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Consensus Sequence Zen
 Thomas D. Schneider 2  
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 runningtitle:  [runningtitle FIX ME] Key words:  consensus sequence, information theory, sequence logo, sequence walker, binding site, genetic control.

Applied Bioinformatics 1(3) 111-119, 2002. [Schneider, 2002]

1.  Abstract.
Consensus sequences are widely used in molecular biology but they have many flaws. As a result, binding sites of proteins and other molecules are missed during studies of genetic sequences and important biological effects cannot be seen. Information theory provides a mathematically robust way to avoid consensus sequences. Instead of using consensus sequences, sequence conservation can be quantitatively presented in bits of information by using sequence logo graphics to represent the average of a set of sites and sequence walker graphics to represent individual sites. 

“All models are wrong but some are useful.” 
— George E. P. Box [Box, 1979]
[commentout FIX ME]"All models are wrong, but some are useful." - G.E.P. Box (1979), Robustness in Statistics.  http: //alltheweb.com/search? t=phrase&q=ALL+MODELS+ARE+WRONGhttp: //cee.engr.ucdavis.edu/faculty/lund/Classes/ECI153/thoughts.htm — http: //www.medill.nwu.edu/faculty/malthouse/quote.html "All Models Are Wrong But Some Are Useful." George E.P. Box ("Robustness is the Strategy of Scientific Model Building" in Robustness in Statistics. eds., R.L. Launer and G.N. Wilkinson, 1979, Academic Press, p.  202.) cannot locate this book at amazon, abebooks or academic press!  — http: //www.biostat.wustl.edu/mailinglists/s-news/199807/msg00085.html According to a posting last year on the Riskanal list, the answer is:  All Models Are Wrong But Some Are Useful (note Caps are as in the quote) George E.P. Box Page 202 of "Robustness is the Strategy of Scientific Model Building" in Robustness in Statistics.  eds., R.L. Launer and G.N. Wilkinson, 1979, Academic Press. — best:  http: //www.google.com/search? q=cache: aNa9PMo729oC: www.alceon.com/AnimatePP.4.pdf+George+E.+P.+Box+
2.  How to be sure to make a mistake.
Genes are controlled by proteins that bind to specific spots on the DNA sequence. Molecular biologists often represent the patterns at these spots by using a consensus sequence. For example, after aligning some binding sites so that they match each other, one position might contain 70% adenine, 10% cytosine, 10% guanine, and 10% thymine. The consensus is the most frequent base, ‘A’. This is the simplest (and possibly the most commonly applied) approach, but there are alternatives [Day  McMorris, 1992]. Various kinds of consensus sequence commonly found in the molecular biology literature will be considered here, while the controversy over the use of consensus trees used in phylogenetic inference [Barrett  et ,Nelson, 1993,Barrett  et ,de ] will not be covered.
The main difficulty with using consensus sequences is that they present distorted pictures of binding sites. In order to locate new binding sites, consensus sequences are compared to various locations in a sequence and the number of matches is tallied. A difficulty arises because a position that is always an ‘A’ in the original set is treated the same as a position that is just 70% A. If we think that a position has A, then when we use this observation to look for additional binding sites, we will find mismatches for 30% of the acceptable sequences. This problem is compounded across the entire binding site, which may be 20 or even 40 bases long [Schneider, 1996,Zheng  et ]. For example, a commonly cited consensus sequence is TAT[Lewin, 1997], which represents the -10 region of bacterial promoters originally discovered by David Pribnow (1975). The most prominent bases for the boxed positions are only 49%, 58%, and 54% respectively [Lisser  Margalit, 1993]. If one demands that a site have all of the consensus bases, one finds only 14 TATAAT sequences out of 291 sequences in the database. To deal with this, people often count mismatches, but it is not obvious from the simple consensus which bases are allowed to be more variable. Sometimes variations such as allowing C or G are indicated but, again, the degree of allowed variation is lost. It is not surprising then, that consensus sequences frequently fail to identify binding sites or that they predict sites where there are none.
Consensus sequences have other serious problems, many of which are revealed by using information theory to measure the amount of conservation in bits. In a set of aligned binding sites, a DNA position that is always an A stays that way during evolution because the molecule that binds to it always selects A from the four possible bases [Schneider, 2000]. Such a selection can be made with a minimum of two yes-no questions:  ‘Is it in the set A or T? ’ and ‘Is it in the set A or C? ’, so the selection takes two bits of information, one to answer each question. Likewise, a position that is either A or T only requires one yes-no question — the other one being ignored — so has one bit of sequence conservation. The late Claude Shannon figured out how to consistently measure the average information when the frequencies are not so simple [Shannon, 1948,Schneider  et ,Schneider, 1995]. One can plot the sequence conservation across all positions in the set of aligned binding sites. This continuous quantitative measure often follows a sine wave, reflecting the binding of a protein to one face of helically twisting B-form DNA [Papp  et ,Schneider, 1996,Schneider, 2001]. This subtle effect cannot be seen by using consensus sequences.
3.  A Paradox:  How can two things be the same but different? 
Intriguingly, the binding sites for human splice junction donor and acceptor sites have the same consensus sequence for a portion of each site around the junction. Yet, when we measured the sequence conservation in bits we found that the information curves are quite different [Stephens  Schneider, 1992]. How could two sites have the same consensus sequence but be different?  This conundrum led us to introduce a computer graphic, called a sequence logo, in order to understand the difference ([fig FIX ME]fig.splicelogo[figmargin FIX ME]fig.splicelogo). A logo depicts an average picture of the set of binding sites by a series of stacks of letters. The height of each stack is the sequence conservation (measured in bits of information; the vertical black bar at each junction is 2 bits high) and the heights of the letters show the relative proportions of the bases, sorted so that the more frequent bases are on top. From the logos shown, it is clear that the donor and acceptor sites have different ‘emphasis’, but this cannot be seen with the consensus sequence CAG|GT, which matches both of them at the junction. The difference in emphasis is important because it shows that there is more information on the intron side of each junction. This allows more freedom during the evolution of the protein-coding exon side, which is a biologically sensible result. The resemblance between the two junctions suggests that the splice machinery that binds to donors and acceptors have a common ancestor [Stephens  Schneider, 1992].
4.  Walking along the genome.
One can depict individual sites using another graphic called a sequence walker, in which the height of a letter above or below zero shows how much that base contributes to the average sequence conservation of the entire collection of sites shown in the logo ([fig FIX ME]fig.walker[figmargin FIX ME]fig.walker) [Schneider, 1997 b,Schneider, 1997 a]. Instead of counting matches to a consensus, one sums the information contributions for a given sequence to obtain the information for an individual binding site. (The Shannon information measure is unique in that it is the only measure that allows addition for statistically independent components [Shannon, 1948]. Generally, binding site bases are independent [Stephens  Schneider, 1992].)
Sequence walkers can be stepped along the sequence (hence the name) to discover positions that match a particular model, and one can predict whether or not a sequence change will destroy the site and cause a genetic disease [Rogan  et ]. In the case shown, splicing is normally accomplished using a 12.7 bit acceptor at position 5154. Nearby, however, is an 8.9 bit ‘cryptic’ acceptor that is not used apparently because the strongest site in any local region normally wins the competition for splice factors. An A to G mutation at 5153 destroys the normal site, making it 4.5 bits while simultaneously raising the cryptic site to 16.5 bits. This results in a single base frame shift, the loss of the protein, and Hunter disease. Cases like this are difficult to understand using consensus sequences because sites are affected by all of their parts and quantitative differences are missed. Using information theory and sequence walkers we have interpreted about 100 mutations in two human workdays (the computer time is only a few seconds).
5.  Statistical effects of making a consensus.
The overall strength of a binding site is found by summing the individual bit contributions. A distribution of these strengths is roughly Gaussian and shows that most natural binding sites have much less information than the consensus sequence [Schneider, 1997 a]. The strict consensus (where only the most frequent base is used) is the strongest possible binding site and is on the far high end of the distribution. For example, only one in 270 acceptor sites matches the strict consensus. For this reason it is generally inappropriate to say that one has a consensus binding site at such-and-such a position on a sequence.
As mentioned earlier, using consensus sequences to find binding sites by counting mismatches can lead to errors. How does this compare to the information theory approach?  If matches to the consensus are assigned to have 1 unit and mismatches 0 units, then the total count is an integer. In contrast, the information theory weights are 2+log2(base frequency)+(a small-sample correction), which includes the real numbers. Summing the information theory weights gives continuous results, while counting mismatches gives blocky results that will often be off the mark. The commonly used ‘percent identity’ between two sequences, such as proteins, is flawed for the same reasons.
Sometimes counting matches or mismatches can give results opposite to the information measure weights so that a base in a site could have a mismatch to the consensus and yet that base could contribute positive information. For example, for a position that has 60% A, 30% T, 5% G, and 5% C the consensus base is A by two-fold, and yet a T in an individual binding site would contribute 2+log20.30=0.26 bits. Only by noting the total distribution can we learn that the T contributes positively to the information. A related effect that is hidden by a consensus is that the diversity of the less frequent bases affects the total sequence conservation. For example, a position with 70% A, 30% T, 0% G, and 0% C has 1.12 bits of conservation, but a position with 70% A and 10% for each of C, G and T has only 0.64 bits. The consensus for both cases, A, does not distinguish between these.
When there are very few sequences, statistical artifacts crop up. Even if there’s no information in the set, it can look like there is. For example, if one has only 6 random sequences, one will frequently observe positions that have 50% or more of one base. If, as is commonly done, one uses 50% as the cutoff for writing the consensus base, then one can get the false impression that there is pretty good sequence conservation. In the example shown in [fig FIX ME]fig.random[figmargin FIX ME]fig.random, 25 of 41 positions would be identified as ‘conserved’ even though the sequences were randomly generated!  In general, 26±3 of the 41 positions in 6 randomly generated sequences would be marked as the consensus.
6.  Missing the trees in the data forest.
As a result of counting mismatches to a consensus it is possible to entirely miss a binding site. One of the most striking examples is a Fis site in the tgt/sec promoter of E. coli ([fig FIX ME]fig.tgtsec[figmargin FIX ME]fig.tgtsec). We carefully collected 60 sequences shown by DNase I footprinting to be bound by Fis, used information theory to maximize the information content of the alignment [Schneider  Mastronarde, 1996] and produced a model of how Fis binds [Hengen  et ]. The model did a good job of predicting where Fis binds in the original footprint regions. However the model also predicted a site in tgt/sec that was not noted by the original authors. Surprisingly, four pieces of data support the existence of a Fis site centered at position -73 relative to the start of transcription [Slany  Kersten, 1992]: 
1. Fis often induces DNase I hypersensitive phosphates; these are seen between bases -53 and -50, corresponding to the Fis site at position -58. In addition there are hypersensitive positions between -80 and -78, which correspond to the site shown by the walker at -73.
2. The 5' end of the original DNase I footprints was not well determined, but could have extended to cover the Fis site at -73.
3. Gel mobility shift assays showed two band shifts when the entire region was used, one band shift when the region 5' to the conveniently (! ) placed ClaI site was removed (which would eliminate the proposed Fis site) and none when both were removed.
4. Assays with these same nested DNAs showed that both sites activate transcription.
The authors were aware that there was a second binding site, but placed its location somewhere in the 69 bases upstream of the ClaI site.  Why did they miss the site?  Two positions (indicated by arrows in the figure) did not match the ‘accepted’ consensus sequence [Hu]. The consensus method gave these positions far more weight than was appropriate. The information for the site at -73 is 10.8 bits, which is 2 bits more than the average. To determine if there is really a site there, we performed a gel shift experiment using a DNA containing only the proposed Fis site at -73 and showed that the sequence is indeed bound by Fis [Hengen  et ]. Because the consensus sequence failed to predict a site that had been documented experimentally, that site could not be seen, and to the scientists it did not exist [Kuhn, 1970].
A more critical example is in the hMSH2 gene, which is associated with familial nonpolyposis colon cancer [Rogan  Schneider, 1995]. A ‘T’ to ‘C’ transition occurred at position -5 of an acceptor site and this change was proposed to be the cause of the disease [Fishel  et ]. Inspection of the logo in [fig FIX ME]fig.splicelogo shows that the consensus at position -5 (base zero is just to the left of the vertical bar, the first base on the intron side) is a T, but that close to half of the bases in the polypyrimidine tract are C. When the transition is made, the individual information changes by only 0.2 bits, which is not significantly different. A study of 20 normal people found that only 2 had this change [Leach  et ], so the change is a polymorphism unrelated to the disease.
Why did this potential ‘misdiagnosis’ happen?  We suppose that T was taken to be the consensus sequence. Given this, one would interpret any change from that consensus to be detrimental. In this case the consensus sequence was so rigid that it could not handle a subtle change and a site disappeared from the scientist’s view even though it was still functional. As DNA sequencing technologies become widely available to doctors, this situation will come up repeatedly. Serious malpractice suits could occur as a result of using the consensus model.
7.  Flipping the light on to see an unseen world.
Two recently published examples demonstrate some of the interesting biology that one can miss by using a consensus sequence. The first example is the RepA binding site ([fig FIX ME]fig.repalogo[figmargin FIX ME]fig.repalogo). The sine wave over the logo represents the twist of B-form DNA. The crests of the wave represent the protein facing a major groove and the troughs represent the protein facing a minor groove. Experimental data indicate that the major groove sides of Gs at positions 1 and 12 (black dots) are facing the protein, while the major groove sides of those base pairs at positions 6 and 8 (open circles) are facing away from the protein [Papp  et ]. Hydroxyl radical and ethylation interference data also support this assignment and indicate that RepA binds to only one face of the DNA [Papp  Chattoraj, 1994].
RepA and other DNA binding proteins show sequence conservation up to 2 bits where they contact the major groove and only 1 bit where they face a minor groove [Papp  et ,Schneider, 2001]. The upper bound of 2 bits is achievable because all 4 bases can be distinguished using contacts in the major groove [Seeman  et ]. In contrast, the minor groove of B-form DNA is essentially symmetrical and can only provide up to 1 bit of sequence conservation.
Intriguingly, as seen in [fig FIX ME]fig.repalogo, RepA violates this rule at positions +7 and +8 where the protein faces a minor groove [Papp  et ]. The violation implies that the DNA is not B-form. To understand this anomaly, we substituted a variety of chemically modified base pairs at position +7 (and its complement +7') and found that the N3 proton on the thymine is responsible for contacting RepA through the minor groove [Lyakhov  et ]. Since the N3 proton is normally sequestered in the center of the DNA helix, the DNA must indeed be distorted, as predicted from the sequence logo. Furthermore, the acceptable contact points for hydrogen bonding vary by several angstroms more than an H-bond could withstand in a rigid structure, suggesting that the base may rotate towards the minor groove for binding to occur. In other words, the T at +7 may be ‘flipping’ out of the DNA.
Base flipping was discovered by Rich Roberts in the co-crystal of the HhaI methyltransferase [Roberts, 1995,Roberts  Cheng, 1998]. This solved a puzzle of how that enzyme functions, since the chemistry of methylation requires attack from above or below the plane of the base. Such an attack is not possible inside the DNA helix. The HhaI methyltransferase solves the problem by flipping the base out of the helix and into a pocket of the enzyme. Other DNA modification proteins also flip bases [Cheng  et ,Klima8.  Just say no! 
log9.  Models and Illusions.
10.  Acknowledgments.