GTfold: A Scalable Multicore Code for RNA Secondary
Structure Prediction
Amrita Mathuriya∗
David A. Bader†
Christine E. Heitsch‡
Stephen C. Harvey§
Georgia Institute of Technology Atlanta, GA, USA
ABSTRACT
1. INTRODUCTION
The prediction of the correct secondary structures of large
RNAs is one of the unsolved challenges of computational
molecular biology. Among the major obstacles is the fact
that accurate calculations scale as O(n4 ), so the computational requirements become prohibitive as the length increases. Existing folding programs implement heuristics and
approximations to overcome these limitations. We present a
new parallel multicore and scalable program called GTfold,
which is one to two orders of magnitude faster than the
de facto standard programs and achieves comparable accuracy of prediction. Development of GTfold opens up a new
path for the algorithmic improvements and application of an
improved thermodynamic model to increase the prediction
accuracy.
In this paper we analyze the algorithm’s concurrency and
describe the parallelism for a shared memory environment
such as a symmetric multiprocessor or multicore chip. In
a remarkable demonstration, GTfold now optimally folds
11 picornaviral RNA sequences ranging from 7100 to 8200
nucleotides in 8 minutes, compared with the two months it
took in a previous study. We are seeing a paradigm shift to
multicore chips and parallelism must be explicitly addressed
to continue gaining performance with each new generation
of systems. We also show that the exact algorithms like
internal loop speedup can be implemented with our method
in an affordable amount of time. GTfold is freely available
as open source from our website.
RNA molecules perform a variety of different biological
functions including the role of “small” RNAs (with tens
or a few hundred of nucleotides) in gene splicing, editing,
and regulation. At the other end of the size spectrum,
the genomes of numerous viruses are lengthy single-stranded
RNA sequences with many thousands of nucleotides. These
single-stranded RNA sequences base pair to form molecular structures, and the secondary structure of viruses like
dengue [3], ebola [18], and HIV [19] is known to have functional significance. Thus, disrupting functionally significant
base pairings in RNA viral genomes is one potential method
for treating or preventing the many RNA-related diseases.
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∗
College of Computing, Georgia Institute of Technology
College of Computing, Georgia Institute of Technology
‡
School of Mathematics, Georgia Institute of Technology
§
School of Biology, Georgia Institute of Technology
†
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SAC’09 March 8-12, 2009, Honolulu, Hawaii, U.S.A.
Copyright 2009 ACM 978-1-60558-166-8/09/03 ...$5.00.
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Figure 1: The optimal secondary structure of an
HIV-1 virus with 9,781 nucleotides predicted using
GTfold in 84 seconds using 16 dual core CPUs. The
minimum free energy of the structure is -2,879.20
Kcal/mole.
Viral sequences range in length from about 1,000 to over
1,000,000 nucleotides in the recently discovered virophage.
Length of the viral sequences poses significant computational challenges for the current computer programs. Free
energy minimization excluding pseudoknots is a conventional
approach for predicting secondary structures. The mfold
[21, 12] and RNAfold [9] programs are the standard programs used by the molecular biology community for the last
several decades. Recently, other folding programs such as
simfold [1] have been developed. These programs predict
structures with good accuracy for the RNA molecules hav-
ing fewer than 1,000 nucleotides. However, for longer RNA
molecules, prediction accuracy is very low.
According to the thermodynamic hypothesis, the structure having the minimum free energy (MFE) is predicted as
the secondary structure of the molecule. The free energy of
a secondary structure is the independent sum of the free energies of distinct substructures called loops. The optimization is performed using the dynamic programming algorithm
given by Zuker and Stiegler in 1981 [22] which is similar to
the algorithm for sequence alignment but far more complex.
The algorithm explores all the possibilities when computing
the MFE structure. There are heuristics and approximations which have been applied to satisfy the computational
requirements in the existing folding programs.
One potential approach to improve the accuracy of the
predicted secondary structures is to implement advanced
thermodynamic details and exact algorithms. However, while
the incorporation of these improvements can significantly
increase the accuracy of the prediction, it also drastically
increases running time and space needs for the execution.
We use shared memory parallelism to overcome the computational challenges of the problem.
We have designed and implemented a new parallel and
scalable program called GTfold for predicting secondary structures of RNA sequences. Our program runs one to two orders of magnitude faster than the current sequential programs for viral sequences on an IBM P5 570, 16 core dual
CPU symmetric multiprocessor system and achieves comparable accuracy in the prediction. We have parallelized the
dynamic programming algorithm at a coarse-grain and the
individual functions which calculate the free energy of various loops at a fine-grain. We demonstrate that GTfold executes exact algorithms in an affordable amount of time for
large RNA sequences. Our implementation includes an exact and optimized algorithm in place of the usually adopted
heuristic option for internal loop calculations, the most significant part of the whole computation. GTfold takes just
minutes (instead of 9 hours) to predict the structure of a
Homo sapiens 23S ribosomal RNA sequence with 5,184 nucleotides. Development of GTfold opens up the path for
applying essential improvements in the prediction programs
to increase the accuracy of the predicted structures.
The algorithm has complicated data dependencies among
various elements, including five different 2D arrays. The energy of the subsequences of equal length can be computed
independently of each other without violating the dependencies pattern introduced by the dynamic programming with
a set of five tables. Our approach calculates the optimal energy of the equal length sequences in parallel starting from
the smallest to the largest subsequences and finally the optimal free energy of the full sequence. We also describe the
nature of individual functions for calculating the energy of
various loops and strategies for parallelization.
2.
RELATED WORK
Several parallel approaches have been taken for RNA secondary structure prediction. Nakaya et al. [14] presented an
approximation approach for generating secondary structures
using minimum free energy criterion. The parallel approach
enumerates all stacking regions of an RNA sequence and
combines the ones which can coexist together to produce
multiple secondary structures. Another parallel approximation approach by Taufer et al. [17] samples the RNA
sequence systematically and extensively and rebuilds the
whole structure by combining the structures of the chunks
according to various criteria. However, these approximation
approaches do not explore the entire search space and can
miss the candidates that do not follow the usual behavior.
Also, the success of these kinds of approaches is dependent
upon the ability of rebuilding methods to identify motifs
correctly by consistently combining the substructures into a
full structure.
Several distributed memory implementations [9, 8, 5] for
RNA secondary structure prediction have been developed
which parallelizes the exact dynamic programming algorithm
for free energy minimization. Hofacker et al. [9, 8] partition
the triangular portion of 2D arrays into equal sectors that
are calculated by different processors and data is reorganized after computing each diagonal in order to minimize
the space requirements. In this implementation the tables
are not stored permanently due to which traceback for all
suboptimal secondary structures is not possible. Fekete et
al. [5] uses a similar technique to parallelize the folding procedure and increases communication to store the tables in
order to facilitate the full traceback. However, these implementations may not be portable to current parallel computers and also the implementation of the optimized algorithms
such as internal loop speedup algorithm whose access pattern differs from the general access pattern become complex
for distributed memory environment. In [8], the authors observe that to fold the HIV virus, memory of 1 to 2GB is
required, dictating the use distributed memory supercomputers; yet in our work, we demonstrate that this can now
be solved efficiently on most personal computers. In our
work, for the first time, we give scientists the ability to solve
very large folding problems on their desktop by leveraging
multicore computing.
Zhou and Lowenthal [20] also studied a parallel, out-ofcore distributed memory algorithm for the RNA secondary
structure prediction problem including pseudoknots. However, their approach does not implement the full structure
prediction but rather studies a synthetic data transformation that improves just one of the dependencies found in the
full dynamic programming algorithm.
3. RNA SECONDARY STRUCTURE
RNA molecules are made up of A, C, G, and U, nucleotides which can pair up according to the rules in {(A,U),
(U,A), (G,C), (C,G), (G,U), (U,G)}. Nested base pairings
result into 2D structures called secondary structures. There
are 3D interactions among the elements of the secondary
structures which result into 3D structures called tertiary
structures. Pairings among nucleotides form various kinds
of loops, which can be classified based on the number of
branches present in them. Nearest neighbor thermodynamic
model (NNTM) provides a set of functions and sequence dependent parameters to calculate the energy of various kinds
of loops. The free energy of a secondary structure is calculated by adding up the energy of all loops and stacking
present in the structure.
Figure 2 shows an MFE secondary structure predicted by
GTfold of a sequence with 79 nucleotides. Various loops
annotated in the figure are named as hairpin loops, internal loops, multiloops, stacks, bulges and external loops.
Loops formed by two consecutive base pairs are called stacks.
Loops having one enclosed base pair and one closing base
pair are called internal or interior loops. Internal loops with
length of one side as zero are called bulges. Loops with two
or more enclosed base pairs and one closing base pair are
called multiloops or multibranched loops. The open loop
which is not closed by any base pair is called an external or
exterior loop.
Hairpin Loop
30
A A
A
A
C
G
C
Internal Loop
G
G
C
G
C
G
20
C
G
A A
C
A
A
A
A
C
A A
G
C
40
G
G
Multiloop
Hairpin Loop
C
C
10
G
G
A
C
G
A A
50
A
C
A
A
A
A
G CA
C G
G C
C G
5’
G C
C G
A
A A
A
C G C G C G
A
G C G C G C
A A
A
A
A
70
60
Stack
External Loop
Figure 2: A sample RNA secondary structure with
79 nucleotides.
4.
THERMODYNAMIC PREDICTION ALGORITHM
Prediction of secondary structures with the free energy
minimization is an optimization problem like the SmithWaterman local alignment algorithm. There is a well-defined
scoring function which can be optimized via dynamic programming, and structures achieving the optimum can be
found through traceback. However, while sequence alignment can be performed with one table and a relatively simple processing order, RNA secondary structure prediction
requires five tables with complex dependencies. Each class
of loop has a different energy function which is dependent
upon the sequence and parameters. For the internal loops
and multiloops with one or more branches, all enclosed base
pairs need to be searched which make the loop optimal for
the closing base pair.
The algorithm can be defined with recursive minimization
formulas. Simplified recursion formulas are reproduced here
from [11] for convenience. Pseudocode of our algorithm that
implements thermodynamically equipped recursion formulas
is presented in the technical report [13]. Consider an RNA
sequence of length N , free energy W (N ), and index values i
and j which vary over the sequence such that 1 ≤ i < j ≤ N .
The optimal free energy of a subsequence from 1 to j is given
with the following formula:
W (j) = min{W (j − 1), min {V (i, j) + W (i − 1)}}
1≤i<j
(1)
In Eq. (1), V (i, j) is the optimal energy of the subsequence
from i to j, if it forms a base pair (i, j). It is defined by the
following equation.
⎧
eH(i, j),
⎪
⎨
eS(i, j) + V (i + 1, j − 1),
V (i, j) = min
⎪
⎩ V BI(i, j),
V M (i, j)
(2)
Eq. (2) considers loops that a base pair (i, j) can close.
The eH(i, j) function returns the energy of a hairpin loop
closed by base pair (i, j). Function eS(i, j) returns the energy of a stack formed by base pairs (i, j) and (i + 1, j − 1).
V BI(i, j) and V M (i, j) are the optimal free energies of the
subsequence from i to j in the case when the (i, j) base pair
closes an internal loop or a multiloop, respectively.
V BI(i, j) =
min
i<i <j <j
where, i − i + j − j − 2 > 0.
The formulation of the multiloop energy function has linear dependence upon the number of single stranded bases
present in the multiloop. The standard is to introduce a 2D
array W M to facilitate the calculation of V M array. Eq. (4)
and (5) shows calculations of W M (i, j) and V M (i, j) respectively.
⎧
V (i, j) + b,
⎪
⎨
W M (i, j − 1) + c,
W M (i, j) = min
⎪
⎩ W M (i + 1, j) + c,
mini<k≤j {W M (i, k − 1) + W M (k, j)}
(4)
{eL(i, j, i , j ) + V (i , j )}
(3)
V M (i, j) =
min
i+1<k≤j−1
{W M (i+1, k −1)+W M (k, j −1)+a}
(5)
These minimization formulas can be implemented recursively as well as iteratively. We implement an iterative formulation of the algorithm in GTfold, described later in this
paper. The implementation uses various 1D and 2D arrays
corresponding to W (j) and V (i, j), V BI(i, j), V M (i, j),
W M (i, j) values. Also we use calcW (j), calcV (i, j), calcV BI(i, j),
calcV M (i, j) and calcW M (i, j) functions to calculate the
values of W (j), V (i, j), V BI(i, j), V M (i, j) and W M (i, j)
array elements.
4.1 Parallelism
The dynamic programming algorithm is computationally
intensive both in terms of running time and space. Its space
requirements are of O(n2 ) as it uses four 2D arrays named
V (i, j), V BI(i, j), V M (i, j) and W M (i, j) that are filled up
during the algorithm’s execution. The main issue is running time rather than memory requirements. For instance,
GTfold has a memory footprints of less than 2GB (common in most desktop PCs) even for sequences with 10,000
nucleotides.
The filled up arrays are traced in the backwards direction
to determine the secondary structures. The traceback for
a single structure takes far less time than filling up these
arrays. Time complexity of the dynamic programming algorithm is O(n3 ) with the currently adopted thermodynamic
model. The two indices i and j are varied over the entire
sequence, and every type of loop for every possible base pair
(i, j) is calculated. This results in the asymptotic time complexity of O(n2 )× maximum time complexity of any type of
loop for a base pair (i, j).
Computations of internal loops and multiloops are the
most expensive parts of the algorithm. We can see from
Eq. (3) that, in the calculation of V BI(i, j), all possible
internal loops with the closing base pair (i, j) are considered
by varying indices i and j over the subsequence from i +
1 to j − 1 such that i < j . This results in the overall
time complexity of O(n4 ). To avoid large running time,
a commonly used heuristic is to limit the size of internal
loops to a threshold k usually set as 30. This significantly
reduces running time from O(n4 ) to O(k2 n2 ). The heuristic
is adopted in most of the standard RNA folding programs.
Lyngsø et al. [11, 10] suggest that the limit is a little
bit small for predictions at higher temperatures and give an
optimized and exact algorithm for internal loop calculations
which has the time complexity of O(n3 ) with the same O(n2 )
space. The algorithm searches for all possible internal loops
closed by base pair (i, j). Practically, this algorithm is far
slower than the heuristic. Choosing one of the options is a
tradeoff of running time versus accuracy. In GTfold we provide an option for the user to select the heuristic or internal
loop speedup algorithm. Also, our parallelization scheme is
valid for both the options.
Thermodynamics of multiloops are still not understood
fully, but improvements continue to be made. Searching for
an optimal multiloop closed by a base pair (i, j) requires
searching for all enclosed base pairs which make the loop
optimal. To make the multiloop energy function feasible
to compute, it may be approximated in O(n3 ) time. This
function has linear dependence upon the number of single
stranded bases in the multiloop. Time complexity of the
algorithm to implement a relatively more realistic multiloop energy function having logarithmic dependence upon
the single stranded bases in the loop is exponential. Also,
many other advanced thermodynamic details such as coaxial
dangling energies are not implemented in the multiloop energy calculations during the optimization, as it significantly
increases the running time.
Both running time and space needs are expected to increase with the use of better thermodynamic models. While
memory requirements can be satisfied with today’s high-end
servers with 256GB or more memory, running time will continue to play as a major prohibitive factor in solving these
grand challenge problems. Our parallelization approach in
GTfold is designed to keep all these factors in consideration.
5.
the sides is less than a constant c. The algorithm scores these
small internal loops as special cases by applying a function
derived from the general internal loop energy function. If an
enclosed base pair (ip , jp ) is better than another candidate
of the enclosed base pair (ip1 , jp1 ) for the closing base pair
(i, j) where jp − ip = jp1 − ip1 , then the enclosed base pair
(ip , jp ) will also be better than (ip1 , jp1 ) for the closing base
pairs of the form (i − b, j + b), where b is a positive integer
such that 1 ≤ b ≤ min{i − 1, N − j}. This algorithm uses
the best enclosed base pair (ip , jp ) for the closing base pair
(i, j) to evaluate all internal loops closed by the base pairs
of the form (i − b, j + b) at the same time. This way, at
element (i, j), the elements of the form (i − b, j + b) are also
accessed. The access pattern of this algorithm is shown in
Figure 4 excluding the calculation of special cases.
j
i
GTFOLD
5.1 Dependencies and Access Patterns
Figure 3 shows a general ij plane. A valid base pair is
defined as (i, j) where j > i. Thus, only the upper right
triangle is valid. Secondary structures can have only nested
base pairings, meaning if there are two base pairs (i, j) and
(i , j ) such that i < i < j then the constraint i < i < j < j
is also satisfied. This assumption of nested base pairings
results in the general dependency of an element (i, j) on the
elements in the Δ ABC as shown in Figure 3. To find the
optimal loop formed by a base pair (i, j), we need to search
for all enclosed base pairs over the subsequence from i + 1 to
j − 1. In the case of internal loops we need to search for one
enclosed base pair while for a multiloop we need to search for
more than one base pair. In this fashion, the computation
of all types of loops for an element (i, j) follows the above
dependency pattern.
j
Figure 4: The access pattern of V BI(i, j) for the internal loop speedup algorithm
5.2 Approach
In the region of the general 2D ij plane corresponding
to j > i, a point (i, j) corresponds to the computation of
energy of the subsequence from i to j. The dependency
pattern shown in Figure 3 allows the calculation of all the
elements existing on a line j−i = k to be independent of each
other, where k is any integer from the set {0, 1, 2, . . . , N −1}.
This way the computation on the line j − i = k can be
performed in parallel, and the whole space can be computed
by considering subsequent lines from k = 0 to k = N − 1.
Note that the points on one of the lines corresponds to the
equal length subsequences.
j
B
A (i, j)
i
i
C
Figure 3: The implicit dependency of point A on
the elements present in the triangle ABC
The speedup algorithm for internal loop calculations follows the same general technique but its access pattern differs. It updates the elements outside the dependency triangle Δ ABC shown in Figure 3 for the element A. It is an
optimized algorithm to reduce the space complexity. We define a small internal loop as a loop in which at least one of
Figure 5: Showing the pattern of computation implemented in GTfold
Algorithm 1 arranges the nested for loops to compute in
the manner described above. The first for loop runs for different lines starting from j = i to j = i + N − 1 and the
second for loop calculates all the points on one line in parallel. Figure 5 shows the sequence of these computations.
This parallelization strategy is suitable for future improvements to the thermodynamic model or to optimizations for
computing the various energy functions. This coarser level
of parallelism enables us to exploit more concurrency while
offering compatibility for possible future improvements.
There are other orderings of the computation that cover
the whole space without violating the dependency pattern.
One way is to compute the elements column-wise, starting
from j = 1 to j = N . On one column the computation is
done for the increasing values of j − i, i.e. from row i = j
to row i = 1. A second way is to compute the elements
row-wise, starting from i = N to i = 1. On one row the
computation is done for the increasing values of j − i, i.e.
from column j = i to column j = N . These two ways achieve
a higher degree of spatial locality but they are inherently
sequential.
input : Sequence of Length N
output: Optimal Energy of the sequence
begin
for b ← 0 to N − 1 do
#pragma omp parallel for schedule (guided)
for i ← 1 to N − b do
j ← i + b;
calcVBI(i, j);
calcVM(i, j);
calcV(i, j);
calcWM(i, j);
end
calcW(b + 1);
end
return W (N );
end
Algorithm 1: Main function to compute the secondary structure of an RNA sequence
5.3 Implementation Details
We use OpenMP [15] to implement shared memory parallelism. All the subsequent diagonals are considered with
the upper for loop and parallelism is implemented by applying an OpenMP for loop pragma over the inner for loop to
parallelize the computation on the diagonal in consideration
as shown in Algorithm 1. The guided scheduling strategy
works best for this parallelization. This is because there
may not be equal amounts of work for every point on the
diagonal. If the bases i and j are not able to make a pair
then it is not necessary to compute for the whole calculation. In this case, for the heuristic option of internal loop
calculations, only W M (i, j) needs to be calculated. And for
the speedup algorithm V BI(i, j) also needs to be calculated
with W M (i, j).
We explore the function level parallelism for the last few
diagonals by deciding a threshold variable A, such that the
parallelism is implemented at the higher level for the diagonals up to j − i = A and implement only the function
level parallelism starting from the diagonal j − i = A + 1
to j − i = N − 1. This facilitates the use of more threads
to exploit more parallelism at the time when there are not
enough points on the diagonals. However this technique did
not give us a performance advantage.
Cache locality
For this algorithm the ratio of computation to the memory
accesses is low. Energy of a secondary structure is calculated by adding up the energies of various loops present in
the structure. Energy of a structure is the sum of various energy terms of which some are read directly from the energy
tables and others are calculated by the program. Therefore,
large cache sizes and locality in reference for accessing various data elements play an important role in reducing the
running time of GTfold. Computing the elements row-wise
or column-wise as described in Section 5.2 provides better
cache locality than computing the elements on the subsequent diagonals. However, these two ways are inherently
sequential.
Parallelism at individual functions
6. EXPERIMENTAL RESULTS
Parallelism can also be exploited at the finer level of individual functions which compute the energies for the various
kinds of loops for a closing base pair (i, j). The general pattern of different functions for calculating the energy of these
kinds of loops is the same except for the function that computes internal loop energies using the speedup algorithm.
The general pattern is to consider various possible options
of the corresponding type of loop and select the option that
gives the minimum energy. In simplified terms, this pattern
of calculation performs minimization over several possible
values. These types of calculations are easily done in parallel by assigning equal-sized chunks of minimization work
to all threads, collecting the results, and taking the global
minimum over all values.
The speedup algorithm for internal loop calculations can
be parallelized only for special cases. The computations of
general internal loops using the extension principle are inherently sequential. However the generally adopted heuristic
option for internal loops has the general minimization pattern, is easily parallelizable, and uses two nested for loops
which results in a complexity of O(k2 ) for a particular (i, j).
Multiloop calculations also follow the general minimization
pattern for the V M and W M arrays, have O(n) time complexity for an element (i, j), and are amenable to parallelization as well.
We have performed several experiments to establish that
GTfold runs faster than competing folding programs such as
mfold and RNAfold and achieves accuracy comparable with
them. For the running time and accuracy comparisons, we
are using RNAfold distributed with Vienna RNA Package
version 1.7.2 and UNAFold version 3.6, which supersedes
mfold.
6.1 Energy and Structure Comparison
To establish the accuracy of GTfold, we compare the structures obtained from GTfold, mfold, and RNAfold, with the
correct structures determined with the more reliable method
of comparative sequence analysis [6, 7]. Please note that the
comparative sequence analysis method requires large data
sets for the prediction of secondary structures, and therefore, its application is limited with the availability of the
required datasets.
Doshi et al. [4] take a phylogenetically diverse dataset
of ribosomal RNA sequences and compare the optimal secondary structures predicted using mfold 2.3 and mfold 3.1
with the correct structures. Here we are using the 16S and
23S ribosomal RNA sequences from Figure 1 and Table 4
of their study [4] for accuracy comparisons and are taken
from the Gutell database [2]. We provide a GenBank accession number for each of these sequences. For predicting
structures with UNAFold and RNAfold their command line
default options are used. Accuracy of the structures is calculated in the same manner as in [4] with one difference.
We include non-canonical base pairs in the comparison instead of excluding them. This affects the accuracy of all
three programs in the same manner because none of them
is able to predict non-canonical base pairs due to the lack
of energy parameters for them. The accuracy is calculated
as a percentage of correctly predicted base pairs which are
also present in the correct secondary structure.
Table 1: Free energy (in Kcal/mole) comparison of
GTfold, UNAFold and RNAfold for 16S rRNA sequences
Sequence
Length GTfold UNAFold RNAfold
X00794
1962 -741.90
-722.70
-746.60
X54253
701 -149.00
-141.30
-149.03
X54252
697 -142.50
-137.50
-142.52
Z17224
1550 -564.80
-549.10
-565.12
X65063
1432 -582.00
-570.80
-581.94
Z17210
1435 -761.90
-626.60
-762.70
X52949
1452 -802.70
-794.50
-804.40
X98467
1295 -487.00
-460.00
-489.31
Y00266/M24612
1244 -325.60
-317.30
-328.80
X59604
1701 -573.00
-491.40
-574.70
K00421
1474 -687.00
-682.10
-687.01
Table 2: Accuracy comparison (in percent) of GTfold, UNAFold and RNAfold for 16S rRNA sequences of Table 1
Sequence
GTfold UNAFold RNAfold
X00794
30.33
31.65
27.91
X54253
25.67
20.32
25.13
X54252
21.16
21.64
21.16
Z17224
26.03
24.57
24.57
X65063
24.09
22.02
23.83
Z17210
24.46
25.98
24.71
X52949
15.07
16.08
15.07
X98467
17.09
10.97
16.33
Y00266/M24612
19.19
17.30
18.11
X59604
27.49
24.83
27.49
K00421
76.42
75.76
76.42
Table 1 shows the optimal free energy of various 16S ribosomal RNA sequences predicted with GTfold, UNAFold
and RNAfold. Table 2 shows the accuracy comparison for
the three programs for the sequences of Table 1. Similarly
Table 3 shows the optimal free energy obtained using the
programs for 23S ribosomal RNA sequences, and Table 4
shows the accuracy comparison for the sequences of Table 3.
The small differences in the optimal free energy scores predicted using the three programs are due to the algorithmic
issues and policies related to thermodynamic aspects. Energy comparisons shown in Tables 1 and 4 demonstrate that
our energy scores for the various sequences lie in the very
small range of these standard programs. Accuracy percentages shown in Tables 2 and 3 establish that GTfold achieves
accuracy comparable with UNAFold and RNAfold for the
diverse dataset chosen. Accuracy comparisons for various
ribosomal sequences show that in general the accuracy of
the prediction programs are very low. The prediction accuracy is expected to improve with the inclusion of advanced
Table 3: Free energy (in Kcal/mole) comparison of
GTfold, UNAFold and RNAfold for 23S rRNA sequences
Sequence Length
GTfold UNAFold RNAfold
X14386
3105
-791.30
-775.10
-792.65
X54252
953
-180.20
-173.50
-179.71
X52392
1621
-395.80
-389.70
-397.85
J01527
3273
-700.60
-684.60
-702.87
K01868
3514 -1328.50
-1294.30 -1333.95
X53361
4052 -1693.60
-1665.60 -1696.49
X52949
2850 -1707.90
-1689.20 -1709.80
M67497
3029 -1666.10
-1647.10 -1668.12
Table 4: Accuracy comparison (in percent) of GTfold, UNAFold and RNAfold for 23S rRNA sequences of Table 3
Sequence GTfold UNAFold RNAfold
X14386
21.77
18.79
18.32
X54252
23.74
21.92
23.29
X52392
24.44
25.56
24.16
J01527
24.72
30.11
25.14
K01868
20.67
22.01
17.85
X53361
22.21
16.59
15.44
X52949
34.44
31.24
26.69
M67497
64.05
63.6
63.94
thermodynamic details that are not presently incorporated
due to high computational cost. The development of GTfold
facilitates the implementation of these improvements.
6.2 Running Time Comparison
GTfold implements parallelism for shared memory multiprocessor and multicore systems. Running time experiments
are performed on an IBM P5-570 server with 16 dual core
1.9 GHz CPUs and 256 GB of main memory with L2 cache
of 1.9 MB per CPU. GTfold is compiled with IBM xlC compiler Enterprise Edition 7.0, with -q64 option for the 64 bit
compilation, -O3 level of optimization and -qsmp=omp option for OpenMP support. RNAfold and UNAFold are compiled with their default compiler and compilation options.
An additional flag -maix64 is set while compiling UNAFold
and RNAfold due to the runtime memory limitations on the
system for 32-bit compilations.
Palmenberg and Sgro in 1997 [16] investigated the optimal and suboptimal secondary structures of 11 picornaviral
RNA sequences using mfold version 2.2. The length of the
sequences varies from 7124 to 8214 nucleotides. They report
that each sequence required 5-7 days of CPU time using a
modern workstation so that all 11 sequences took 2 to 3
months of time. In stark comparison, GTfold finishes the
execution of this set of sequences in approximately 8 minutes using 32 threads.
In Figure 6 we compare the running time of GTfold with
32 threads, UNAFold, and RNAfold, for all the picornaviral
sequences using the same machine. We can see that GTfold
runs one to two orders of magnitude faster than the standard
sequential programs UNAFold and RNAfold.
Figure 7 compares the running time of GTfold, UNAFold,
and RNAfold, for an HIV-1 sequence (accession number
Z11530) with 9,781 nucleotides. The secondary structure
predicted with GTfold of the viral sequence is shown in Figure 1. All three programs implement the heuristic option
Figure 6: Comparison of running times for predicting the RNA secondary structures of 11 picornaviral
sequences. The sequences are arranged in increasing
order of length from 7124 to 8214 nucleotides.
and limit the internal loop size to 30. It is clear from the
graph that GTfold with one thread performs much better
than UNAfold and is comparable with RNAfold. The running time of GTfold decreases with the increasing number
of threads. Even with two threads GTfold runs 2.06 times
faster than RNAfold. GTfold folds the entire HIV viral sequence in 84 seconds with 32 threads in comparison to UNAFold and RNAfold which take approximately 2.4 hours and
27 minutes, respectively.
GTfold also implements an exact algorithm for finding the
optimal internal loops called Internal Loop Speedup Algorithm (ILSA). Though internal loops with sizes longer than
30 are observed, they are usually rare. ILSA can catch these
exceptional cases occurring with rarity in nature. It is far
more expensive to run this algorithm than the commonly
used heuristic. Figure 8 shows the running time of GTfold with the varying number of threads for a 5,184 length
23S Ribosomal RNA sequence of Homo sapiens with accession number J01866/M11167. GTfold is able to reduce the
running time from 512 minutes (approximately 9 hours) to
21.5 minutes by using 32 threads. This way, we show that
optimized algorithms such as internal loop speedup algorithm can be executed with GTfold in an affordable time.
Please note that UNAFold and RNAfold do not implement
the ILSA algorithm and can miss the rare possibilities.
Figure 7: Comparison of running times for predicting the RNA secondary structure of the HIV1 virus. The dashed horizontal lines represent the
sequential running time of UNAFold and RNAfold.
Figure 9: Speedups obtained by GTfold with the
heuristic algorithm for the HIV-1 virus and with
the internal loop speedup algorithm for the Homo
sapiens ribosomal RNA sequence. Speedup is with
respect to GTfold running on one processor for each
series.
Figure 8: GTfold running time statistics for a Homo
sapiens 23S ribosomal RNA sequence with accession
number J01866/M11167 using the Internal Loop
Speedup Algorithm
Figure 9 shows the speedup achieved using 2, 4, 8, 16, and
32, threads for GTfold using the internal loop speedup algorithm (ILSA) option for the sequence with accession number
J01866/M11167 and the heuristic options with an HIV viral
sequence. The maximum speedup achieved in the first and
second cases is approximately 23.8 and 19.8, respectively.
We have achieved slightly superlinear speedups for 2, 4, and
8, threads in the case of the heuristic option due to the better cache locality when the number of threads is more than
one.
7.
CONCLUSIONS
We have developed GTfold, a parallel and multicore code
for predicting RNA secondary structures that achieves 19.8
fold speedups over the current best sequential program for
large, important RNA sequences and has accuracy comparable to the existing standard folding programs. We have recognized the computational requirements of the problem and
implemented shared memory parallelism for the problem.
Development of GTfold solves the problem of prohibitive
running time factor for large RNA sequences and will help
the molecular biology community to work towards the problem of predicting more accurate secondary structures. We
have reduced the running time from approximately 2 months
to 8 minutes for folding 11 picornaviral sequences.
8.
ACKNOWLEDGMENTS
This work was supported in part by National Institutes
of Health grant NIH NIGMS R01 GM083621, and by NSF
Grants CNS-0614915 and DBI-04-20513. Additionally, the
research of Christine E. Heitsch, Ph.D., is supported in part
by a Career Award at the Scientific Interface (CASI) from
the Burroughs Wellcome Fund (BWF). We acknowledge visiting students Gregory Nou, Sonny Hernández, and Manoj
Soni, who helped with the implementation of GTfold.
9.
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