In a previous paper, we introduced MUSCLE, a new program for creating multiple alignments of protein sequences, giving a brief summary of the algorithm and showing MUSCLE to achieve the highest scores reported to date on four alignment accuracy benchmarks. Here we present a more complete discussion of the algorithm, describing several previously unpublished techniques that improve biological accuracy and / or computational complexity. We introduce a new option, MUSCLE-fast, designed for high-throughput applications. We also describe a new protocol for evaluating objective functions that align two profiles.We compare the speed and accuracy of MUSCLE with CLUSTALW, Progressive POA and the MAFFT script FFTNS1, the fastest previously published program known to the author. Accuracy is measured using four benchmarks: BAliBASE, PREFAB, SABmark and SMART. We test three variants that offer highest accuracy (MUSCLE with default settings), highest speed (MUSCLE-fast), and a carefully chosen compromise between the two (MUSCLE-prog). We find MUSCLE-fast to be the fastest algorithm on all test sets, achieving average alignment accuracy similar to CLUSTALW in times that are typically two to three orders of magnitude less. MUSCLE-fast is able to align 1,000 sequences of average length 282 in 21 seconds on a current desktop computer..MUSCLE offers a range of options that provide improved speed and / or alignment accuracy compared with currently available programs. MUSCLE is freely available at  Multiple alignments of protein sequences are important in many applications, including phylogenetic tree estimation, secondary structure prediction and critical residue identification. Many multiple sequence alignment (MSA) algorithms have been proposed; for a recent review, see . Two attLN) in the sequence length L and number of sequences N  / [min  - k + 1 ].     (1)τ is a k-mer, LX, LY are the sequence lengths, and nX(τ) and nY(τ) are the number of times τ occurs in X and Y respectively. This definition can be motivated by considering an alignment of X to Y and defining the similarity to be the fraction of k-mers that are conserved between the two sequences. The denominator of F is the maximum number of k-mers that could be aligned. Note that if a given k-mer occurs more often in one sequence than the other, the excess cannot be conserved, hence the minimum in the numerator. The definition of F is an approximation in which it is assumed that (after correcting for excesses) common k-mers are always alignable to each other. MUSCLE also implements a binary approximation FBinary, so-called because it reduces the k-mer count to a present / absent bit:Here FBinary = Στ δXY(τ) / [min  - k + 1 ].     (2)δXY(τ) is 1 if τ is present in both sequences, 0 otherwise. As multiple instances of a given k-mer in one sequence are relatively rare, this is often a good approximation to F. The binary approximation enables a significant speed improvement as the size of the count vector for a given sequence can be reduced by an order of magnitude. This allows the count vector for every sequence to be retained in memory, and pairs of vectors to be compared efficiently using bit-wise instructions. When using an integer count, there may be insufficient memory to store all count vectors, making it necessary to re-compute counts several times for a given sequence.Here, mutation distance, i.e. the number of mutations that occurred on the historical path between the sequences. The historical path through the phylogenetic tree extends from one sequence to the other via their most recent common ancestor. The mutation distance is trivially additive. The fractional identity D is often used as a similarity measure; for closely related sequences 1 - D is a good approximation to a mutation distance . As sequences diverge, there is an increasing probability of multiple mutations at a single site. To correct for this, we use the following distance estimate .     (6)Following MAFFT, we also implemented a weighted mixture of minimum and average linkage:dMixPC = (1 - s) dMinPC + s dAvgPC,     (7)s is a parameter set to 0.1 by default. Clustering produces a pseudo-root (the last node created). We implemented two other methods for determining a root: minimizing the average branch weight .     (15)Dk is the last position in X that is aligned to a letter in Y. Extract the special case of a gap of length 1:Here, xy = max { max(k<x-1) [Mky + bXk+1], Mx-1y + bXx}.     (16)DHence,xy = max { Dx-1y, Mx-1y + bXx }.     (17)DSimilarly,xy = max { Ixy-1, Mxy-1 + bYy }.     (18)Ix and the inner loop over increasing y. For fixed x, define vectors Mcurry = Mxy, Mprevy = Mx-1y, Dcurrx = Dxy, Dprevx = Dx-1y; for fixed x, y define scalars Icurr = Ixy, Iprev = Ixy-1. Now we can re-write (14), (17) and (18) to obtain the following recursion relations:Let the outer loop iterate over increasing curry = Sxy + max { Mprevy-1, Dprev y-1 + tXx-1, Iprevy-1 + tYy-1 }     (19)Mcurry = max { Dprevy, Mprevy + bXx }     (20)Dcurr = max { Iprev, Mprevy + bYy }.     (21)ILX × LY matrix is needed for the trace-back that produces the final alignment.An i Σj f xi f yj Sij = Σi f xi Wyi, where Wyi = Σj f yj Sij. The vector Wyi is used LX times, and it therefore pays to compute it once and cache it. Observe that a typical profile column contains << 20 different amino acids. We sort the frequencies in decreasing order; the summation Σi f xi Wyi is terminated if a frequency f xi = 0 is encountered. This typically reduces the time spent in the summation, especially when sequences are closely related. As with Wyi, the sort order is computed once and cached. Observe that the roles of the two profiles are not symmetrical. It is most efficient to choose X, for which frequency sort orders are computed, to be the profile with the lowest amino acid diversity when averaged over columns. With this choice, the summation terminates earlier on average then if the other profile is identified as X. Note that out of N - 1 iterations of progressive alignment, a minimum of  and maximum of N - 1 profile-profile alignments will include at least one profile containing one sequence only, and in the refinement phase exactly N of the 2N - 1 edges in the tree terminate on a leaf. At least half of all profile-profile alignments created in the MUSCLE algorithm therefore include a profile of one sequence only. Special cases where one or both profiles is a single sequence can be handled in separate subroutines, saving overhead due to unneeded loops that are guaranteed to execute once only. This optimization is especially useful for the LE function as it enables the logarithm to be incorporated into the W vector.The inner-most dynamic programming loop, which computes the profile function, deserves careful optimization. We will consider the case of PSP; similar optimizations are possible for LE. PSP = Σk-mers in common between two sequences, then attempt to extend the match. In the case of diagonal identification we found compressed alphabets to significantly out-perform the standard amino acid alphabet  be the amino acid of sequence s in column x, and S be the substitution score of amino acids i and j. It is convenient to impose an (arbitrary) ordering on the sequences and amino acid types. Then,Notice that computation of the SP score dominates the time complexity of refinement and of MUSCLE overall, introducing O.     (23)SPδ = 1 if s [x] = i, 0 otherwise, and ni [x] = Σs δ. We say ni [x] is the count of amino acid type i in column x. We can now transform the sum over pairs of sequences into a sum over pairs of amino acid types:Define a = Σx Σi ni Σj nj>i S + 1/2 Σx Σi (ni2 - ni) S.     (24)SPFrequencies are computed as:f xi = ni [x]/ N.     (25)Using frequencies,N2LP) but (25) and (26) are O(NLP). For N >> 20, this is a substantial improvement. Let SPg be the contribution of gap penalties to SP, so SP = SPa + SPg. It is natural to seek an O(NLP) expression for SPg analogous to (26), but to the best of our knowledge no solution is known. Note that in MUSCLE refinement, the absolute value of the SP score is not needed; rather, it suffices to determine the difference in the SP scores before and after re-aligning a pair of profiles. Let SP be the contribution to the SP score from a pair of sequences s and t, so SP = Σs Σt>s SP, and denote the two profiles by X and Y. Then we can decompose SP into intra-and inter-profile terms as follows:For simplicity, we have neglected sequence weighting; it is straightforward to show that (26) applies unchanged if weighting is used. Note that (23) is O + Σs∈Y Σt∈Y:t>s SP + Σs∈X Σt∈Y SP     (27)SP = ΣNote that the intra-profile terms are unchanged in any alignment that preserves the columns of the profile intact, which is true by definition in profile-profile alignment. This follows by noting that any indels added to align the profiles are guaranteed to be external gaps with respect to any pair of sequences in the same profile. It therefore suffices to compute the change in the inter-profile term:XY = Σs∈X Σt∈Y SP.     (28)SPπ- be the alignment path before re-alignment and π+ the path after re-alignment. The change in alignment can be specified as the set of edges in π- or π+, but not both; i.e., by considering a path to be a set of edges and taking the set symmetric difference Δπ = (π- ∪ π+) - (π- ∩ π+). The path π+ after re-alignment is available from the dynamic programming traceback. The path π- before re-alignment can be efficiently computed in O(LP) time. Note that in order to construct the profile of a subset of sequences extracted from a multiple alignment, those columns that contain only indels in that subset must be deleted. The set of such columns in both profiles is therefore available as a side effect of profile construction, and this set immediately implies π-. It is a simple O(LP) procedure to compute Δπ from π- and π+. Note that SPa is a sum over columns, and there is a one-to-one correspondence between columns and edges in π. The change in SPa can therefore be computed as a sum over columns in Δπ, with a negative sign for edges from π-, reducing the time complexity from O(NLP) to O(N|Δπ|). We now turn our attention to SPg. We say that a gap G intersects Δπ if and only if any indel in G is in a column in Δπ, and denote by Γ the set of gaps that intersect Δπ. If a gap does not intersect Δπ, i.e. does not have an indel in a changed column, its contribution to SPg is unchanged. It therefore suffices to consider penalties for gaps in Γ, again with negative signs for edges from π-. The construction of Γ is straightforward in O(NLP) time. Finally, a sum over pairs in Γ is needed, reducing the O(N2) component to the smallest possible set of terms.This reduces the average time by a factor of about two. We can further improve on this by noting that in the typical case, there are few or no changes to the alignment. This suggests computing the change in SP score by looking only at differences between the two alignments. Let NLP) time. Define a two-symbol alphabet {X, -} in which X represents any amino acid and - is the indel symbol. There are four dimers in this alphabet: XX, X-, -X and --, which denote by no-gap, gap-open, gap-close and gap-extend respectively. Re-write a multiple alignment in terms of these dimers, adopting the convention that dimer ab composed of symbol a in column x-1 and symbol b in column x is written in column x. Now consider the contribution to SPg of an aligned pair of dimers, written as ab↔cd. Clearly XX↔X- adds a gap-open penalty; XX↔-X adds a gap-close , and compute the change in SP by considering only those columns in Δπ. We find use of the dimer approximation to marginally reduce benchmark scores. By default, MUSCLE therefore uses the exact SP score for N ≤ 100 and the dimer approximation for N > 100, where the higher time complexity of the exact score becomes more noticeable.We next describe an approximation to SP that can be computed in O against the number of false pairs with score ≤ S (y axis); we call the resulting graph a discrimination plot. Ideally, all true pairs would score higher than all false pairs, in which case the profile function would be a perfect discriminator and would always produce perfect alignments. A function that perfectly discriminates will appear as a Γ-shaped plot; a function that has no ability to discriminate will appear as a diagonal plot along the line x = y. If a function F has a discrimination plot that is always above another function G  > DG(x) ∀ x, where DF is the discriminator plot for F as a function of x), then F has a superior ability to discriminate true from false pairs compared with G. If the plots intersect, the situation is ambiguous and neither function is clearly superior. We used sets of structural alignments from [m(F) where m(x) is a monotonically increasing function). However, a monotonic transformation may change the alignments produced by a profile function, so we can regard high discrimination as a necessary but not sufficient condition for a good profile function. One can turn this into a virtue by noting that the discrimination plot allows the relative probability of true versus false to be determined from a score. It is therefore possible to numerically determine a log-odds function from the discrimination plot, which can be evaluated by table look-up. Using discrimination plots for PP2, we found the optimal transformation for LE to be close to linear, in contrast to other functions we tried, including PSP (results not shown). This observation further encouraged us to explore the performance of LE in an MSA algorithm. Testing on multiple alignment benchmarks we find LE to give superior results on BAliBASE, but statistically indistinguishable results on other databases (results not shown). MUSCLE therefore uses LE as the default choice as it sometimes gives better results but has not been observed to give lower average accuracy on any of our tests. It is also useful to introduce a method with a distinctively different scoring scheme as an alternative that may give better results on some input data and may provide unique features for incorporation into jury or consensus systems. One drawback of LE is its relatively slow performance due to the need to compute a logarithm for each cell of the dynamic programming matrix.We have previously attempted a systematic comparison of profile functions . The metnts from  (PP) andnts from  (PP2). Pnts from  with ≤ 3nts from , LAMA [4nts from . Using PLP = O(L + N), the e-string construction for the root alignment, and a fixed number of refinement iterations.The complexity of MUSCLE is summarized in Table FBinary is used as a distance measure (Equation 2), the PSP profile function is used, and diagonal finding is enabled.MUSCLE offers a variety of options that offer different trade-offs between speed and accuracy. In the following, we report speed and accuracy results for three sets of options: (1) the full MUSCLE algorithm including Stages 1, 2 and 3 with default options; (2) Stages 1 and 2 only, using default options (MUSCLE-prog); and (3) Stage 1 only using the fastest possible options (MUSCLE-fast), which are as follows: Q, the number of residue pairs correctly aligned divided by the length of the reference alignment. For more discussion of the reference data, assessment methodology and a comparison of MUSCLE with T-Coffee and NWNSI, the most accurate MAFFT script, see [In Tables ipt, see .LP is O(L). Results are shown in Figure To compare speeds for a larger number of sequences, we created a test set by using PSI-BLAST to search the NCBI non-redundant protein sequence database for hits to dienoyl-coa isomerase (1dci in the Protein Data Bank ), selectMUSCLE demonstrates improvements in accuracy and reductions in computational complexity by exploiting a range of existing and new algorithmic techniques. While the design–typically for practical multiple sequence alignment tools–arguably lacks elegance and theoretical coherence, useful improvements were achieved through a number of factors. Most important of these were selection of heuristics, close attention to details of the implementation, and careful evaluation of the impact of different elements of the algorithm on speed and accuracy. MUSCLE enables high-throughput applications to achieve average accuracy comparable to the most accurate tools previously available, which we expect to be increasingly important in view of the continuing rapid growth in sequence data..MUSCLE is a command-line program written in a conservative subset of C++. At the time of writing, MUSCLE has been successfully ported to 32-bit Windows, 32-bit Intel architecture Linux, Solaris, Macintosh OSX and the 64-bit HP Alpha Tru64 platform. MUSCLE is donated to the public domain. Source code and executable files are freely available at


The hit criterion is a key component of heuristic local alignment algorithms. It specifies a class of patterns assumed to witness a potential similarity, and this choice is decisive for the selectivity and sensitivity of the whole method.group criterion combining the advantages of the single-seed and double-seed approaches used in existing algorithms. Second, we introduce transition-constrained seeds that extend spaced seeds by the possibility of distinguishing transition and transversion mismatches. We provide analytical data as well as experimental results, obtained with the YASS software, supporting both improvements.In this paper, we propose two ways to improve the hit criterion. First, we define the .Proposed algorithmic ideas allow to obtain a significant gain in sensitivity of similarity search without increase in execution time. The method has been implemented in YASS software available at  The central problem is therefore to improve the trade-off between those opposite requirements.Sequence alignment is a fundamental problem in Bioinformatics. Despite of a big amount of efforts spent by researchers on designing efficient alignment methods, improving the alignment efficiency remains of primary importance. This is due to the continuously increasing amount of nucleotide sequence data, such as EST and newly sequenced genomic sequences, that need to be compared in order to detect similar regions occurring in them. Those comparisons are done routinely, and therefore need to be done very fast, preferably instantaneously on commonly used computers. On the other hand, they need to be precise, i.e. should report all, or at least a vast majority of interesting similarities that could be relevant in the underlying biological study. The latter requirement for the alignment method, called the hit) a potential similarity. Those patterns are formed by seeds which are small strings  that appear in both sequences. FASTA  = S2[j..j + k - 1] for some i ≤ m and j ≤ n, then these two k-words form a seed denoted <i, j>. Two functions on seeds are considered: For a seed <i, j>, the seed diagonal d is m + j - i. It can be seen as the distance between the k-words S1[i..i + k - 1] and S2[j..j + k - 1] if S2 is concatenated to S1, For two seeds <i1, j1> and <i2, j2>, where i1 <i2 and j1 <j2, the inter-seed distance D is the maximum between |i2 - i1| and |j2 - j1|. The problem considered in this Section is to derive conditions under which two seeds are likely to be a part of the same alignment, and therefore should be grouped together. More precisely, we want to be able to compute parameters ρ and δ such that two seeds <i1, j1> and <i2, j2> have a probability (1 - ε) to belong to the same similarity iffWe first introduce some notations used in this section. Let D ≤ ρ,     (1)d - d| ≤ δ.     (2)|k-words occur at close diagonals.The first inter-seed condition insures that the seeds are close enough to each other. The second seed diagonal condition requires that in both seeds, the two ρ and δ.We now describe statistical models used to compute parameters p for a match and (1 - p) for a mismatch. To estimate the inter-seed shift Dk, we have to estimate the distance between the starts of two successive runs of at least k matches in the Bernoulli sequence. It obeys the geometric distribution of order k called the Waiting time distribution [Consider two homologous DNA sequences that stem from a duplication of a common ancestor sequence, followed by independent individual substitution events. Under this assumption, the two sequences have an equal length and their alignment is a sequence of matched and mismatched pairs of nucleotides. We model this alignment by a Bernoulli sequence with the probability ribution ,26:ρ such that the probability  is (1 - ε) for some small ε.Using this formula, we compute k inside a Bernoulli sequence of length x. In a Bernoulli sequence of length x, the probability of the event Ip,x,r of having exactly r non-overlapping runs of matches of length at least k is given by the following recursive formula:Note that the Waiting time distribution allows us to estimate another useful parameter: the number of runs of matches of length at least r non-overlapping seeds of length at least k inside a repeat of size x. The recurrence starts with r = 0, in which case  and is computed through the Waiting time distribution.This gives the probability of having exactly  allows us to infer a lower bound on the number of non-overlapping seeds expected to be found inside a similarity region. In particular, we will use this bound as a first estimate of the group criterion introduced later.The distribution Indels (nucleotide insertions/deletions) are responsible for a diagonal shift of seeds viewed on a dotplot matrix. In other words, they introduce a possible difference between d and d. To estimate a typical shift size, we use a method similar to the one proposed in [posed in  for the q at each of l nucleotides separating two consecutive seeds. Under this assumption, estimating the diagonal shift produced by indels is done through a discrete one-dimensional random walk model, where the probability of moving left or right is equal to q, and the probability of staying in place is 1 - 2q. Our goal is to bound, with a given probability, the deviation from the starting point.Assume that an indel of an individual nucleotide occurs with an equal probability i after l steps is given by the following sum:The probability of ending the random walk at position  and consider the power Pl(x) = al.xl +…+ al-.xl-. Then the coefficient ai computes precisely the above formula, and therefore gives the probability of ending the random walk at position i after l steps. We then have to sum up coefficients ai for i = 0,1, -1, 2, -2,..., l, -l until we reach a given threshold probability (1 - ε). The obtained value l is then taken as the parameter δ used to bound the maximal diagonal shift between two seeds.A direct computation of multi-monomial coefficients quickly leads to a memory overflow, and to circumvent this, we use a technique based on generating functions. Consider the function


Multiple genome alignment is an important problem in bioinformatics. An important subproblem used by many multiple alignment approaches is that of aligning two multiple alignments. Many popular alignment algorithms for DNA use the sum-of-pairs heuristic, where the score of a multiple alignment is the sum of its induced pairwise alignment scores. However, the biological meaning of the sum-of-pairs of pairs heuristic is not obvious. Additionally, many algorithms based on the sum-of-pairs heuristic are complicated and slow, compared to pairwise alignment algorithms.An alternative approach to aligning alignments is to first infer ancestral sequences for each alignment, and then align the two ancestral sequences. In addition to being fast, this method has a clear biological basis that takes into account the evolution implied by an underlying phylogenetic tree.In this study we explore the accuracy of aligning alignments by ancestral sequence alignment. We examine the use of both maximum likelihood and parsimony to infer ancestral sequences. Additionally, we investigate the effect on accuracy of allowing ambiguity in our ancestral sequences.We use synthetic sequence data that we generate by simulating evolution on a phylogenetic tree. We use two different types of phylogenetic trees: trees with a period of rapid growth followed by a period of slow growth, and trees with a period of slow growth followed by a period of rapid growth.We examine the alignment accuracy of four ancestral sequence reconstruction and alignment methods: parsimony, maximum likelihood, ambiguous parsimony, and ambiguous maximum likelihood. Additionally, we compare against the alignment accuracy of two sum-of-pairs algorithms: ClustalW and the heuristic of Ma, Zhang, and Wang.We find that allowing ambiguity in ancestral sequences does not lead to better multiple alignments. Regardless of whether we use parsimony or maximum likelihood, the success of aligning ancestral sequences containing ambiguity is very sensitive to the choice of gap open cost. Surprisingly, we find that using maximum likelihood to infer ancestral sequences results in less accurate alignments than when using parsimony to infer ancestral sequences. Finally, we find that the sum-of-pairs methods produce better alignments than all of the ancestral alignment methods. Multiple genome alignment is an important problem in bioinformatics. It is used in comparative studies to help find new genomic features such as genes and regulatory elements. Current multiple genome alignment programs ,2 use prThe primary operation of progressive alignment is the alignment of two multiple alignments. Most genome aligners have two main phases: anchoring and aligning between the anchors. Here, we focus on the algorithms used to align between anchors. Many popular alignment algorithms for DNA use the sum-of-pairs heuristic, where the score of a multiple alignment is the sum of the induced pairwise alignment scores. However, Just  has showThe biological meaning of the sum-of-pairs of pairs heuristic is not obvious. Additionally, many heuristic algorithms are complicated and slow, compared to pairwise alignment algorithms -7. An alBray and Pachter use this approach to align alignments in MAVID . MAVID uIn this study, we explore this idea as well as other aspects of aligning alignments by ancestral sequence inference. We compare four ancestral alignment methods with two sum-of-pairs alignment algorithms. We infer ancestral sequences using parsimony and maximum likelihood, and study the effect of allowing ambiguity in these sequences. Since we are interested in the performance of these methods under optimal conditions, we use data generated by a very simple evolution simulation. For aligning full alignments with the sum-of-pairs heuristic, we use ClustalW  and a neWe find that alignment algorithms based on the sum-of-pairs heuristic are more accurate than all of our methods based on ancestral sequence alignment. However for alignment of inferred ancestral sequences, parsimony outperforms maximum likelihood in this application. Using maximum likelihood to infer ancestral sequences results in final alignment accuracies that are more unpredictable. Also, computing log-odds for ancestral sequences inferred with maximum likelihood is far more computationally intensive than computing log-odds scores for ancestral sequences inferred with parsimony. Finally, we find that allowing ancestral sequences to have ambiguity does not result in more accurate final alignments.To determine whether using ambiguous symbols in ancestral sequences inference improves multiple alignment, we have performed experiments on simulated sequences. We propose five hypotheses, explain our experimental method, and finally discuss results and give conclusions.The first hypothesis is that by using ancestral sequences with ambiguity, we obtain more accurate multiple alignments. Ambiguous symbols may allow us to retain more information about the underlying multiple alignments, which may make it easier to identify matching positions. Combined with an appropriate log-odds scoring system, this extra information may allow for more accurate alignment of ancestral sequences, and by extension, for more accurate multiple alignments.Our second hypothesis is that alignment of ancestral sequences is more sensitive to gap open costs than alignment of alignments using the sum-of-pairs heuristic. When aligning ancestral sequences, existing gaps in the underlying alignment are not considered when inserting a new gap, so the first position of a new gap always costs the gap open cost. Incorrect gap penalties may cause too many gaps to be inserted between ancestral sequences. During progressive alignment, errors at each step propagate leading to an incorrect final alignment. In contrast, when aligning alignments using the sum-of-pairs heuristic, the cost of adding a new gap depends on all underlying gaps as well as the gap open cost. An incorrect gap open cost affects the cost of gaps less and new gaps may still be correctly inserted based on the structure of the existing gaps. Our third hypothesis is that the function used to estimate the gap open cost during progressive alignment is important to alignment accuracy when aligning ancestral sequences. When aligning ancestral sequences, the frequencies of gaps in the ancestral sequences depends on the amount of mutation between the sequences. Therefore, it is important to modify the gap open cost based on the distance between the ancestral sequences being aligned.Our fourth hypothesis is that we expect that aligning alignments using the sum-of-pairs heuristics gives more accurate multiple alignments than aligning inferred ancestral sequences. There are two reasons for this. First, as stated in hypothesis two, using an incorrect gap open cost affects the sum-of-pairs heuristic less than it affects the alignment of ancestral sequences. Since choosing the correct gap open cost can be difficult in practice, we expect that aligning alignments using the sum-of-pairs heuristic results in a more accurate final alignment because it is less sensitive to this parameter. Additionally, the ancestral sequences we infer are not completely accurate, which compounds the errors made in the process of progressive alignment. Thus, the much slower run times of algorithms based on the sum-of-pairs heuristic are acceptable.Finally, we expect that the maximum likelihood methods result in better multiple alignments than the parsimony methods. Unlike parsimony, maximum likelihood uses the edge distances on the phylogenetic tree. Thus we expect maximum likelihood to better infer ancestral sequences.We use synthetic data in order to have correct alignments to test our methods against. Additionally, by generating our own data we ensure that the data is generated from the same model of evolution that is required for the alignment algorithm. Therefore, we consider the performance of the algorithms on this data to be the the best possible for algorithms of their type.Despite our use of synthetic data, we want our data to mimic the basic properties of real biological sequence. Thus we generate random trees that resemble real trees, and assign mutation rates based on analysis of real sequences.Specifically, we are interested in algorithm performance on two different types of random trees: trees with a period of heavy growth followed by a period of no growth, and trees with a period of light growth followed by a period of heavy growth. The first type of tree, which we refer to as early growth, is similar to the tree of placental mammals from Eizirik, Murphy, and O'Brien . The secWe generate these in two steps. First, we generate one large tree of each type using a random birth-death process implemented in Phyl-O-Gen v1.2 . To geneFrom each final tree, using the method of Kearney, Munro, and Phillips , we randFor each of the eight taxon trees in our two data sets, we generate twenty random sequence sets by simulating evolution over the tree. We use a program written by us, but similar to ROSE , to geneSince we wanted our mutations, insertions, and deletions to be as close to real sequences as possible, we calibrated our simulator with parameters estimated through analysis of homologous human and baboon sequences. This gives us two sets of 400 random input sequences, one for each set of trees.We chose the CFTR region in human and baboon for this parameter estimation, so that we could ensure that our alignment is mostly correct. We obtained human and baboon sequences with repeats masked out from the NISC Comparative Vertebrate Sequencing project . We alig-rαte),     (1)Pr[mutation] = 3/4  accuracy. Tables It is not clear that including ambiguity in ancestral sequences improves alignment. Looking at Tables Table Our experiment confirmed our hypothesis that the gap cost scaling function is very important to the resulting alignment accuracies. When changing from scaling based on the largest value in the ancestral sequence scoring matrix to the expected cost for related positions, we see a significant increase in alignment accuracy on all data sets and all methods. See Table In Tables Surprisingly, the maximum likelihood methods performed worse than parsimony methods in the context of ancestral alignment. In Tables We have tested four ancestral alignment methods as well as two sum-of-pairs alignment methods on simulated data. The data mimics evolution on two types of evolutionary trees: trees with a period of rapid growth followed by a period of slow growth, and trees with a period of slow growth followed by a period of rapid growth. The four ancestral alignment methods we have tested are unambiguous parsimony, ambiguous parsimony, unambiguous maximum likelihood, and ambiguous maximum likelihood. The sum-of-pairs alignment methods we have tested are the ClustalW  algorithWe have found that, contrary to our hypotheses, allowing ambiguity in ancestral sequences does not lead to better alignments. When we use ambiguous ancestral sequences, we find that the multiple alignment is more sensitive to our choice in gap costs than to the form of ancestral sequence chosen. Reinforcing this conclusion, we find that the gap open cost scaling function is also extremely important to obtaining good scores when aligning ancestral sequences. Finally, to our surprise, using maximum likelihood to infer ancestral sequences resulting in less accurate alignments than using parsimony. The reason for this is that the maximum likelihood method is far more sensitive to the underlying data and therefore resulting in alignments accuracies that have a large amount of variation. Also, on the data set generated from the tree that has a small amount of growth followed by a large amount of growth, the maximum likelihood based methods did particularly poorly compared to the parsimony based methods.Finally, both the sum-of-pairs approaches did better than all the ancestral alignment methods, as expected. Additionally, we found that Ma, Wang, and Zhang's algorithm  outperfoOur multiple alignment framework uses progressive alignment up a specified phylogenetic tree. At each internal node we perform an alignment of two multiple alignments. We test six different algorithms for aligning alignments: ClustalW, the recent algorithm of Ma, Wang, and Zhang, and four algorithms that align inferred ancestral sequences. Our four ancestral alignment algorithms explore the use of both parsimony and maximum likelihood to infer ancestral sequences, and also allow the use of both ambiguous and unambiguous ancestral sequences. We include ClustalW, as it is widely used in practice, and the algorithm of Ma, Wang, and Zhang, whose output more accurately approximates the optimal alignment under sum-of-pairs scoring.In this section, we describe how we align two alignments using ancestral sequence inference, as well as our four ancestral sequence inference techniques and associated log-odds scoring frameworks.n alignment of k sequences, we can infer an ancestral sequence in time Θ(kn). To align two length n alignments, one with k sequences and with ℓ sequences, requires Θ((k + ℓ)n + n2) time; this contrasts with the much larger run time of O (n2 (k + ℓ)) required by Ma, Wang and Zhang's algorithm  of seeing the observed symbols d1 and d2 in ancestral sequence positions related by evolution according to the given tree, and Pr, the probability of seeing observed symbols d1 and d2 in ancestral sequences positions unrelated by evolution. Here, d1 and d2 are either from the ambiguous or unambiguous alphabet depending on the type of ancestral sequence we use.We compute two probabilities: the probability PrCv     (3)cv at node v given the true value is bv, times the probability that the true value at node v is bv given that the true value at node u is b. We compute Pr[Tv(bv)|Tu(b)] using the length of the edge  and the probabilistic mutation model. We compute Cv for each node of the tree, and obtain Cz in O(m) time, where m is the number of leaves.     □is the the probability of seeing c1 and c2 at a positions arising from a common ancestor isWhen we use ambiguous ancestral sequences, the probability of seeing consensus letters x, y, and z, we compute the conditional probabilities of seeing consensus pair  at positions related by ancestry.That is, over all choices of the true value at c1 and c2 isAt positions unrelated by ancestry, we assume that the true value for the ancestral sequence is equally likely to be any of the four DNA bases, though this model can be made more complex to model more realistic sequences. Therefore, the probability of seeing consensus letters c1 and c2 isFinally, the log-odds score for aligning consensus symbols S = log2 ,     (6)in bits.V be the probability that we randomly choose base b from the set of bases represented by consensus symbol c ∊ Γ. That is, if c is a symbol corresponding to a set of k symbols from Σ, and b is in this set, then V = 1/k, otherwise it is zero. Then,If we use unambiguous ancestral sequences, the situation is similar. Let b1 at x and DNA base b2 at y given that we are looking at positions arising from a common ancestor.is the probability that we see DNA base b1 and b2 isAt positions unrelated by ancestry, we assume that the true value for the ancestral sequence is equally likely to be any of the four DNA bases. Therefore, the probability of seeing DNA base b1 and b2 isThe log-odds score for DNA symbols S = log2 ,     (9)in bits.For a given alignment column, we compute the most likely ancestral DNA base as in Felsenstein , but modW represents an A or a T. We define associated vector , scaled to a probability distribution, where the numbers in the vector refer to A,T,C, and G, in that order. We then map the likelihood vector, also scaled to a probability vector , to an IUPAC symbol, choosing the IUPAC symbol with associated vector that has the closest euclidean distance to the likelihood vector scaled to a probability distribution.Upon completion of the basic inference algorithm, we have a vector at the root that gives, for each position and each DNA base, the likelihood of that base. Assuming independence, the likelihood of a given ancestral sequence is the product of these. We obtain an unambiguous ancestral sequence from this by taking the base with maximum likelihood, randomly choosing between ties. To obtain an ambiguous maximum likelihood, which in effect is an approximation of the posterior Bayesian distribution of the ancestral symbol at that site, where we assume a uniform prior distribution over all alphabet symbols, we map the vector to an IUPAC symbol as follows. For each IUPAC symbol, we define a vector over the DNA alphabet where we have a one for each DNA symbol described by the IUPAC symbol, and a zero for all other DNA symbols. For example, the IUPAC symbol We desire a log-odds scoring framework similar to that developed for parsimonious ancestral sequences. While an appropriate scoring matrix can be obtained by sampling, we want to eliminate any sampling error from our study and so choose to compute the log-odds scoring framework directly.z with children x and y in a tree rooted at node r. First, consider the sub tree rooted at x. Let G to the leaves of the sub tree rooted at x such that maximum likelihood infers symbol d at x.Assume we are at node Ox(a) be the probability that a particular assignment a of bases to the leaves of sub tree x occurs by evolution. We compute the probability that the true DNA base at node x is b by considering the probability all possible bases for the root of the tree, and simulating evolution down to b.Let d1 and d2 refer to symbols from ΣG. To compute scores for ambiguous ancestral sequences, d1 and d2 refer to symbols from Γ. We compute Pr, the probability of seeing symbols d1 and d2 in positions related by a common ancestor, asIn the following, as we compute scores for unambiguous ancestral sequences, z is b aswhere we compute the probability that the true value at node r, we compute the probability that the value mutates to b on the path from r to z.That is, for each possible true value at root d1,d2|unrelated], the probability that we see symbols d1 and d2 in positions unrelated by a common ancestor. First, let MLx(d) be the event that maximum likelihood infers symbol d at node x. We compute MLx(d) asWe now compute Pr[Tx(b)] in the same way as Pr[Tz(b)] previously. We compute Pr aswhere we compute Pr = Pr[MLx(d1)] Pr[MLy(d2)].     (13)Pr[d1 and d1 isFinally, the log-odds score of symbols S = log2 .     (14)k be the maximum of the number of taxa below the left child of the root and the right child of the root. We require time O(|ΣG|k) to compute the log-odds scores for this tree.To compute the above scores, we must examine every possible assignment of bases to leaves for both the left and right children of the root. For a particular tree, let When aligning alignments with ancestral sequences, gap open costs play a major role. In a tree with differing edge lengths, the gap open cost should also be able to vary. Since existing gaps are not considered when inserting a new gap in ancestral alignments, the gap open cost has a large influence over the quality of the resulting alignment. It is therefore important that we estimate an appropriate gap open cost for each node of the tree.z with children x and y. We scale the gap open cost according to the distance between the nodes x and y. We test two slightly different scaling functions. The first function multiplies the gap open cost by the largest score value in the log-odds scoring matrix for node z. We call this the Max method. The second function computes the expected score of two symbols from unrelated positions and uses this value to scale the gap open cost. We call this the Expected method.Consider creating alignment for node Both AH and DB developed the fundamental ideas and hypotheses and the mathematical framework for deriving the alignment scores. AH developed and ran the experiments and implemented the alignment algorithms.


Topological descriptors, other graph measures, and in a broader sense, graph-theoretical methods, have been proven as powerful tools to perform biological network analysis. However, the majority of the developed descriptors and graph-theoretical methods does not have the ability to take vertex- and edge-labels into account, e.g., atom- and bond-types when considering molecular graphs. Indeed, this feature is important to characterize biological networks more meaningfully instead of only considering pure topological information.In this paper, we put the emphasis on analyzing a special type of biological networks, namely bio-chemical structures. First, we derive entropic measures to calculate the information content of vertex- and edge-labeled graphs and investigate some useful properties thereof. Second, we apply the mentioned measures combined with other well-known descriptors to supervised machine learning methods for predicting Ames mutagenicity. Moreover, we investigate the influence of our topological descriptors - measures for only unlabeled vs. measures for labeled graphs - on the prediction performance of the underlying graph classification problem.Our study demonstrates that the application of entropic measures to molecules representing graphs is useful to characterize such structures meaningfully. For instance, we have found that if one extends the measures for determining the structural information content of unlabeled graphs to labeled graphs, the uniqueness of the resulting indices is higher. Because measures to structurally characterize labeled graphs are clearly underrepresented so far, the further development of such methods might be valuable and fruitful for solving problems within biological network analysis. Major reasons for the emergence of biological network analysis -4 are thANG et al. [UBER et al. [Taking into account that a large number of graph-theoretical methods have been developed so far, approaches to process and meaningfully analyze labeled graphs are clearly underrepresented in the scientific literature. In particular, this holds for chemical graph analysis where various graph-theoretical methods and topological indices have been intensely used, see, e.g., -34. Yet,G et al.  recentlyG et al. . FinallyR et al.  reviewedR et al. .In this paper, we restrict our analysis to a set of bio-chemical graphs which have already been used for predicting Ames mutagenicity, see . To perfSHANNON's entropy for predicting Ames mutagenicity, see [As already mentioned, topological indices have been proven to be powerful tools in drug design, chemometrics, bioinformatics, and mathematical and medicinal chemistry ,34,41-43ity, see ,45-47. Gity, see ,45-47.HANNON's entropy to characterize graphs by determining their structural information content [Further, topological descriptors have often been combined with other techniques from statistical data analysis, e.g., clustering methods ,48 to in content ,52-54. U content ,52,54-60 content ,52,61. R content ,62. More content . In this content ,52,53. N content ,53,54,63The contribution of our paper is twofold: First, we develop some novel information-theoretic descriptors having the ability to incorporate vertex- and edge-labels when measuring the information content of a chemical structure. Because we already mentioned that there is a lack of graph measures which can process vertex-and edge-labeled graphs meaningfully, such descriptors need to be further developed. In terms of analyzing chemical structures, that means they can only be adequately represented by graphs if different types of atoms (vertices) and different types of bonds (edges) are considered. Hence, there is a strong need to exploring such labeled networks. Besides developing the novel information-theoretic measures for vertex- and edge-labeled graphs, we will investigate some of their properties thereof (see section 'Properties of the Novel Information-Theoretic Descriptors') [ESHPANDE et al. [UE et al. [UE et al. [AHÉ et al. [iptors') ,47. Secoiptors') ,65 (RF) iptors') ,66 (SVM)iptors') . Furtheriptors') -69. For E et al.  developeE et al.  that deaE et al. . ParticuE et al.  determinE et al. . The lasÉ et al. ,69 was vÉ et al. ,71-74.To present the novel information-theoretic measures for labeled (weighted) graphs, we express some graph-theoretical preliminaries ,57,75-77Definition 1 is a finite, undirected graph. In this paper, we always assume that the considered graphs are connected and do not have loops.Definition 2 Let G be a finite and undirected graph. δ(v) is called the degree of a vertex v ∈ V and equals the number of edges e ∈ E which are incident with v.Definition 30 d stands for the distance between u ∈ V and v ∈ V expressed as the minimum length of a path between u,v. Further, the quantity σ(v) = maxu∈V d is called the eccentricity of v ∈ V. ρ(G) = maxv∈V σ(v) is called the diameter of G.Definition 4 We callthe j-sphere of a vertex vi regarding G.Definition 5 Letandbe unique (finite) vertex and edge alphabets, respectively. and are the corresponding edge and vertex labeling functions. G :=  is called a finite, labeled graph.Definition 6 LetClearly, vi, are equal to j and possess the vertex label denotes the cardinality of the set of vertices whose distances, starting from G. In the following, we will use this definition to derive an advanced information functional for incorporating edge- and vertex-labels when measuring the structural information content of a labeled network.To finalize this section, we repeat the definition  of a so-Definition 7 Let G =  be an undirected graph. For a vertex vi ∈ V, we calculate and the induced shortest paths,kj stands for the number of shortest paths of length j. Their edge sets are defined byFurther, letandThe local information graph ℒG of G regarding vi is finally defined byFig. EHMER et al. [f E (see next section), we also deal with certain graph partitions for quantifying the information content of a vertex- and edge-labeled graph because we have to compute all local information graphs . But nonetheless, the construction of our information measures basically differs from the ones mentioned in [As already outlined, the majority of classical information measures for graphs are based on determining partitions by using an arbitrary graph invariant and an equivalence criterion, see, e.g., ,48,53,54R et al. ,62 recenR et al. ,62. Thisioned in   be an arbitrary finite graph. The vertex probabilities for each v i ∈ V are defined by the quantitiesf represents an arbitrary information functional.Definition 9 Let G =  be an arbitrary finite graph. Then, the entropy of G is defined byNow, we repeat the definition of an information functional for quantifying the structural complexity of unlabeled and unweighted chemical graphs . GeneralDefinition 10 Let G =  be an undirected finite graph. For a vertex v i ∈ V, the information functional f V is defined asRemark 1 We want to point out that further information functionals have been developed so far [The appropriateness of such a functional that captures structural information of a graph strongly depends on the graph class and on the specific problem under consideration.d so far . The appAnother measure to determine the structural information content is the following one. Until now, it has been used  to perfoDefinition 11 Let G =  be an undirected finite graph. We define the family of information measureswhereis a scaling constant.λ > 0 In this section, we present novel information measures to quantify structural information of labeled (weighted) chemical structures by adapting the just shown approach. Because the majority of the developed topological indices is only defined for the underlying skeleton of a chemical structure, the further development of descriptors for processing chemical graphs containing heteroatoms and multiple bonds is generally of great importance. Before we start expressing the new definitions, we first point out some related work in this area.VANCIUC et al. [IENER index [U, V, X, Y [VANCIUC et al. [Note that earlier contributions to infer measures for labeled graphs are often based on special distance matrices and polynomial methods -80. AnotC et al.  where thC et al. . For exaC et al. . Then, sER index  for vertER index . Further V, X, Y  have bee V, X, Y . As a reC et al. ,84 obtaij-spheres) for all involved atoms (vertices) of the molecule. By now considering labeled graphs, our first attempt results in an information functional with the property that every vertex in each j-sphere possessing a certain vertex label (atom type) will be weighted differently.We now start by stating the novel partition-independent information-based descriptors to determine the information content of vertex- and edge-labeled graphs. The first definition represents an information functional to account for vertex labels of a chemical structure. For this, we adapt the idea ,62 of deDefinition 12 Let G =  be an undirected finite vertex-labeled graph, We defineExample 2 To demonstrate the calculation of  exemplarily, we consider Fig. 1and set O, C and N denote the atom types of the molecule. The edge type s represents a single bond whereas d represents a double bond within the chemical structure. For example, if we now calculate  for G shown in Fig. we yieldBecause it is not always clear how to choose the involved parameter in practice, we further derive an information functional to overcome this problem.Definition 13 Let G =  be an undirected finite vertex-labeled graph, If we determine all local information graphs ℒG of G for the vertices vi ∈ V, we then define the quantitiesThis quantity denotes the number of vertices of ℒG possessing vertex label Definition 14 Let G =  be an undirected finite vertex-labeled graph, We define the information functionalwhere Remark 3 We note thatThe expressionquantifies the number of occurrences of vertex label inExample 4 Fig. 2shows the calculated local information graphs of G regarding v3. For example, this leads toBy determining all local information graphs for the remaining vertices of G, the just shown calculation can be performed analogously.G into account. The main idea is to use weighted paths which can be directly determined by calculating the local information graphs.Next, we are able to derive an information functional that takes the edge labels of a graph Definition 15 Let G =  be an undirected finite edge-labeled graph, , and assume that there exists a correspondence between the edge labels and numerical values. We definewhereandNow, we present an example how to apply this definition to the local information graphs shown in Fig. Example 5 We exemplarily apply the information functional f E to G and v3 as the starting vertex and recall that s = 1, d = 2. The edge labeled local information graphs for this vertex are depicted in Fig. We yield,andThus,In order to incorporate both edge and vertex labels when determining the topological entropy of a labeled graph, we also deriveDefinition 16Finally, we obtain the following entropy measures for measuring the structural information content of labeled graphs.Definition 17 Let G =  be an undirected finite labeled graph, We now straightforwardly define the information-theoretic descriptors (graph entropy measures) as follows:Remark 6 We emphasize that according to the above stated definition and the definitions of the underlying information functionals, the resulting information measures are obviously parametric. This property generalizes classical information measures which have often been used in mathematical chemistry, see, e.g., [As already pointed out in [such measures establish a link to machine learning because the parameters could be learned using appropriate datasets. However, we won't study this problem in the present paper.e, e.g., ,29,53,83d out in , such meThis section aims to evaluate the just presented (see previous section) information measures for labeled graphs numerically. Also, we will calculate some known information indices to tackle the second part of our study when applying these measures to machine learning algorithms. Our study will be twofold: First, we examine some properties of the measures for labeled graphs when applying them to a large set of real chemical structures. Second, we analyze a QSAR problem by applying supervised machine learning methods ,85 usingV| ≤ 109; 1 ≤ ρ(G) ≤ 47 ∀ G ∈ AG 3982. To evaluate the novel descriptors for labeled graphs, we then considered these structures as vertex- and edge-labeled graphs. Evidently, for calculating the descriptors of the unlabeled graph versions (skeletons), the corresponding descriptors were used which take only topological information into account.We created the database AG 3982 from the benchmark database called Ames mutagenicity ,47 origiTo generate and process the underlying graph structures, we used the known Molfile format . The graBefore starting to evaluate our novel molecular descriptors, we define some concrete information measures by choosing special weighting schemes for the coefficients.Definition 18 We define a special weighting scheme for the coefficients to determine as follows: Starting fromwhere ma denotes the atomic mass of the atom a (in the i-th sphere), we also defineThe scheme starts with the lightest element Hydrogen (H) and ends with the heaviest one, namely Uranium (U). If the underlying ci will be chosen byand by using Definition (11) and Definition (17), the concrete information-theoretic descriptors are called and If the underlying ci will be chosen bythe measures and follow correspondingly. Further, if the underlying ci will be chosen linearly or exponentially decreasing ; see also that the measures and follow correspondingly (Equation (50), (35), (52), (53)).Definition 19 Let G =  be an undirected finite labeled graph, If we choose the coefficients of information functional (see Equation (29)) linearly or exponentially decreasing, we call the resulting information measures and λ = 1000 to perform the entire numerical calculations in this paper. In order to interpret some of these measures, we consider Fig. k-regular graphs, the measure Ifv always leads to maximum entropy. By definition, it then follows that G0, G3 and G6, all three measures vanish. Because the graphs G1, G2 and G4, G5 have different label configurations - based on the different weighting schemes - and, therefore, the line between these points is not exactly horizontal as shown by the zoomed region depicted in Fig. Ifv and the higher the value of G0, G3 and G6 leads to descriptor values equal to zero. Again, we obtain maximal values for the calculated indices when applying them to G7 because the edge and vertex configurations are most disordered.Note, that we set j-spheres. To determine the corresponding descriptor values, we first considered the graphs of AG 3982 as only vertex-labeled graphs  into account. If we use the information functional j-sphere cardinalities , the resulting measure captures nearly the same structural information than fV, that is, we only considered the skeleton versions. The plot shows that in this case, Another problem we want to investigate relates to determine the information loss when computing the structural information content by truncating the cardinalities of the see Fig. . The notEHMER et al. [ONSTANTINOVA et al. [In order to evaluate the uniqueness  oR et al.  utilizedI. In general, I. In Table Iorb denotes the well-known topological information content developed by RASHEVSKY[W is the WIENER index [to evaluate the discrimination power of an index RASHEVSKY that is RASHEVSKY,53. W isER index  and [55,ER index (55)(56) based oNow, before discussing the classification results, we first state some definitions.Definition 20 Let I1,..., Im be topological indices. The superindex of these measures is defined as [fined as (61)Definition 21 Let G =  be an undirected finite labeled graph, Then, each graph will be represented bySI such that it consists of the twelve indices from Table m = 16. The measure IU is defined as [To perform the graph classification, we chose fined as (63)and by Equation (58). Further, we state the definitions  for(64)(whereandSI-representation (see Equation (62)) of a chemical graph, we tackle the mentioned graph classification problem using RF and SVM. The main steps were as follows:Now, based on the • We performed 10-fold crossvalidation for both classification methods.• When doing cross validation, we did a parameter optimization on the corresponding training sets. By using different kernels like linear polynomials, polynomials of higher degree etc., we found that the RBF kernel give the best results.• The random forest was composed by fifty different trees.• We performed the classification both with all features (information measures) and with only seven features gression .The classification results are shown in Table Taking into account that we classified only with (i) sixteen and (ii) seven information measures, we consider the classification results as feasible. One clearly sees that for both classifiers, the Precision and Sensitivity values - which are important quantities to evaluate the performance of the classification - are relatively high. Precision is the probability that the cases classified as positives are correctly identified where Sensitivity is the probability of positive examples which were correctly identified as such. The F-Measure defined as the harmonic mean of Precision and Sensitivity represents a single measure to evaluate the performance of the classifiers. By definition, the F-Measure varies between zero and one whereas one would represent the perfect and zero the worst classification result. We clearly see that by using SVM's, we reached values of F-Measure of over seventy percent which are the highest among all calculated ones. In order to examine the influence of incorporating vertex- and edge-labeled graphs on the prediction performance, we first present the following procedure and, then, the obtained results, see Table • Note that in our previously presented classification, we used eleven indices for unlabeled graphs and five for vertex- and edge-labeled graphs. From this feature set, we generated ten subsets composed of seven randomly selected measures for unlabeled graphs (among the eleven), and ten subsets composed of five randomly selected measures for unlabeled graphs and two measures for vertex- and edge-labeled graphs (among five available).• Based on these sets, we again performed 10-fold cross validation with RF and SVM and averaged the classification results.As a result, Table To finalize our numerical section, we also present results when choosing a different representation model of the graphs. In the following, we do not characterize a graph by its structural information content and by its superindex. In contrast, we now represent every graph by a vector that indicates if the given graphs contains certain substructures. To achieve this, we used a database  of 1365 By looking at the performance evaluation in Table This paper dealt with investigating several aspects of information-theoretic measures for vertex- and edge-labeled chemical structures. We now summarize the main results of the paper as follows:• We already mentioned that the majority of the topological indices which have been developed so far are only suitable to characterize unlabeled graphs. By adapting the approach of deriving partition-independent information measures, we developed families of information-theoretic descriptors to incorporate vertex- and edge labels when measuring the structural information content of graphs. First, we did this by calculating spherical neighborhoods and distinguishing atom types for every sphere. For the resulting measures, we presented a weighting scheme for the vertices which takes chemical information of the graphs into account. Second, to reduce the number of parameters, we developed a simplified version based on the so-called local information graphs. Generally, these graphs are induced by shortest paths and provide information about the local information spread in a network. We here assume that information spreads out via shortest paths in the network . By usin• Using the benchmark database AG 3982, we evaluated the novel information-theoretic descriptors to see how they capture structural information of the chemical graphs. Based on some characteristic properties  of the m• Another aim was to predict Ames mutagenicity when using supervised machine learning methods (RF and SVM) and representing the graphs by a vector consisting of topological descriptors (superindex). First, we performed the graph classification based on 10-fold crossvalidation and evaluated the quality of the learned models. Taking into account that we only used (i) 16 and (ii) 7 information measures for classifying the graphs, we obtained feasible results . However, another goal was to examine the influence of incorporating vertex- and edge-labels when measuring the prediction performance of the underlying graph classification problem. Here, we obtained the result that the prediction performance  was very similar to the one we obtained by only measuring skeletal  information. From this, interesting future work arises as follows: Because of the obtained results, it would be important to explore the developed measures for determining the structural information content  of the underlying vertex- and edge-labeled graphs in depth. This aims to investigate the measures such that the prediction performance could be significantly improved when applying them to the machine learning methods we have used in this paper. Another reason for the results shown in Table • As already mentioned (see section 'Introduction'), labeled graphs play an important role when analyzing biological networks. But because the theory of labeled graphs is not well developed so far (compared to the contributions which have been done towards unlabeled graphs), see, e.g., , a thoroInspired from this study, we think that especially the development of further measures for labeled graphs can be an interesting and valuable attempt not merely to analyze QSPR/QSAR problems. Besides applying these measures to machine learning methods, we believe that the measures itself might be valuable for those who will investigate biological networks, see, e.g., . In fact• As a conclusive remark, we argue from a mathematical point of view that a further development of the theory of labeled graphs will surely help to develop more sophisticated methods for analyzing biological networks, see, e.g., ,22,38,39All authors contributed equally to all aspects of the article. All authors read and approved the final manuscript.


This paper aims to investigate information-theoretic network complexity measures which have already been intensely used in mathematical- and medicinal chemistry including drug design. Numerous such measures have been developed so far but many of them lack a meaningful interpretation, e.g., we want to examine which kind of structural information they detect. Therefore, our main contribution is to shed light on the relatedness between some selected information measures for graphs by performing a large scale analysis using chemical networks. Starting from several sets containing real and synthetic chemical structures represented by graphs, we study the relatedness between a classical (partition-based) complexity measure called the topological information content of a graph and some others inferred by a different paradigm leading to partition-independent measures. Moreover, we evaluate the uniqueness of network complexity measures numerically. Generally, a high uniqueness is an important and desirable property when designing novel topological descriptors having the potential to be applied to large chemical databases. The problem to quantify the complexity of a network appears in various scientific disciplines hannon's entropy Before sketching the aims of our paper, we start with a brief review about classical and more recent approaches to measure the complexity of networks. However, for performing the numerical results, we mainly restrict our analysis to information-theoretic measures which are based on Scomplexity and, even, structural complexity is generally not uniquely defined because it is in the eye of a beholder onstantineuknainoliolmogorovolmogorov-complexity of labeled and unlabeled graphs were obtained in In general, it seems clear that Size of the giant connected component Degree distributions Exponent of degree distributions Total number of vertices and edges Path-based quantities Distance-based quantities, e.g., Degree, degree statistics and edge density Clustering coefficient, modularity and network motifs Eigenvector measures im et al. da Costa et al. laussenim et al. im et al. Further, various measures have been developed to characterize the complexity of networks where many of the recent ones were summarized by KIn this paper, we investigate information-theoretic network complexity measures which are particularly relevant for enhancing empirical QSAR/QSPR models partition-based measure that relies on symmetry with respect to topologically equivalent vertices having the same degrees. The latter is a partition-independent information measure that is based on using a special information functional capturing structural features of the networks. In order to perform this study, we evaluate these measures numerically by using several large datasets containing real and synthetic chemical graphs. To our best knowledge, such a large scale analysis involving the classical topological information content has not been done so far. Note that in this study, we only consider skeletons of the chemical structures, that is, all atoms are equal and all bonds are equal. Another problem we want to address in this paper is to investigate the uniqueness of complexity measures. This relates to examine their discrimination power, that means, their ability to discriminate non-isomorphic graphs as unique as possible. For this, we also use the mentioned databases - real and synthetic chemical structures - and calculate a special sensitivity measure To tackle this problem, we select a few measures from two different paradigms for inferring such indices: The so-called topological information content This section aims to present the information-theoretic topological descriptors we want to investigate in this paper. In the following, we briefly shed light on the two main procedures (resulting in partition-based and partition-independent measures) to infer information-theoretic complexity measures for characterizing chemical network structures. Afterwards, we express their concrete definitions for performing our numerical analysis.ruccoashevskyowshowitzApplying information-theoretic methods for exploring complex networks is a still challenging and ongoing problem ruccoashevskyashevskyhannon's entropy formulas More precisely, TowshowitzEquation (2) is the total information content of Now, we give a sketch of the second procedure for inferring graph entropy measures that results in obtaining partition-independent measures As follows, we start with the definition of some concrete partition-based entropy measures to be applied to real and synthetic chemical structures. Note that in this paper, we only evaluate the mean information contents. For the sake of simplicity, we write Let  be a graph.is called topological information content of . Here,  denotes the number of topologically equivalent vertices in the -th vertex orbit of  where  is the number of different orbits.Let  be a graph. We recall the definition  for two vertices  being topologically equivalent: For each -th neighboring vertex of  there exists an -th neighboring vertex of  which possesses the same degree. A vertex orbit is a set of vertices that only contains topologically equivalent vertices.Let  be a graph.whereiener index  is called the W and  denotes the shortest distance between .  and  are so-called magnitude-based information indices, see . It is assumed that the distance of a value  in the distance matrix  appears  times.  stands for the diameter of a graph .Let  be a graph.whereSee .  equals the number of vertices having distance  starting from . Also,  equals the corresponding -sphere cardinality.  is the eccentricity of .  denotes the cyclomatic number, see .Let  be a graph.where is a local vertex entropy . Finally, the entropy of  can be defined byIn particular, we define special information measures for characterizing graphs by choosing concrete coefficients Let  be a graph. We definewhereFinally,whereehmer et al. To finalize this section, we now express the definitions of some partition-independent entropy measures for graphs introduced by DLet  be a graph. The following partition-independent entropy measures based on a special information functional were defined as where  is a scaling constant.are vertex probabilities. The special information functional  was defined as Here,  denotes the -sphere  of a vertex , that is, the set of vertices having shortest distance  starting from .  are positive coefficients for emphasizing certain structural of a graph, e.g., high vertex degrees, also see, .To perform the numerical calculations in this paper, we set .Let  be a graph. The measure  becomes to  by choosing the coefficients  according to Equation (19), i.e., linearly decreasing. Correspondingly,  becomes to  when choosing the coefficients  according to Equation (21), i.e., exponentially decreasing.iener index is In the following, we briefly comment on the computational complexity of the discussed information measures without giving proofs. Obviously, the measures whose definitions are based on calculating matrices can be often computed in polynomial time . For instance, it has been proven As stated in the Let  be a graph and let  be the vertex-vertex link correlation matrix, see .  denotes the number of all neighbors with degree  of all vertices with degree .  stands for the maximum degree of . The normalized version of  can be defined as whereandLet  be a graph and let  be the largest eigenvalue computed from its adjacency matrix.whereBefore discussing numerical results, we describe the databases and our developed software in brief.MS 2265: This database has been extracted by own software from the commercially available mass spectral database NIST AG 3982: The original freely available database called Ames Genetoxicity contains 6512 chemical compounds, see APL 91075: The ASINEX Platinum Collection is a freely available, in-house designed and synthesized collection of 126615 drug-like compounds CCCIn order to generate and process our chemical graphs, we used the known Molfile format We implemented all used topological measures in Python using freely available libraries like Networkx, Openbabel and Pybel packages In this section, we will apply the complexity measures presented in the previous section. As stated before, we mainly put the emphasis on exploring the relatedness between the topological information content In the following, we discuss and interpret numerical results when applying the selected descriptors to sets containing real chemical structures. Our study involves calculating and interpreting dependency plots, cumulative entropy distributions, and the so-called uniqueness of the used topological indices We start to examine how the entropies lots see  for explIf  is vertex transitive , then If  is -regular , then  and, hence, .The graph vertices . BecauseThe interrelation between the entropies ): The main idea is to partition the vertex set in equivalence classes according to the criterion that each such class contains topologically equivalent vertices iener index is also very low We now start discussing the results shown in ower see .To interpret the sensitivity values when applying our information measures to synthetical chemical graphs, we look at However for the tree class, our The cumulative entropy distributions are illustrated by We start by observing that about 80% of the graphs of MS 2265 possess relatively small entropy values when evaluating Equally, the cumulative entropy distributions of AG 3982 are depicted in In particular, we have found that for all three chemical databases, the evaluation of the topological information content (see Equation (4)) and the partition-independent measures (see Equation (23)) led to clearly different cumulative entropy distributions that is obviously in accordance with the results of the preceding sections.In the present paper, we studied interrelations between classical and novel entropy measures to quantify the structural information content of networks. Here, these measures served as graph complexity measures which take certain structural features of the networks under consideration into account. In the following, we express the main findings of the paper in brief:We explored the relatedness between information measures for graphs. In particular, we examined the correlation between the topological information content is large , the invAnother important aspect of our numerical study was to examine the discrimination power of the used network measures. We found that the topological information content For the real chemical databases, the cumulative entropy distributions of some measures were calculated. This approach can be considered as an important preprocessing step to learn how the measures capture structural information of networks. Particularly, it is suitable to explore certain correlations between the measures and, finally, to learn whether the complexity indices capture structural information differently or similarly.As a conclusive remark, we emphasize that the presented information-theoretic methods to analyze complex networks bear a considerable potential. Our study aimed to get a better understanding towards the problem of characterizing chemical graphs using information-theoretic complexity measures. In this paper, we put the emphasis on such measures which have already been applied in the context of mathematical chemistry and drug design. We think that our results can help to apply the measures to more complex network classes and to interpret the results more adequately than before.In the future, we want to extend our measures for determining the structural complexity of weighted chemical graphs  and test their ability to tackle QSAR/QSPR problems. Further, we would like to test novel information indices by combining existing ones and evaluate their discrimination power. Moreover, an interesting task would be to classify molecules by using this approach and to apply it to special problems in drug design.


Helicobacter pylori, a gram-negative bacterial pathogen that expresses a strong urease activity, is associated with the development of gastroduodenal disease. Urease B subunit, one of the two structural subunits of urease, was expressed in E. coli BL21 (DE3) strain. The objective of this study was to evaluate the effects of Helicobacter pylori urease B subunit on the immune responses in mice by subcutaneous immunization.Helicobacter pylori urease B subunit antigen subcutaneously three times with 2-wk intervals between the immunizations and boosters. The mice in the control group were immunized with PBS. The adjuvant group received PBS containing complete/incomplete freund’s adjuvant identical to antigen group without Helicobacter pylori urease B subunit antigen. Four weeks after the final booster, all the mice were sacrificed. Blood was collected on d 0, 14, 28 and 56 before immunization, booster and sacrifice, respectively. Immediately after sacrifice, gastric liquid and spleen were collected for antibody and cytokine analyses.The mice were immunized and boosted with P < 0.05).Urease B subunit increased the concentrations of serum and gastric anti-urease B antigen specific IgG, and the levels of interleukin-4 and interferon-γ in splenocytes of the mice (Helicobacter pylori.This study demonstrated that recombinant urease B subunit can induce systemic and local immune responses in mice by subcutaneous immunization, which might be used as the effective component of vaccine against  Helicobacter pylori, a gram-negative bacterial pathogen that expresses a strong urease activity, is associated with the development of chronic gastritis, peptic ulceration and gastric carcinoma [H. pylori infection, negative effects concerning antibiotic-resistant strains always limit the treatment [H. pylori from the upper gastrointestinal tract of the patients, utilization of a safe and effective vaccine might be the ideal strategy to immunologically prevent H. pylori infection [arcinoma . It is oreatment -4. In canfection -7.H. pylori, is expressed broadly on the surface of H. pylori and contributes a lot to colonization of H. pylori[Urease, a recognized virulence factor of H. pylori,9. The mH. pylori.H. pylori or other bacteria containing urease or its subunits antigens via oral, nasal, rectal or other routes has previously been reported as effective ways to protect human or animals from H. pylori infection [H. pylori infection as effectively as oral immunization because of the stimulation of relatively higher specific antibody concentrations [Mucosal immunization with the attenuated nfection -13. Howetrations .H. pylori urease B subunit was obtained after plasmid-encoded urease B from H. pylori was expressed in E. coli BL21 (DE3) strain. To evaluate the effects of H. pylori urease B subunit on the immune responses by subcutaneous immunization in mice, the indices of specific antibodies in serum and stomach as well as the splenocyte-secreted cytokines were determined in this study.Here E. coli SE 5000 containing the urease expression genes used in this study were kindly provided by H. T. Mobley from Department of Biology, Kilgore College, Kilgore, TX, USA. E. coli BL21 (DE3) strain was used as a recipient for the recombinant urease plasmid constructs that expressed urease B from the plasmid in which urease B genes from H. pylori had been cloned. After the E. coli BL21 (DE3) strain containing the urease B genes was activated overnight at 37°C, the bacteria were inoculated and cultured in Luria broth medium. Then the culture was induced by isopropyl-β-D-thiogalactopyranoside  and centrifuged at 6,000 × g for 15 min. Cells were harvested and catabolizated by lysozyme and nuclease. After sonication, the cell lysate was centrifuged at 12,000 × g for 10 min. Urease B was expressed and obtained after washing with Buffer B  and resolving in urea-Tris–HCl solution. Urease B antigen was purified by using a Ni-NTA kit  according to manufacturer’s instructions. Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) was applied to identify the expressed protein and the purity of urease B was measured by Bradford method .ad libitum under pathogen-free conditions.All the mice used in this experiment were maintained in accordance with the principles of Chinese Academy of Agricultural Sciences Animal Care and Use Committee. Fifteen specific-pathogen-free, six-week-old female Kunming mice weighing 17–20 g were purchased from the Beijing Laboratory Animal Research Centre  and divided into three treatments at random. Each treatment had five replicates with one mouse per replicate. All animals were housed in plastic cages in a mechanically ventilated nursery room where 12 h light: 12 h dark was set, and constant temperature remained at 23–25°C and relative humidity at 50-60%. All the mice had sterilized commercial chow  and water H. pylori urease B subunit antigen was mixed with complete/incomplete freund’s adjuvant , and 40 μg of H. pylori urease B subunit antigen in a volume of 100 μL of emulsion was injected into the lower back of the antigenic mice on d 0 and 14. The mice in the control group were immunized with PBS. The adjuvant group received PBS containing complete/incomplete freund’s adjuvant identical to antigen group without H. pylori urease B subunit antigen. The mice were boosted with 80 μg of urease B antigen and incomplete freund’s adjuvant on d 28. Blood was collected retro-orbitally before immunization and booster on d 0, 14 and 28, respectively.Three groups of 5 mice were used as control, adjuvant and antigen. After seven days of adaptation, mice were immunized and boosted subcutaneously three times with 2-wk intervals between the immunizations and booster. Four wk after the final booster, all the mice were sacrificed after final blood collection from the heart. Mice were immersed into 75% ethanol for 5 min immediately after sacrifice. Thereafter, the peritoneal cavity was opened, and the spleen was removed from each mouse followed by the recovery of gastric fluid flushed with 1 mL of phosphate buffered saline (PBS) containing protease inhibitor.6 cells per well with or without 10 μg/mL H. pylori lysate for 48 h at 37°C and 5% CO2. The supernatant was harvested on d 0, 7 and 14 and stored at −70°C for cytokine assay.Immediately after sacrifice, the spleen was aseptically removed from mice and the tissue was minced by syringe and washed twice with RPMI 1640 containing 10% fecal bovine serum (HyClone Laboratories Inc. Logan UT), 10 mmol/L Hepes, 100 μg/mL penicillin and 100 μg/mL streptomycin . After erythrocyte lysis, splenocytes were separated and cultured in 24-well plates at a density of 1 × 10g at 4°C for 10 min, and the supernatant was obtained for IgA and IgG determination. The measurement of serum anti- H. pylori urease B specific IgA, IgG, IgE, gastric IgA and IgG levels was performed by enzyme-linked immunosorbent assay (ELISA) using an indirect ELISA as described by Weltzin et al. [3/Na2CO3, pH 9.6) and then free binding sites were blocked with PBS containing 1% bovine serum albumin (BSA) and 0.1% Tween 20 (ELISA buffer) for 1 h at 37°C. Duplicate serum and gastric fluid samples were diluted in ELISA buffer and incubated for 2 h at 37°C. The plates were then incubated with ELISA buffer containing IgA, IgG or IgE antibodies conjugated with horseradish peroxidase  for 1 h at 37°C. Samples were washed five times with PBS containing 0.1% Tween 20 between each incubation step. Then the plates were developed with tetramethylbenzidine (TMB) to measure the absorbance at 450 nm. The data were expressed as optical density (OD) units.The gastric fluid sample was centrifuged at 10,000 × n et al.  with somin vitro with or without H. pylori lysate at the concentration of 10 μg/mL in a total volume of 1 mL. The supernatants were collected on d 0, 7 and 14, respectively. Interleukin-4 (IL-4) and interferon-γ (IFN-γ) concentrations in the cultured splenocyte supernatant were measured by the mouse ELISA Kit  following the manufacturer’s instructions.Single splenic cell suspension was inoculated P values less than 0.05 were considered statistically significant.All data were analyzed using the ANOVA procedure of SAS system . E. coli BL21 (DE3) strain was assayed using SDS-polyacrylamide gels for expression of urease B. As shown in Figure H. pylori can be obtained through mostly expression in E. coli without negative effects and is at least suitable for animal protection experiments in further studies.E. coli was capable of inducing specific antibody responses and what type of antibody production against urease B was stimulated by systemic immunization, specific IgA, IgE and IgG in the serum and IgA and IgG in gastric liquid were measured by ELISA. It is reported that parenteral immunization tends to induce low levels of secretary IgA but often increase a low level IgE production in most mice [Mice were immunized and boosted with recombinant urease B subunit and freund’s adjuvant by subcutaneous route. To investigate whether urease B subunit expressed by ost mice . As presost mice . Serum IP < 0.05). Furthermore, gastric IgG to urease B subunit antigen was statistically higher (P < 0.05) in urease B antigen-immunized mice than in the control and adjuvant-injected mice . Simultaneously, the production of IFN-γ was also enhanced by splenic cells while the concentrations of IFN-γ secteted by splenocytes from control mice or adjuvant mice were very low and nearly zero  cells, Th2 cells and Th17 cells are different types of helper T cells resulting in the secretion of different patterns of cytokines [It has been well documented that T lymphocytes can be classified as CD4 markers . Recentl markers ,30. CD4+ytokines ,29,31. Tytokines ,16. Th 1Helicobacter antigens and Freund’s adjuvant induced protective anti-Helicobacter immunity resulting in production of IFN-γ. The analysis for cytokines in the cultured splenocyte supernatant also showed that concentrations of IL-4 and IFN-γ was significantly elevated in the supernatant of spleen cell culture of urease B immunized mice compared to the PBS control and adjuvant control groups, which suggested that Th1/Th2 type responses were both stimulated by the recombinant urease B subunit antigen. Similar results were also obtained by Guy et al. [H. pylori infection than that of a predominantly Th2 type response.Previous studies have proven that IL-4 is a Th2-type representative cytokine which plays a key role in allergic inflammation . ProductH. pylori.The current study demonstrates that recombinant urease B subunit induced higher concentrations of serum and gastric IgG as well as an increase of IL-4 and IFN-γ in splenocytes of the immunized mice by subcutaneous immunization. The application of urease B subunit in parenteral inoculation strategies might enlighten us to use it as the effective component of vaccine against The authors declare that they have no competing interests in relation to this study.PS carried out the experiment and drafted the manuscript. JQW conceived the study, participated in its design and coordination, and helped draft the manuscript. YTZ performed the animal experiments. All authors read and approved the final manuscript.


Musa (banana and plantains). The expression stability of six candidate reference genes was tested on six different sample sets, and the results were analyzed using the publicly available algorithms geNorm and NormFinder. Our results show that variety, plant material, primer set, and gene identity can all influence the robustness and outcome of RT-qPCR analysis. In the case of Musa, a combination of three reference genes  can be used for normalization of gene expression data from greenhouse leaf samples. In the case of shoot meristem cultures, numerous combinations can be used because the investigated reference genes exhibited limited variability. In contrast, variability in expression of the reference genes was much larger among leaf samples from plants grown in vitro, for which the best combination of reference genes (L2 and ACT genes) is still suboptimal. Overall, our data confirm that the stability of candidate reference genes should be thoroughly investigated for each experimental condition under investigation.Gene expression analysis by reverse transcriptase real-time or quantitative polymerase chain reaction (RT-qPCR) is becoming widely used for non-model plant species. Given the high sensitivity of this method, normalization using multiple housekeeping or reference genes is critical, and careful selection of these reference genes is one of the most important steps to obtain reliable results. In this study, reference genes commonly used for other plant species were investigated to identify genes displaying highly uniform expression patterns in different varieties, tissues, developmental stages, fungal infection, and osmotic stress conditions for the non-model crop The online version of this article (doi:10.1007/s11032-012-9711-1) contains supplementary material, which is available to authorized users. Arabidopsis thaliana  is a routinely used technique for gene expression analysis because of its main advantages of relatively low cost, good speed, a wide dynamic range, and feasibility in non-model organisms  species provide a staple food in many developing countries and with an annual production of more than 130 million tons per year it is the fourth most important food crop worldwide (FAO wide FAO . Diseasewide FAO  as well wide FAO  are amonMusa Genome Consortium (GMGC)  sequences available. The Global A summary of the different types of cultures, tissues, and varieties used is provided in Table –1 ascorbic acid, 0.09 M sucrose, and 3 g l−1 Gelrite®] at 26 ± 2 °C under a 16-h photoperiod with a photosynthetic photon flux density of 50 μE m−2 s−1 provided by Cool White fluorescent lamps . After 5.5 weeks of growth, the plants were transferred to a liquid REG medium. After 2 months, fresh liquid REG medium was added, and to half of the plants, acetone was supplemented to a final concentration of 0.5 % (v/v). Acetone treatment was tested since acetone is used to dissolve certain biologically active compounds in the author’s laboratory. Leaves were harvested 2 days after the addition of acetone from six and seven plants grown on the REG medium without and with 0.5 % (v/v) acetone, respectively.Plants of the variety Grand Nain , were grown on semi-solid regeneration medium  ] Pfaffl  (Table 2M), defined as the average pair-wise variation between a particular reference gene and all of the other candidate reference genes, was determined using geNorm v3.4 . ANOVA was used to determine whether differences in the Ct levels between the different experimental treatments within each experiment were significant.The Ct values were converted into relative quantities or expression levels according to the data obtained for the samples of the dilutions series, which are used to create standard curves. Next, the reference gene stability factor (Musa (banana and plantains). Nine genes from different functional groups were chosen: 18S rRNA, glyceraldehyde-3-phosphate dehydrogenase (GAPDH), elongation factor-1α (EF1), polyubiquitin, actin11 (ACT11), α-tubulin, β-tubulin (TUB), cyclophilin, and ribosomal protein L2 (L2) genes. Banana genes and EST fragments belonging to these gene families were identified by conducting similarity searches (BlastX). The identity of the coding sequence between Arabidopsis or rice and Musa varied between 80 and 97 % . At the time this study was performed, no orthologous Musa sequences of sufficient length could be identified for GAPDH, 18S rRNA, and α-tubulin.Reference genes commonly used for other plant species were investigated to identify genes displaying highly uniform expression patterns in different varieties, tissues, developmental stages, and stress conditions for the non-model crop ACT11, cyclophilin, EF1, L2, TUB, and polyubiquitin genes . Using data from other plant species, a primer pair spanning an intron was designed for ACT11; this was not possible for the other genes. To ensure that each primer pair resulted in the production of a single PCR product, gradient PCR was performed on genomic DNA (gDNA) and on cDNA from leaves. For ACT11, EF1, L2, and TUB, a suitable primer pair was identified (Table 25S) and actin (ACT) genes that have been previously used in other banana gene expression studies .The leaves of each of the six and seven plants grown on the REG medium without and with 0.5 % (v/v) acetone, respectively, were pooled and used for RNA isolation and cDNA synthesis. The Ct values of the six different candidate reference genes exhibited broad variability between the samples, irrespective of the acetone treatment . This result was confirmed by the Ct values obtained, as ACT11 (ΔCt = 3.6) showed larger variability than ACT (ΔCt = 2.6), and by Normfinder analysis , a measure that is used to determine how many additional reference genes should be included in the calculation of the normalization factor for gene expression. A cut-off V-value of 0.15, below which the inclusion of additional reference genes is not required, has been proposed by Vandesompele et al.  between leaf samples harvested at different time points, although the differences between the average Ct values of the different groups were smaller than 1.05 Ct except for ACT with a difference of 1.4 between time points Ta and Tc. In contrast, the Ct values for 25S, L2, and TUB showed significant differences between the two varieties and the Ct differences between the average values for L2, 25S, and TUB in the two different varieties is 4.3, 4.1, and 2.0 Ct’s, respectively  Fig. b. AdditiACT is the most stable reference gene, and subsequently 25S, EF1, TUB, L2 and EF1, TUB, L2,25S with and without grouping function, respectively , thus excluding the other reference genes from further analysis. For the greenhouse development experiment, the most stable reference genes were TUB and EF1 and the least stable genes were 25S and L2  was optimal as the V-value of 0.15 was obtained  was sufficient as the combination yielded a V-value of 0.09  was possible as this combination yielded a V-value of 0.14  and the least variation for ACT  although the differences between the average Ct values were less than 1 Ct. For the variety experiment, five meristems of Cachaco, Mbwazirume, and Williams grown under standard conditions on the P4 medium were harvested. The Ct value showed the maximum variation for ACT (ΔCt = 2.2) and the least variation for EF1 (ΔCt = 0.7) over all three varieties  experiment, RNA was isolated from five meristems grown either on control medium or on a medium containing a higher sucrose concentration. Additionally, a third group of meristems was harvested 24 h after subculturing to investigate the effect of wounding associated with the cutting and subculturing process. The Ct values of the six different reference genes exhibited the largest variation for TUB was the most stable reference gene followed by L2,ACT11, EF1, and ACT, although the order interchanged depending on the grouping factor. 25S was the least stable reference gene  and 0.09 (ACT11 and TUB) was obtained for the samples of the sucrose and varieties experiments, respectively  as well as two previously reported reference genes in Musa (ACT and 25S RNA). Recently, Chen et al.  resulted in even more unacceptable reference gene combinations for normalization, indicating that suitable reference genes for in vitro gene expression studies are scarce. These results also suggest that plants grown in vitro might be stressed and show variable expression levels of genes involved in basic biological processes. Analysis of the expression of genes of interest in such samples is thus difficult and requires careful examination of candidate reference genes prior to any analysis.Our study showed that the expression levels of all reference genes investigated exhibited high Ct variability in the leaf samples of the EF1/TUB/ACT and L2/ACT, respectively, allow reliable normalization despite the occurrence of significant Ct differences between different sample groups in the former for all but one (L2) reference gene. NormFinder analyses resulted in similar results and indicated that the combinations ACT/EF1 and ACT/TUB are optimal for normalization of leaf samples at different developmental stages and leaf discs, respectively. For leaf samples from different varieties the ANOVA indicated significant differences for L2, 25S, and TUB with large differences (ΔCt > 4.0) in average Ct’s for L2 and 25S, which were both excluded from the geNorm analysis, resulting in the inability to identify a suitable combination of reference genes. NormFinder identified ACT as most stable and L2 and 25S as least stable reference genes, but surprisingly indicated L2 and 25S as the most suitable reference gene pair. A glance at the raw Ct’s shows that these genes are relatively stable within each variety, but the level of L2 and 25S is ±4 Ct’s higher and ±4 Ct’s lower, respectively, in Km5 than in TG . In our study, ACT was found to be one of the most stable reference genes whereas Chen et al. Below is the link to the electronic supplementary material.