target folding energy e(r, S

∗

) be low, indicat-

ing the presence of thermodynamic stability. Here r is the RNA primary sequence, S

∗

is

the target structure for design, and e(r, S

∗

) computes the thermodynamic free energy for

sequence r while adopting structure s

∗

.

• Structure speciﬁcity imposes the primary constraint on the design problem, requiring that

a designed sequence r must have its native MFE structure S

o

r

similar to the target S

∗

. In

other words, the structural distance d(S

o

r

, S

∗

) shall be small.

An early attempt, namely RNAinverse [16], conducts an adaptive local search within the sequence

space, minimizing a cost function deﬁned as the distance between a native fold and the target fold.

A native fold is deﬁned as the MFE structure of the sequence currently investigated by the local

search algorithm. A target fold is the objective secondary structure conﬁguration for the design

problem. The performance of the algorithm, however, is found to be quite sensitive to initial search

points (sequence). In the presence of local optima, RNAinverse fails to return a solution. Recent

works [2] [1] (RNA-SSD) apply stochastic local search to tackle the problem. In RNA-SSD, a

hierarchical decomposition procedure is ﬁrst applied that recursively breaks down the full length

structure into structural components; a local search is then employed on each substructure in an

attempt to ﬁnd the desired subsequence. In a ﬁnal step, the full length sequence is assembled to

form the solution. Empirical results [1] show that RNA-SSD outperforms RNAinverse in terms of

both algorithm runtime and problem solvability. Another hybrid algorithm, namely INFO-RNA [6],

combines a dynamic programming procedure (DP) with an adaptive random walk. The DP is used

for heuristic generation of initial sequence(s) and the subsequent local search is applied for solution

quality improvement. The designability aspect of the design problem over different alphabet set was

studied in [5], and a branch-and-bound deterministic was presented thereafter.

3.1 Local Search for RNA Design

We recently proposed a novel local search algorithm, rnaDesign for solving the problem [9]. Exist-

ing algorithms utilized local search as a supplementary procedure for solution quality improvement

[16] [6]. Although empirical results show that combining local search with ad-hoc design methods

lead to improved performance, it also inevitably increases the model complexity of both algorithm

design and implementation. We demonstrated that applying an adaptive local search procedure is

capable of solving the design problem and show that in certain cases it outperformsa hybrid method.

Our rnaDesign algorithm represents a special case of Simulated Annealing [20] where a combina-

tion of three heuristic schemes is used and the annealing temperature level is ﬁxed throughout the

optimization process. Another motivation of this work is that, since local search performance is

closely correlated to the optimization problem being investigated, a performance analysis of the

algorithm will provide insights into studying the empirical hardness of RNA design as well as its

combinatorial properties.

4 Performance Prediction for RNA Design

It is empirically known that many hard combinatorial optimization problems can be efﬁciently

solved using metaheuristic algorithms [3]. Metaheuristics (alternatively stochastic local search, or

SLS [17]) refers to an abstract algorithm framework that employs a high level search strategy aim-

ing at exploring the solution space of the target problem in an effective and efﬁcient manner. The

No Free Lunch Theorem [29] states that one algorithm may outperform another over one particular

problem or problem instance, however in general, there exists no such algorithmic solution that is

optimal for all cases. Therefore, the best strategy for optimization is to develop algorithms special-

ized to the speciﬁc problem under consideration. In the literature, it is experimentally shown that

the performanceof a given metaheuristics is affected by its meta-parameter setting (MPS) [17] [11].

Thus part of the research effort involves tuning MPS in order to achieve an optimal algorithm per-

formance. However, this process is time consuming as one has to explore a combinatorial parameter

space which itself may be exponential in the parameter size. Furthermore, algorithm performance

may vary across problem instances: an optimal meta-parameter setting under one particular problem

4

may shift under another. As a result, a growing interest in the literature involves applying machine

learning techniques to achieve an automatic MPS tuning (either online or prior conﬁguration). This

leads to more robust algorithm design, as the typical behavior of the SLS would be self-adjusted on

a per-problem or per-instance base. An accurate prediction of algorithm performance therefore is a

prerequisite to achieve this goal. Another motivation of this topic involves studying the empirical

hardness of the optimization problem itself: Since a prediction model correlates characteristics of

the problem (input features) with the performance benchmarks of the algorithm (target values), we

expect to answer questions such as: under what circumstances a given problem(or probleminstance)

is hard to solve, or what particular factor(s) govern the performance of the underlying algorithms

being studied.

One recent work [19] demonstrates using linear regression models for predicting SLS algorithm on

the SAT problem. [18] applies similar methods to achieve an optimal parameter conﬁguration on a

per-instance base for SAT. [23] studied the empirical hardness of a combinatorial auction problem.

It applies linear & nonlinear regression models to identify key problem features that affect algorithm

performance. Our previous work [10] demonstrated using parametric & non-parametric regression

models for algorithm performance prediction on the RNA design problem. Our regression models

include ridge regression (linear regression with regularization), Nadaraya kernel method and Clas-

siﬁcation & Regression Tree (CART) model [4] [25]. In that work, we focused on predicting the

performance of the RNAinverse algorithm [16], and evaluated our models against various secondary

structure instances including two biological data sets and one random set. We showed that the

non-parametric model (kernel & CART) outperforms the parametric method (ridge) on biological

data sets, and found that the selection of input features are of crucial importance towards prediction

accuracy

1

. We also found that the CART model identiﬁes key structure features that affect algo-

rithm performance, and it is an intuitive tool for investigating the empirical hardness of RNA design

solving.

The result of our work enables an accurate prediction of algorithm performance on RNA design

problem. Applying the prediction model to a given design algorithm, we are able to predict structure

designability (solvability)

2

on unforeseen secondary structures; furthermore, analyzing the model

itself may help us answer questions such as why some structures are hard/easy to design, or what

structure components are contributing to the overall design difﬁculty.

5 Multi-Objective Optimization for RNA Design

There is an increasing interest in the literature applying computational methods to design RNA

molecules that satisfy speciﬁc constraints. For example, considering the physical aspects of RNA,

it is found that natural RNAs differ from random RNA sequences in a number of physical mea-

surements [13] including thermodynamic stability, mutational robustness, linguistic complexity and

folding efﬁciency (kinetics). Therefore one of these goals is to design RNA sequences whose char-

acteristics resemble that of a naturally existing one.

In the thermodynamic stability requirement, a desirable RNA sequence shall fold into its MFE struc-

ture with low energy level; it may also has fewer suboptimal structure alternatives such that the

designed sequence is stable; furthermore, we may search for sequences whose MFE structures are

insensitive to parameter perturbations in the free energy model.

In mutational robustness, we look for RNA sequences whose MFE structure remain unchanged

under one-point or k-point mutation. The degree of neutrality measures the average portion of the

structure that remains intact after one-point mutation and could be used towards this measurement.

Alternatively, deleterious effect [28] provides an overall measurement of robustness by investigating

the k-point mutation landscape of a given RNA sequence in the sequence space.

In linguistic complexity, we look at the content of the RNA sequences and search for repeated pat-

terns that conform to a given requirement. It is found that linguistic complexity in various natural

RNAs is lower comparing to random sequences, therefore is an informativecriteria for RNA design.

1

in the literature, the feature selection problem (FSP) itself was studied, for example [21].

2

Designability is often used to characterize the performance property of a given stochastic local search algo-

rithm, where the algorithm isissued multiple independent runs against the problem instance, and the percentage

of successful runs which returns valid results within a given runtime bound are recorded.

5

We may also consider the kinetic aspects of RNAs and aim at designing RNA sequences achieving

certain level of folding efﬁciency. In folding efﬁciency [14], we measure the total number of ele-

mentary steps required to fold a given RNA sequence into its native MFE structure. It is found that

natural RNAs have persistent meta-stable states with relatively small folding time. Empirical ex-

periments also show that arbitrary RNA sequences usually have frustrated energy landscape where

there exist high energy barriers and rugged landscape regions.

Therefore, the research question is how to design RNA sequences satisfying various physical con-

straints while also folded into a predescribed secondary structure. Since multiple objectives exist in

this optimization process, we are solving a Multi-Objective Optimization Problem (MOOP). MOOP

refers to the problem of simultaneous optimization of several possibly conﬂicting and incompatible

objective functions. The typical solution of MOOP would be a set of non-dominated solution can-

didates. In Operations Research, decisions are made from a set of candidate strategies. The choice

of one strategy over another represents a trade-off among various objectives. In MOOP, the optimal

solution is referred to as the Pareto Optimality or Pareto Efﬁciency. A solution s

∗

is said to be a

Pareto Optimum if there exists no feasible solution s that further improvesat least one function value

within the objective functions set, without simultaneously decreasing the function value of another.

6 Conclusion and Discussion

In this survey, we presented the RNA design problem and its algorithmic solutions. We showed

that RNA design is an empirically hard problem and heuristic search algorithm is established as

the standard technique for tackling it. Solving the RNA design problem requires an efﬁcient and

effective exploration of the high-dimensional sequence space in search for candidates that fold into

a given target structure. Therefore understanding the underlying sequence-structure combinatorics

is of crucial importance, both for interpreting the algorithm performance, as well as developing new

algorithms for the problem. We also introduced the algorithm performance prediction model and

addressed its importance in terms of empirical hardness study. We identiﬁed a number of newly

emerged RNA design criteria and proposed solving RNA design as a multi-objective optimization

problem. In summary, RNA design is a promising research topic with various open issues to be

further explored.

Acknowledgments

This work is supported and funded by Dr. Kay C. Wiese (senior supervisor) and the School of

Computing Science at Simon Fraser University.

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