Methylation analysis with Methyl-IT. Part 2

 Methylation analysis with Methyl-IT is illustrated on simulated datasets of methylated and unmethylated read counts with relatively high average of  methylation levels: 0.15 and 0.286 for control and treatment groups, respectively. In this part, potential differentially methylated positions are estimated following different approaches.

1. Background

Only a signal detection approach can detect with high probability real DMPs. Any statistical test (like e.g. Fisher’s exact test) not based on signal detection requires for further analysis to distinguish DMPs that naturally can occur in the control group from those DMPs induced by a treatment. The analysis here is a continuation of Part 1.

2. Potential DMPs from the methylation signal using empirical distribution

As suggested from the empirical density graphics (above), the critical values $H_{\alpha=0.05}$ and $TV_{d_{\alpha=0.05}}$ can be used as cutpoints to select potential DMPs. After setting $dist.name = “ECDF”$ and $tv.cut = 0.926$ in Methyl-IT function getPotentialDIMP, potential DMPs are estimated using the empirical cummulative distribution function (ECDF) and the critical value $TV_{d_{\alpha=0.05}}=0.926$.

DMP.ecdf <- getPotentialDIMP(LR = divs, div.col = 9L, tv.cut = 0.926, tv.col = 7,
                            alpha = 0.05, dist.name = "ECDF")

3. Potential DMPs detected with Fisher’s exact test

In Methyl-IT Fisher’s exact test (FT) is implemented in function FisherTest. In the current case, a pairwise group application of FT to each cytosine site is performed. The differences between the group means of read counts of methylated and unmethylated cytosines at each site are used for testing (pooling.stat=”mean”). Notice that only cytosine sites with critical values $TV_d$> 0.926 are tested (tv.cut = 0.926).

ft = FisherTest(LR = divs, tv.cut = 0.926,
                     pAdjustMethod = "BH",  pooling.stat = "mean", 
                     pvalCutOff = 0.05, num.cores = 4L,
                     verbose = FALSE, saveAll = FALSE) 

ft.tv <- getPotentialDIMP(LR = ft, div.col = 9L, dist.name = "None",
                          tv.cut = 0.926, tv.col = 7, alpha = 0.05)

There is not a one-to-one mapping between $TV$ and $HD$. However, at each cytosine site $i$, these information divergences hold the inequality:

$TV(p^{tt}_i,p^{ct}_i)\leq \sqrt{2}H_d(p^{tt}_i,p^{ct}_i)$  [1].

where $H_d(p^{tt}_i,p^{ct}_i) = \sqrt{\frac{H(p^{tt}_i,p^{ct}_i)}w}$ is the Hellinger distance and $H(p^{tt}_i,p^{ct}_i)$ is given by Eq. 1 in part 1.
So, potential DMPs detected with FT can be constrained with the critical value $H^{TT}_{\alpha=0.05}\geq114.5$

4. Potential DMPs detected with Weibull 2-parameters model

Potential DMPs can be estimated using the critical values derived from the fitted Weibull 2-parameters models, which are obtained after the non-linear fit of the theoretical model on the genome-wide $HD$ values for each individual sample using Methyl-IT function nonlinearFitDist [2]. As before, only cytosine sites with critical values $TV>0.926$ are considered DMPs. Notice that, it is always possible to use any other values of $HD$ and $TV$ as critical values, but whatever could be the value it will affect the final accuracy of the classification performance of DMPs into two groups, DMPs from control and DNPs from treatment (see below). So, it is important to do an good choices of the critical values.

nlms.wb <- nonlinearFitDist(divs, column = 9L, verbose = FALSE, num.cores = 6L)
# Potential DMPs from 'Weibull2P' model
DMPs.wb <- getPotentialDIMP(LR = divs, nlms = nlms.wb, div.col = 9L, 
                            tv.cut = 0.926, tv.col = 7, alpha = 0.05, 
                            dist.name = "Weibull2P")
nlms.wb$T1 
##         Estimate   Std. Error  t value Pr(>|t|))      Adj.R.Square
## shape  0.5413711 0.0003964435 1365.570         0 0.991666592250838
## scale 19.4097502 0.0155797315 1245.833         0                  
##                     rho       R.Cross.val              DEV
## shape 0.991666258901194 0.996595712743823 34.7217494754823
## scale                                                     
##                     AIC               BIC     COV.shape     COV.scale
## shape -221720.747067975 -221694.287733122  1.571674e-07 -1.165129e-06
## scale                                     -1.165129e-06  2.427280e-04
##       COV.mu     n
## shape     NA 50000
## scale     NA 50000

5. Potential DMPs detected with Gamma 2-parameters model

As in the case of Weibull 2-parameters model, potential DMPs can be estimated using the critical values derived from the fitted Gamma 2-parameters models and only cytosine sites with critical values $TV_d > 0.926$ are considered DMPs.

nlms.g2p <- nonlinearFitDist(divs, column = 9L, verbose = FALSE, num.cores = 6L,
                            dist.name = "Gamma2P")
# Potential DMPs from 'Gamma2P' model
DMPs.g2p <- getPotentialDIMP(LR = divs, nlms = nlms.g2p,  div.col = 9L, 
                             tv.cut = 0.926, tv.col = 7, alpha = 0.05, 
                             dist.name = "Gamma2P")
nlms.g2p$T1
##         Estimate   Std. Error  t value Pr(>|t|))      Adj.R.Square
## shape  0.3866249 0.0001480347 2611.717         0 0.999998194156282
## scale 76.1580083 0.0642929555 1184.547         0                  
##                     rho       R.Cross.val                 DEV
## shape 0.999998194084045 0.998331895911125 0.00752417919133131
## scale                                                        
##                    AIC               BIC     COV.alpha     COV.scale
## shape -265404.29138371 -265369.012270572  2.191429e-08 -8.581717e-06
## scale                                    -8.581717e-06  4.133584e-03
##       COV.mu    df
## shape     NA 49998
## scale     NA 49998

Summary table:

data.frame(ft = unlist(lapply(ft, length)), ft.hd = unlist(lapply(ft.hd, length)),
ecdf = unlist(lapply(DMPs.hd, length)), Weibull = unlist(lapply(DMPs.wb, length)),
Gamma = unlist(lapply(DMPs.g2p, length)))
##      ft ft.hd ecdf Weibull Gamma
## C1 1253   773   63     756   935
## C2 1221   776   62     755   925
## C3 1280   786   64     768   947
## T1 2504  1554  126     924  1346
## T2 2464  1532  124     942  1379
## T3 2408  1477  121     979  1354

The graphics for the empirical (in black) and Gamma (in blue) densities distributions of Hellinger divergence of methylation levels for sample T1 are shown below. The 2-parameter gamma model is build by using the parameters estimated in the non-linear fit of $H$ values from sample T1. The critical values estimated from the 2-parameter gamma distribution $H^{\Gamma}_{\alpha=0.05}=124$ is more ‘conservative’ than the critical value based on the empirical distribution $H^{Emp}_{\alpha=0.05}=114.5$. That is, in accordance with the empirical distribution, for a methylation change to be considered a signal its $H$ value must be $H\geq114.5$, while according with the 2-parameter gamma model any cytosine carrying a signal must hold $H\geq124$.

suppressMessages(library(ggplot2))

# Some information for graphic
dt <- data[data$sample == "T1", ]
coef <- nlms.g2p$T1$Estimate # Coefficients from the non-linear fit
dgamma2p <- function(x) dgamma(x, shape = coef[1], scale = coef[2])
qgamma2p <- function(x) qgamma(x, shape = coef[1], scale = coef[2])

# 95% quantiles 
q95 <- qgamma2p(0.95) # Gamma model based quantile
emp.q95 = quantile(divs$T1$hdiv, 0.95) # Empirical quantile

# Density plot with ggplot
ggplot(dt, aes(x = HD)) + 
  geom_density(alpha = 0.05, bw = 0.2, position = "identity", na.rm = TRUE,
               size = 0.4) + xlim(c(0, 150)) +
  stat_function(fun = dgamma2p, colour = "blue") +
  xlab(expression(bolditalic("Hellinger divergence (HD)"))) + 
  ylab(expression(bolditalic("Density"))) +
  ggtitle("Empirical and Gamma densities distributions of Hellinger divergence (T1)") +
  geom_vline(xintercept = emp.q95, color = "black", linetype = "dashed", size = 0.4) +
  annotate(geom = "text", x = emp.q95 - 20, y = 0.16, size = 5,
           label = 'bolditalic(HD[alpha == 0.05]^Emp==114.5)',
           family = "serif", color = "black", parse = TRUE) +
  geom_vline(xintercept = q95, color = "blue", linetype = "dashed", size = 0.4) +
  annotate(geom = "text", x = q95 + 9, y = 0.14, size = 5,
           label = 'bolditalic(HD[alpha == 0.05]^Gamma==124)',
           family = "serif", color = "blue", parse = TRUE) +
  theme(
    axis.text.x  = element_text( face = "bold", size = 12, color="black",
                                 margin = margin(1,0,1,0, unit = "pt" )),
    axis.text.y  = element_text( face = "bold", size = 12, color="black", 
                                 margin = margin( 0,0.1,0,0, unit = "mm")),
    axis.title.x = element_text(face = "bold", size = 13,
                                color="black", vjust = 0 ),
    axis.title.y = element_text(face = "bold", size = 13,
                                color="black", vjust = 0 ),
    legend.title = element_blank(),
    legend.margin = margin(c(0.3, 0.3, 0.3, 0.3), unit = 'mm'),
    legend.box.spacing = unit(0.5, "lines"),
    legend.text = element_text(face = "bold", size = 12, family = "serif")
  1. Steerneman, Ton, K. Behnen, G. Neuhaus, Julius R. Blum, Pramod K. Pathak, Wassily Hoeffding, J. Wolfowitz, et al. 1983. “On the total variation and Hellinger distance between signed measures; an application to product measures.” Proceedings of the American Mathematical Society 88 (4). Springer-Verlag, Berlin-New York: 684–84. doi:10.1090/S0002-9939-1983-0702299-0.
  2. Sanchez, Robersy, and Sally A. Mackenzie. 2016. “Information Thermodynamics of Cytosine DNA Methylation.” Edited by Barbara Bardoni. PLOS ONE 11 (3). Public Library of Science: e0150427. doi:10.1371/journal.pone.0150427.

Methylation analysis with Methyl-IT. Part 1

 Methylation analysis with Methyl-IT is illustrated on simulated datasets of methylated and unmethylated read counts with relatively high average of  methylation levels: 0.15 and 0.286 for control and treatment groups, respectively. The main Methyl-IT downstream analysis is presented alongside the application of Fisher’s exact test. The importance of a signal detection step is shown.

Methyl-IT R package offers a methylome analysis approach based on information thermodynamics (IT) and signal detection. Methyl-IT approach confront detection of differentially methylated cytosine as a signal detection problem. This approach was designed to discriminate methylation regulatory signal from background noise induced by molecular stochastic fluctuations. Methyl-IT R package is not limited to the IT approach but also includes Fisher’s exact test (FT), Root-mean-square statistic (RMST) or Hellinger divergence (HDT) tests. Herein, we will show that a signal detection step is also required for FT, RMST, and HDT as well.

For the current example on methylation analysis with Methyl-IT we will use simulated data. Read count matrices of methylated and unmethylated cytosine are generated with Methyl-IT function simulateCounts. Function simulateCounts randomly generates prior methylation levels using Beta distribution function. The expected mean of methylation levels that we would like to have can be estimated using the auxiliary function:

bmean <- function(alpha, beta) alpha/(alpha + beta)
alpha.ct <- 0.09
alpha.tt <- 0.2
c(control.group = bmean(alpha.ct, 0.5), treatment.group = bmean(alpha.tt, 0.5), 
  mean.diff = bmean(alpha.tt, 0.5) - bmean(alpha.ct, 0.5)) 
##   control.group treatment.group       mean.diff 
##       0.1525424       0.2857143       0.1331719

This simple function uses the α and β (shape2) parameters from the Beta distribution function to compute the expected value of methylation levels. In the current case, we expect to have a difference of methylation levels about 0.133 between the control and the treatment.

Methyl-IT function simulateCounts will be used to generate the datasets, which will include three group of samples: reference, control, and treatment.

suppressMessages(library(MethylIT))

# The number of cytosine sites to generate
sites = 50000 
# Set a seed for pseudo-random number generation
set.seed(124)
control.nam <- c("C1", "C2", "C3")
treatment.nam <- c("T1", "T2", "T3")

# Reference group 
ref0 = simulateCounts(num.samples = 4, sites = sites, alpha = alpha.ct, beta = 0.5,
                      size = 50, theta = 4.5, sample.ids = c("R1", "R2", "R3"))
# Control group
ctrl = simulateCounts(num.samples = 3, sites = sites, alpha = alpha.ct, beta = 0.5,
                      size = 50, theta = 4.5, sample.ids = control.nam)
# Treatment group
treat = simulateCounts(num.samples = 3, sites = sites, alpha = alpha.tt, beta = 0.5,
                       size = 50, theta = 4.5, sample.ids = treatment.nam)

Notice that reference and control groups of samples are not identical but belong to the same population.

The estimation of the divergences of methylation levels is required to proceed with the application of signal detection basic approach. The information divergence is estimated here using the function estimateDivergence. For each cytosine site, methylation levels are estimated according to the formulas: $p_i={n_i}^{mC_j}/({n_i}^{mC_j}+{n_i}^{C_j})$, where ${n_i}^{mC_j}$ and ${n_i}^{C_j}$ are the number of methylated and unmethylated cytosines at site $i$.

If a Bayesian correction of counts is selected in function estimateDivergence, then methylated read counts are modeled by a beta-binomial distribution in a Bayesian framework, which accounts for the biological and sampling variations [1,2,3]. In our case we adopted the Bayesian approach suggested in reference [4] (Chapter 3).

Two types of information divergences are estimated: TV, total variation (TV, absolute value of methylation levels) and Hellinger divergence (H). TV is computed according to the formula: $TV=|p_{tt}-p_{ct}|$ and H:

$H(\hat p_{ij},\hat p_{ir}) = w_i[(\sqrt{\hat p_{ij}} – \sqrt{\hat p_{ir}})^2+(\sqrt{1-\hat p_{ij}} – \sqrt{1-\hat p_{ir}})^2]$  (1)

where $w_i = 2 \frac{m_{ij} m_{ir}}{m_{ij} + m_{ir}}$, $m_{ij} = {n_i}^{mC_j}+{n_i}^{uC_j}+1$, $m_{ir} = {n_i}^{mC_r}+{n_i}^{uC_r}+1$ and $j \in {\{c,t}\}$

The equation for Hellinger divergence is given in reference [5], but
any other information theoretical divergences could be used as well. Divergences are estimated for control and treatment groups in respect to a virtual sample, which is created applying function poolFromGRlist on the reference group.

.

# Reference sample
ref = poolFromGRlist(ref0, stat = "mean", num.cores = 4L, verbose = FALSE)

# Methylation level divergences
DIVs <- estimateDivergence(ref = ref, indiv = c(ctrl, treat), Bayesian = TRUE, 
                           num.cores = 6L, percentile = 1, verbose = FALSE)

The mean of methylation levels differences is:

unlist(lapply(DIVs, function(x) mean(mcols(x[, 7])[,1])))
##            C1            C2            C3            T1            T2 
## -0.0009820776 -0.0014922009 -0.0022257725  0.1358867135  0.1359160219 
##            T3 
##  0.1309217360

Likewise for any other signal in nature, the analysis of methylation signal requires for the knowledge of its probability distribution. In the current case, the signal is represented in terms of the Hellinger divergence of methylation levels (H).

divs = DIVs[order(names(DIVs))]

# To remove hd == 0 to estimate. The methylation signal only is given for  
divs = lapply(divs, function(div) div[ abs(div$hdiv) > 0 ])
names(divs) <- names(DIVs)

# Data frame with the Hellinger divergences from both groups of samples samples 
l = c(); for (k in 1:length(divs)) l = c(l, length(divs[[k]]))
data <- data.frame(H = c(abs(divs$C1$hdiv), abs(divs$C2$hdiv), abs(divs$C3$hdiv),
                           abs(divs$T1$hdiv), abs(divs$T2$hdiv), abs(divs$T3$hdiv)),
                   sample = c(rep("C1", l[1]), rep("C2", l[2]), rep("C3", l[3]),
                              rep("T1", l[4]), rep("T2", l[5]), rep("T3", l[6]))
)

Empirical critical values for the probability distribution of H and TV can be obtained using quantile function from the R package stats.

critical.val <- do.call(rbind, lapply(divs, function(x) {
  hd.95 = quantile(x$hdiv, 0.95)
  tv.95 = quantile(x$TV, 0.95)
  return(c(tv = tv.95, hd = hd.95))
}))

critical.val
##       tv.95%    hd.95%
## C1 0.7893927  81.47256
## C2 0.7870469  80.95873
## C3 0.7950869  81.27145
## T1 0.9261629 113.73798
## T2 0.9240506 114.45228
## T3 0.9212163 111.54258

3.1. Density estimation

The kernel density estimation yields the empirical density shown in the graphics:

suppressMessages(library(ggplot2))

# Some information for graphic
crit.val.ct <- max(critical.val[c("C1", "C2", "C3"), 2]) # 81.5
crit.val.tt <- min(critical.val[c("T1", "T2", "T3"), 2]) # 111.5426

# Density plot with ggplot
ggplot(data, aes(x = H, colour = sample, fill = sample)) + 
  geom_density(alpha = 0.05, bw = 0.2, position = "identity", na.rm = TRUE,
               size = 0.4) + xlim(c(0, 125)) +   
  xlab(expression(bolditalic("Hellinger divergence (H)"))) + 
  ylab(expression(bolditalic("Density"))) +
  ggtitle("Density distribution for control and treatment") +
  geom_vline(xintercept = crit.val.ct, color = "red", linetype = "dashed", size = 0.4) +
  annotate(geom = "text", x = crit.val.ct-2, y = 0.3, size = 5,
           label = 'bolditalic(H[alpha == 0.05]^CT==81.5)',
           family = "serif", color = "red", parse = TRUE) +
  geom_vline(xintercept = crit.val.tt, color = "blue", linetype = "dashed", size = 0.4) +
  annotate(geom = "text", x = crit.val.tt -2, y = 0.2, size = 5,
           label = 'bolditalic(H[alpha == 0.05]^TT==114.5)',
           family = "serif", color = "blue", parse = TRUE) +
  theme(
    axis.text.x  = element_text( face = "bold", size = 12, color="black",
                                 margin = margin(1,0,1,0, unit = "pt" )),
    axis.text.y  = element_text( face = "bold", size = 12, color="black", 
                                 margin = margin( 0,0.1,0,0, unit = "mm")),
    axis.title.x = element_text(face = "bold", size = 13,
                                color="black", vjust = 0 ),
    axis.title.y = element_text(face = "bold", size = 13,
                                color="black", vjust = 0 ),
    legend.title = element_blank(),
    legend.margin = margin(c(0.3, 0.3, 0.3, 0.3), unit = 'mm'),
    legend.box.spacing = unit(0.5, "lines"),
    legend.text = element_text(face = "bold", size = 12, family = "serif")
  )
The above graphic shows that with high probability the methylation signal induced by the treatment has H values $H^{TT}_{\alpha=0.05}\geq114.5$. According to the critical value estimated for the differences of methylation levels, the methylation signal holds $TV^{TT}_{\alpha=0.05}\geq0.926$. Notice that most of the methylation changes are not signal but noise (found to the left of the critical values). This situation is typical for all the natural and technologically generated signals. Assuming that the background methylation variation is consistent with a Poisson process and that methylation changes conform to the second law of thermodynamics, the Hellinger divergence of methylation levels follows a Weibull distribution probability or some member of the generalized gamma distribution family [6].
  1. Hebestreit, Katja, Martin Dugas, and Hans-Ulrich Klein. 2013. “Detection of significantly differentially methylated regions in targeted bisulfite sequencing data.” Bioinformatics (Oxford, England) 29 (13): 1647–53. doi:10.1093/bioinformatics/btt263.
  2. Hebestreit, Katja, Martin Dugas, and Hans-Ulrich Klein. 2013. “Detection of significantly differentially methylated regions in targeted bisulfite sequencing data.” Bioinformatics (Oxford, England) 29 (13): 1647–53. doi:10.1093/bioinformatics/btt263.
  3. Dolzhenko, Egor, and Andrew D Smith. 2014. “Using beta-binomial regression for high-precision differential methylation analysis in multifactor whole-genome bisulfite sequencing experiments.” BMC Bioinformatics 15 (1). BioMed Central: 215. doi:10.1186/1471-2105-15-215.
  4. Baldi, Pierre, and Soren Brunak. 2001. Bioinformatics: the machine learning approach. Second. Cambridge: MIT Press.
  5. Basu, A., A. Mandal, and L. Pardo. 2010. “Hypothesis testing for two discrete populations based on the Hellinger distance.” Statistics & Probability Letters 80 (3-4). Elsevier B.V.: 206–14. doi:10.1016/j.spl.2009.10.008.
  6. Sanchez R, Mackenzie SA. Information Thermodynamics of Cytosine DNA Methylation. PLoS One, 2016, 11:e0150427.

Group operations on the set of five DNA bases

General biochemical background. The genetic information on how to build proteins able to perform different biological/biochemical functions is encoded in the DNA sequence. Code-words of three letters/bases, called triplets or codons, are used to encode the information that will be used to synthesize proteins. Every codon encodes the information for one amino acid and every amino acid can be encoded by one or more codons. The genetic code is the biochemical system that establishes the rules by which the nucleotide sequence of a protein-encoding gene is transcribed into mRNA codon sequences and then translated into the amino acid sequences of the corresponding proteins. The genetic code is an extension of the four-letter alphabet found in DNA molecules. These “letters” are the DNA bases: Adenine, Guanine, Cytosine, and Thymine, usually denoted A, G, C, and T respectively (in an RNA molecule, T is changed to U, uracil).

In the DNA molecules, nucleotide bases are paired according to the following rule (Watson-Crick base pairings): G:C, A:T. That is, base G is the complementary base of C, and A is the complementary base of T (or U) in the DNA (or in the RNA) molecule and vice-versa.

Each DNA/RNA base can be classified into three main classes according to three criteria: chemical type (purines: A and G,  or pyrimidines: C and T), number of hydrogen bonds (strong or weak) and the whether the base has an amino group or a keto group [1]. Each criterion produces a partition of the set of bases [2].

Base pair AT
Two hydrogen bonds are established in the pairing of Adenine (T) and Thymine (T) two. Yikrazuul [Public domain], from Wikimedia Commons
Base pair GC
The pairing between guanine and cytosine involved three hydrogen bonds. Yikrazuul [Public domain], from Wikimedia Commons

The relationships between the DNA nucleotide bases, quantitatively expressed throughout the Watson-Crick base-pairings, permit the representation of the standard genetic code as a cube inserted in the Euclidean three-dimensional vector space $\mathbb{R}^3$ [3,4]. In particular, it is plausible that the present standard genetic code was derived from an ancestral code architecture with five or more bases (see main text for full discussion). The algebraic and biological model suggests the plausibility of the transition from a primeval code with an extended DNA alphabet $\mathfrak{B}$ ={D,A,C,G,U} to the present standard code, where the symbol “D” represents one or more hypothetical bases with unspecific pairings. It is important to observe that though the evidence from organic chemistry experiments supports the necessity of five or more DNA bases in the primordial genetic system apparatus, the formal development of the algebraic theory necessarily leads to an extension of the DNA base alphabet. In fact, the additional base was implicit in the multiple sequence alignments of the DNA sequences as gaps representing insertion and deletion mutations (indel mutations). The importance of considering the indel mutations in the phylogenetic analysis was analyzed in [2]. Perhaps the most significant role of the fifth base in the current DNA molecules is played by the epigenetics role of cytosine DNA methylation (CDM). CDM patterning represents one feature of the epigenome that is highly responsive to environmental stress and associates with trans-generational adaptation in plants and in animals.

If the Watson-Crick base pairings are symbolically expressed by means of the sum “+” operation in such a way that the following relationships hold: G + C = C + G = D, and A + U = U + A = D, then this requirement leads to the definition of an additive group or Abelian group on the set of five RNA (DNA) bases. Explicitly, it will be required that bases with the same number of hydrogen bonds in the DNA molecule and different chemical types be algebraic inverses of each other in the additive group defined on the set of DNA bases. In other words, the complementary RNA bases G:C and A:U (or G:C and A:T in the DNA) are, respectively, algebraic complements. This definition also reflects the non-specific pairings of the ancient hypothetical base(s) D, which is taken as the neutral element of the sum operation. Next, there is only one possible definition for the multiplication operation ($\times$) (with base A as the neutral element for this operation) in such a way that it completes a finite (Galois) field structure isomorphic to the field $\mathbb{Z_5}$ and defined over the set of integers modulo 5 (GF(5)). The simplicity of these operations can be noticed in Table 1.

Table 1. Operation tables of the Galois field (GF(5)) on the ordered set of the extended bases alphabet $\mathfrak{B}$={D, A, C, G, U}, and on $\mathbb{Z_5}$.

Sum Product
+ D A C G U × D A C G U
D D A C G U D D D D D D
A A C G U D A D A C G U
C C G U D A C D C U A G
G G U D A C G D G A U C
U U D A C G U D U G C A
SumProduct
+01234 ×01234
001234000000
112340101234
223401202413
334012303142
440123404321

Further readings are provided in the Computable Document Format (CDF) named IntroductionToZ5GeneticCodeVectorSpace.cdf. In this CDF, readers will gain a better comprehension of the biological and algebraic background on the genetic code algebras presented in reference [4]. This is a didactic and interactive visualization to  introduce the Genetic code $\mathbb{Z}_5$-Vector Space.

  1. Cornish-Bowden, A. (1985). Nomenclature for incompletely specified bases in nucleic acid sequences: recommendations 1984. Nucleic Acids Research, 13(9), 3021–3030.
  2. Jimenez-Montano MA, de la Mora-Basanez CR, Poschel T. The hypercube structure of the genetic code explains conservative and non-conservative aminoacid substitutions in vivo and in vitro. Biosystems, 1996, 39:117–25
  3. Sanchez R, Grau R, Morgado E (2006) A novel Lie algebra of the genetic code over the Galois field of four DNA bases. Math Biosci 202: 156-174.
  4. Sánchez R, Grau R (2009) An algebraic hypothesis about the primeval genetic code architecture. Math Biosci 221 : 60 – 76.