Background
==========

The introduction of small subunit ribosomal RNA as a tool in microbial taxonomy by Carl Woese and George Fox \[[@B1]\] led most microbiologists to assume that the concepts of animal and plant taxonomy could be extended to the realm of prokaryotes. In particular, it was assumed that a natural taxonomic system for microorganisms was feasible \[[@B2]\]. The goal of a natural taxonomic system is the formation of taxonomic groups that are defined by shared ancestry \[[@B3]\]. By definition, an ancestor that defines a monophyletic group can only give rise to members of this group. No organism outside this group has a lineage that traces back to the same ancestor (paraphyletic group); however, there might be earlier ancestors that define more inclusive monophyletic groups. The metaphor for organismal evolution that underlies a natural taxonomic system is a strictly bifurcating tree of species. A decade ago ribosomal RNA promised that one day it might be possible to place every extant organism on a universal tree of life, and the hope was that more genomic sequences would make this placement more accurate.

However, the analyses of completely sequenced genomes initiated a reassessment of concepts in microbial evolution \[[@B4]\]. While some molecular markers were found to agree with one another e.g., \[[@B5]\], others do not \[[@B6]-[@B12]\]. Transfer of genetic information between divergent organisms has turned the tree of life into a net or web \[[@B13]\], and genomes into mosaics. Different parts of genomes have different histories, and representing the history of genome evolution as a single tree appears inconsistent with the data. Nevertheless, the assumption of a tree-like process still underlies many approaches. Genome content trees have been calculated based on the presence and absence of genes \[[@B14]-[@B16]\] or types of protein folds \[[@B17]\]. While there is limited agreement between genome and rRNA phylogeny, at present it remains unclear whether this similarity is based on shared ancestry of part of a less frequently exchanging genome core \[[@B18]\], or if the apparent congruence is itself the result of horizontal gene transfer \[[@B19]\].

Overall genome content is not best represented on a single tree. Fig. [1](#F1){ref-type="fig"} gives an example of an alternative depiction, where thickness of a line reflects percentage of genes shared between two genomes. The coherence among the three domains of life (Bacteria, Archaea, Eucarya \[[@B20]\]), is clearly reflected in genome content; i.e., Archaea share more genes with other Archaea than with Bacteria, but many features are incompatible with representing the relationships between different genomes as a tree. For example, the mesophilic euryarchaeon *Halobacterium* sp. has more genes in common with the mesophilic Bacteria than does the thermophilic crenarchaeote *Aeropyrum pernix*. However, the extremophilic euryarcheote *Archaeoglobus fulgidus* shares many more genes with the extremophilic bacteria, *Aquifex aeolicus* and *Thermotoga maritima* than does *Halobacterium*. While this example illustrates the web-like relationships among genomes, recent phylogenetic reconstructions from molecular data have explored only few alternatives to the tree-paradigm (e.g. \[[@B21],[@B22]\]).

![Star Like Representation of Genome Relationships. The diagram depicts pairwise comparisons among thirteen genomes. Every genome is represented as a point on the perimeter of a circle. The thickness of the line connecting two genomes reflects the percentage of shared genes between the genomes. The thickest line connecting *Aquifex aeolicus* and *Thermotoga maritima* corresponds to 51% shared genes, and the thinnest line connecting *Aeropyrum pernix* and *Borrelia burgdorferi* corresponds to 9% shared genes. A gene is considered shared when it had a BLAST hit in the other genome with an E-value below 10^-8^. The percentage of genes shared between genomes A and B is calculated as ((\#of genes in A shared with B/total \# of genes in A)+(\# of genes in B shared with A/total \# of genes in B))/2. Bacteria are depicted in green, Archaea in red and Eukaryotes in blue. The domain affiliation is also indicated by a letter following the species name (A: Archaea, B: Bacteria, and E: Eukaryotes).](1471-2164-3-4-1){#F1}

One obvious drawback of the star-like representation in Fig. [1](#F1){ref-type="fig"} is that it utilizes BLAST search results only. Any phylogenetic information retained in the sequences is not utilized beyond the presence or absence decision based on a single expectation value cut-off. Because of recombination, individual genes themselves might be mosaic \[[@B23]\]; however, within-gene recombination of protein coding genes occurs mostly between closely related organisms. The redundancy of the genetic code greatly reduces recombination between divergent proteins. Even if a region is 100% conserved on the amino acid level, the encoding DNA can be so different as to allow the mismatch repair system to prevent recombination. For studies of single divergent orthologous protein encoding genes the assumption of a tree-like evolutionary history remains a reasonable expectation. In this manuscript we focus on methods that utilize the phylogenetic information that is retained in molecular sequence data, while not presuming that genomes as a whole evolved in a tree-like fashion.

In an elegant approach Korbinian Strimmer and Arndt von Haeseler \[[@B24]\] utilized Bayesian posterior probabilities to assess the phylogenetic information contained in an alignment of four homologous sequences. With four sequences there are only three possible tree topologies, and thus the three posterior probabilities corresponding to these three trees must sum up to one. Utilizing a barycentric coordinate system, the resulting probability vector is represented as a point in an equilateral triangle (Fig. [2](#F2){ref-type="fig"}), where the distances of the point P to the three sides represent the three probabilities. Strimmer and von Haeseler applied this approach to depict the phylogenetic information content present in a multiple sequence alignment. They plot the results from the analyses of all possible quartets, where the four sequences are selected from a single multiple sequence alignment in the same coordinate system. If there is a lot of phylogenetic information in the alignment, then most probability vectors will fall close to one of the corners; conversely datasets containing little phylogenetic information will mainly result in vectors falling into the center of the triangle. Here we explore the application of this and similar approaches in comparative genome analyses. In particular, we compare different approaches to calculate Bayesian posterior probabilities, and we compare these probabilities to the more widely used bootstrap support values. We assess the reliability of the different probability mapping approaches through their application to mitochondrial genomes, and we illustrate their usefulness by mapping selected interphylum and interdomain relationships.

![Mapping of the probability vector onto an equilateral triangle. Each QuartOP is represented as a probability vector P inside an equilateral triangle. The position of P is determined by the barycentric coordinates (p~1~, p~2~, p~3~), which correspond to the posterior probabilities or bootstrap support values of the three possible tree topologies. The vertices of the triangle T~1~, T~2~ and T~3~ represent the three possible unrooted tree topologies. Geometrically, each of the coordinates (p~1~, p~2~, p~3~) equals the distance between P and the side of the triangle opposite the corresponding vertex. Points closer to a vertex T~i~ have a larger corresponding probability p~i~ and represent a more probable tree topology than the two alternatives. All the points are classified by their position in one of three zones: \"total\" zone, \"90%\" zone and \"99%\" zone, which are depicted schematically and not drawn to scale. In this diagram, point P corresponds to a dataset which has highest probability for the topology T~3~, but the probability is below 90%, so the point P is located in the \"total\" zone, but not in the 99% or 90% zone. Figure adapted from \[[@B24]\].](1471-2164-3-4-2){#F2}

Results and Discussion
======================

Overview of data flow in probability mapping
--------------------------------------------

An outline of our approach to genome probability mapping is given in Figure [3](#F3){ref-type="fig"}. Using SEALS \[[@B25]\] and MySQL we developed scripts that identify and retrieve **quart**ets of **o**rthologous **p**rotein-encoding open reading frames (QuartOPs) from four selected genomes. We use the term genome to denote the collection of all ORFs identified in a genome. (In the case of genomes that are not well annotated, it is feasible to use a very wide definition of ORF, e.g, all amino acid sequences encoded between two stop codons in any of the six possible reading frames. As long as one of the genomes included in the analyses is properly annotated, only those identified ORFs that are actually homologous to an identified ORF will become part of a quartet of orthologs.) We utilize an operational definition of an ortholog: two open reading frames are considered orthologous, if and only if they are each other\'s top scoring BLAST hit when one is used as a query to search the other genome. A QuartOP is formed when each of the open reading frames picks the other members of the quartet as the top scoring hit in searches of the respective genomes. QuartOPs are similar to the clusters of orthologous groups (COGs) maintained by the NCBI \[[@B26]-[@B28]\], but differ in that COGs require only unidirectional, circular best hit relationships for three of the reference genomes, whereas we require the reciprocal top hit relationship for the four genomes included in a quartet, and we do not limit our identification of QuartOPs to a number of reference genomes. Montague and Hutchison utilized a comparable approach in their definition of congruent COGs \[[@B29]\]. So far we have analyzed 68 genome quartets (see supplementary material). The number of QuartOPs identified per genome quartet ranges from 82 (for genome quartet \#6: *Deinococcus radiodurans, Treponema pallidum, Escherichia coli,* and *Halobacterium sp*.) to 1182 (for genome quartet \#63: *Agrobacterium tumifaciens, Sinorhizobium meliloti, Mezorhizobium loti* and *Caulobacter crescentus*).

![Data flow for the genome quartet analysis. See Materials and Methods for details.](1471-2164-3-4-3){#F3}

Each of the aligned QuartOPs from a genome quartet was analyzed with respect to the posterior probability of the three possible tree topologies given the aligned QuartOP. Routinely we calculated these probabilities using Strimmer\'s and von Haeseler\'s approach \[[@B24]\]: Using each of the three topologies as a usertree, we calculated the maximum likelihood for each of the three topologies given the data. We then use the three maximum likelihoods to calculate the probability for topology *i* according to the formula: *P*~*i*~ = *L*~*i*~/(*L*~1~+*L*~2~+*L*~3~), where *L*~*i*~ is the likelihood for the best tree given topology *i*. Other types of reliability measures used to evaluate QuartOPs were bootstrap support values and Bayesian posterior probabilities estimated using MrBayes program (see below).

An example for the comparison of four genomes from different phyla is given in Fig. [4A](#F4){ref-type="fig"}. Surprisingly, each of the three tree topologies is strongly supported by more than 40 QuartOPs, and most of the QuartOPs appear to strongly support one of the trees. None of the three possibilities has majority support. Figure [5](#F5){ref-type="fig"} lists the functional categories of those QuartOPs that strongly support the different tree topologies. None of the categories shows a preference for a particular tree topology. For each tree topology more than 50% of the strongly supporting QuartOPs belong to the category \"information storage and processing\", while this category contains only about 1/3 of the genes present in the genomes. While the genes in this category appear more conserved and phylogenetically informative, the strong support that the genes in this category provide is nearly evenly split between the three possibilities.

![Maps of a genome quartet with organisms from four different bacterial phyla: *Escherichia coli* (Gram negative), *Deinococcus radiodurans* (Deinococcales), *Bacillus subtilis* (Gram positive) and *Treponema pallidum* (spirochete). Tree topologies assigned to the vertices are depicted in New Hampshire tree format near the corresponding vertex of the triangle and they are equivalent to the unrooted tree topologies as depicted in Figure [2](#F2){ref-type="fig"}. The three numbers associated with each tree topology indicate how many QuartOPs fall into each of the three zones: \"total\", 90% and 99% respectively. For definition of zones see figure [2](#F2){ref-type="fig"}. **A)** Probabilities are calculated according to Strimmer and von Haeseler \[[@B24]\]. There is no single topology that is supported by the majority of the QuartOPs and all three possible tree topologies are supported by roughly equal number of QuartOPs at the different probability levels. **B)** Probabilities are calculated with MrBayes program \[[@B31]\]. **C)** Bootstrap support values are plotted. For this case the zones are \"total\", 70% and 90% support, respectively. Bootstrapping appears to provide a more conservative reliability estimate than the posterior probabilities used in cases A and B. Nevertheless, each tree topology is still supported by a roughly equal number of bootstrapped datasets.](1471-2164-3-4-4){#F4}

![Distribution among different functional categories for those datasets that support one of the three topologies with better than 99% posterior probability. Tree topologies are indicated by column numbers 1, 2 and 3. Column 1 corresponds to topology ((1,4),2,3), columns 2 and 3 correspond to topologies ((1,3),2,4) and ((1,2),3,4) respectively. Divisions into functional categories are adopted from the COG database \[[@B27]\]. Functional categories are aggregated into four broad functional meta-categories. Distributions of datasets among the meta-categories are plotted as pie charts for each tree topology. In this case all three topologies are supported by roughly equal number of datasets from each meta-category.](1471-2164-3-4-5){#F5}

Impact of model parameters and sequence conservation
----------------------------------------------------

To test if ill-aligned sequences might have had an impact on the analyses, we repeated the analysis of genome quartet \# 8 (see Figure [4](#F4){ref-type="fig"}) using only QuartOPs that contained very similar sequences. By default we only excluded top hits with an E-value larger than 10^-4^. We repeated the example given in Fig. [4A](#F4){ref-type="fig"} with a cut-off of 10^-20^, i.e., we not only required the sequences in a QuartOP to be each others top hit, but in addition we asked for a high similarity between the two sequences. As a result the support for the three topologies in genome quartet \#8 dropped to 54 (44, 38), 51 (45, 32), 39 (29, 28) (the numbers in parenthesis are the numbers of quartets that support the topology with posterior probability larger than 90% and 99%, respectively). To access the level of sequence conservation within the QuartOPs\' sequences, we calculated the average percentage of pairwise identity per QuartOP. It varied from 40.53 10.54% to 43.84 ± 9.7% when the E-value cut-off was varied between 10^-2^ and 10^-20^ (see supplementary material for the summary table). While pairwise sequence identity is not a universally dependable measure of phylogenetic information content, these values illustrates that the sequences within a QuartOP are neither identical to one another, nor so divergent as to be saturated with substitutions and of questionable homology \[[@B30]\]. Using only the most conserved QuartOPs does not change the qualitative result: each of the three possible tree topologies is supported by about an equal number of QuartOPs (see supplementary material).

We recalculated the likelihoods for all QuartOPs in genome quartet \#8 using a model that incorporates **a**mong **s**ite **r**ate **v**ariation (ASRV). The posterior probabilities calculated according to Strimmer and von Haeseler did not change dramatically and each of the three tree topologies is still supported by roughly equal number of QuartOPs. The maps for this analysis are available in the supplementary material.

Estimating Bayesian Posterior Probabilities
-------------------------------------------

The formula used by Strimmer and von Haeseler \[[@B24]\] to calculate posterior probabilities (i.e. the probability that tree topology *T*~*i*~ is true given an aligned set of four sequences) considers only three trees (i.e. branch lengths and topology), each with the same prior probability. These three trees are those that have the highest likelihood for the three possible topologies. However, there are infinitely many other trees that differ from the three chosen ones only by differences in branch lengths. What is the effect on the calculated posterior probability of using only the single best tree as a representative of all the trees with the same topology? There is no *a priori* reason to exclude the other trees that have slightly lower likelihoods.

A different approach that does not make these assumptions is the use of Markov Chain Monte Carlo methods to explore tree space. We used the program MrBayes written by Huelsenbeck and Ronquist \[[@B31]\]. Using a QuartOP with posterior probabilities of .76, .10 and .13 we explored different parameter choices for the biased random walk through tree-space. We chose two chains with 5,000 burn-in cycles, and 25,000 cycles with sampling after every cycle as a compromise between increased precision of the probability estimate and computation time (see Materials and Methods for more details).

The result of calculating the posterior probabilities of all QuartOPs in genome quartet \#8 is given in figure [4B](#F4){ref-type="fig"}. Again all of the three tree topologies are strongly supported by some QuartOPs. When we repeated this analysis using the same settings, none of the probabilities changed by more than a few percent. The support for the three tree topologies changed from 67/47/37, 69/40/22 and 86/55/32 (the three numbers indicate total support and QuartOPs that supported a topology with more than .90 and .99 respectively) in the first run to 67/47/37, 70/41/22 and 85/55/32 in the second, indicating that the chosen parameters provided satisfactory reproducibility. Plots of both analyses are available in the supplementary material. Comparing figure [4B](#F4){ref-type="fig"} with [4A](#F4){ref-type="fig"} it is clear that in this case the Bayesian posterior probabilities estimated with MrBayes are more conservative assessments of reliability than the ones calculated according to \[[@B24]\]. The 99% support level calculated according to \[[@B24]\] approximately corresponds to the 90% support level calculated with MrBayes.

Bootstrap support values versus posterior probabilities
-------------------------------------------------------

To facilitate comparison of Bayesian posterior probabilities with a more widely used confidence measure, we generated 100 bootstrapped samples \[[@B32]\] from each QuartOP in case \#8. Each of the bootstrapped samples was analyzed using maximum likelihood with the same model of substitution as before. Each of the bootstrapped samples supports one of the three possible topologies, thus the sum of the bootstrap support values for the three topologies adds up to 100%, and the percentage of bootstrapped samples for each QuartOP that best supported each tree was again plotted in a barycentric coordinate system (Fig. [4C](#F4){ref-type="fig"}). Many more QuartOPs map into the central region of the triangle as compared to Figure [4A](#F4){ref-type="fig"} and [4B](#F4){ref-type="fig"}. Clearly, for this test bootstrap support values are more conservative measures of support than either of the posterior probabilities calculated above. Nevertheless, there are still several QuartOPs that strongly support each of the three tree topologies; however, there are 22 QuartOPs that support grouping *E. coli* with *Treponema pallidum* with better than 90% bootstrap support, whereas the alternatives are supported by only 12 and 13 QuartOPs, respectively. Comparing Figures [4A](#F4){ref-type="fig"} and [4C](#F4){ref-type="fig"} it appears that in analyzing quartets 70% bootstrap support is comparable to .99 posterior probability calculated according to \[[@B24]\].

Comparison of the different reliability assessment tools
--------------------------------------------------------

ML-mapping according to \[[@B24]\] is the least conservative of the tools explored. For the test cases analyzed a posterior probability of .99 according to \[[@B24]\], corresponds to a Bayesian posterior probability of .90 calculating using a Markov chain exploration of tree-space using \[[@B31]\] and about 70% bootstrap support. We did not find a strong dependence of the results on the substitution models used in calculating likelihoods and separate runs indicated satisfactory precision of the calculated probabilities and bootstrap values. Given that we only analyzed about 300 QuartOPs using all three approaches it would be premature to generalize our findings; however, other analyses that utilized both bootstrapping and Bayesian posterior probabilities also found bootstrapping to be more conservative than posterior probabilities calculated using Bayesian methods with Markov chain Monte Carlo sampling (e.g., \[[@B33]-[@B35]\]).

Mitochondrial genomes
---------------------

While gene transfer into the mitochondrial genomes has been inferred \[[@B36]-[@B39]\], mitochondrial genomes are expected to have undergone many fewer legitimate and illegitimate recombination events than free-living prokaryotes. Clearly, if probability mapping is to be considered a reliable approach, we expect that when analyzing quartets of mitochondrial genomes, the different genes should all support the same tree topology.

In most instances, this expectation is fulfilled (see Table [1](#T1){ref-type="table"}), even though we selected instances in which the splits could be expected to be ill resolved, e.g., echinoderm, mammal, insect, mollusk (m4), or protist, fungus, animal, plant (m7). The only exception was an ORF in quartet m7 that encodes the cytochrome oxidase subunit II. This ORF did not support grouping the animal with the fungal homolog as expected; rather it grouped the protist and animal homologs together (posterior probability according to \[[@B24]\] was " 0.99). Inspection of the aligned sequences (Fig. [6](#F6){ref-type="fig"}) revealed that there are more residues shared between the homologs from *Cafeteria roenbergensis* and *Homo sapiens* thanbetween the homologs from *Cafeteria roenbergensis* and *Arabidopsis thaliana*. No artifact that could be responsible for this unexpected grouping was detected. The same high support for this unexpected grouping is also recovered in bootstrap analysis and in posterior probabilities calculated with MrBayes \[[@B31]\].

![Alignment of mitochondrial cytochrome oxidase subunit II. The alignment for the control mitochondrial quartet m7 (see Table [1](#T1){ref-type="table"}) that supports the unexpected ((Homo sapiens, Cafeteria), Saccharomyces, Arabidopsis) topology. The exact matches for each tree topology are colored in three different colors. Blue corresponds to the ((Homo sapiens, Cafeteria), Saccharomyces, Arabidopsis), yellow corresponds to the ((Homo sapiens, Arabidopsis), Saccharomyces, Cafeteria) and green corresponds to the ((Homo sapiens, Saccharomyces), Arabidopsis, Cafeteria) tree topology. As can be seen, the majority of the matches are in favor of ((Homo sapiens, Cafeteria), Saccharomyces, Arabidopsis) tree topology. There are nine parsimony informative positions favoring the latter topology, and only three for each of the other two topologies.](1471-2164-3-4-6){#F6}

::: {#T1 .table-wrap}
::: {.caption}
######

Results of analyses for the control mitochondrial genome quartets \#m1-m7.
:::

  **\#**   **Genome 1**        **Genome 2**     **Genome 3**         **Genome 4**     **((1,2),3,4)**   ** **   **((1,3),2,4)**   ** **   **((1,4),2,3)**
  -------- ------------------- ---------------- -------------------- ---------------- ----------------- ------- ----------------- ------- ----------------- -------- -------- ------- --- --- ---
  **m1**   **Drosophila**      **Drosophila**   Ceratitis capitata   Apis mellifera   **9**             **8**   **8**             ** **   0                 0        0        ** **   1   0   0
           **melanogaster**    **yakuba**                            ligustica        **10**            **8**   **8**             ** **   0                 0        0        ** **   0   0   0
                                                                                **9**             **7**   **5**             ** **   0                 0        0        ** **   1   1   0
                                                                                ** **             ** **   ** **             ** **                                       ** **
  **m2**   **Alligator**       Opossum          **Stork**            Donkey           0                 0       0                 ** **   **11**            **11**   **11**   ** **   0   0   0
                                                                                0                 0       0                 ** **   **11**            **11**   **11**   ** **   0   0   0
                                                                                0                 0       0                 ** **   **11**            **11**   **11**   ** **   0   0   0
                                                                                ** **             ** **   ** **             ** **                                       ** **
  **m3**   **Turtle**          Opossum          **Stork**            Donkey           0                 0       0                 ** **   **12**            **12**   **12**   ** **   0   0   0
                                                                                0                 0       0                 ** **   **12**            **12**   **12**   ** **   0   0   0
                                                                                0                 0       0                 ** **   **12**            **12**   **12**   ** **   0   0   0
                                                                                ** **             ** **   ** **             ** **                                       ** **
  **m4**   **Starfish**        **Donkey**       Fruit Fly            Doorsnail        **8**             **7**   **7**             ** **   2                 1        1        ** **   0   0   0
                                                                                **8**             **7**   **7**             ** **   1                 0        0        ** **   1   0   0
                                                                                **8**             **7**   **5**             ** **   1                 1        0        ** **   1   0   0
                                                                                ** **             ** **   ** **             ** **                                       ** **
  **m6**   **Reclinomonas**    Saccharomyces    **Arabidopsis**      Homo sapiens     0                 0       0                 ** **   **4**             **3**    **3**    ** **   0   0   0
           **americana**       cerevisiae       **thaliana**                          1                 0       0                 ** **   **3**             **3**    **3**    ** **   0   0   0
                                                                                0                 0       0                 ** **   **4**             **3**    **2**    ** **   0   0   0
                                                                                ** **             ** **   ** **             ** **                                       ** **
  **m7**   **Cafeteria**       Saccharomyces    **Arabidopsis**      Homo sapiens     0                 0       0                 ** **   **3**             **2**    **2**    ** **   1   1   1
           **roenbergensis**   cerevisiae       **thaliana**                          1                 0       0                 ** **   **2**             **2**    **1**    ** **   1   1   1
                                                                                1                 0       0                 ** **   **2**             **1**    **0**    ** **   1   1   1

The groupings corresponding to the expected organismal phylogenies are given in bold. The three numbers in each table cell correspond to the three approaches used. The top number corresponds to results obtained using Strimmer and von Haeseler\'s approach \[24\], the middle number corresponds to results obtained using MrBayes program \[31\], and the bottom number corresponds to results of bootstrap support values calculation. Column \"Tot.\" lists the number of QuartOPs from \"total\" zone, column A lists the number of QuartOPs from \"90%\" zone (70% for bootstrap support), and column B lists the number of QuartOPs from \"99%\" zone (90% for bootstrap support). For definition of zones see Fig. 2. With the exception of the one dataset for quartet \#m7, the analyses proved to be consistent with organismal tree topologies. The alignment for the exceptional dataset is presented in Fig. 6. The common names for the organisms listed correspond to the following scientific names: alligator corresponds to *Alligator mississippiensis*, opossum to *Didelphis virginiana*, stork to *Ciconia ciconia*, donkey to *Equus asinus*, turtle to *Chelonia mydas*, starfish to *Asterina pectinifera*, doorsnail to *Albinaria caerulia*, fruit fly to *Drosophila melanogaster*.
:::

The finding of a QuartOP in a mitochondrial genome quartet that supports a non-traditional grouping could either reflect a rare recombination event, selection pressures that led to convergent evolution in two lineages, or a chance event -- if one looks at enough samples one will find some that (considered by themselves) appear significant. At present it is not possible to decide between these three possible explanations. Our analysis of mitochondrial genomes shows that in most instances the calculated probabilities (ML-mapping, Bayesian posterior probabilities, or bootstrap values) support the expected tree topologies, albeit with surprisingly strong support values. Rarely, unexpected groupings can be recovered and support for these probably erroneous groupings can be high. In most instances the ML-mapping approach accurately revealed the expected relationships between the mitochondrial genomes. This confirms the suitability of this approach in genome analyses.

Interphylum genome quartets
---------------------------

Here we focus on examples that illustrate the utility of the probability mapping approach. Focusing on the relationships between the cyanobacteria with other bacterial phyla we calculated several genome quartets that include the *Synechocystis* sp. genome and three members each of other bacterial phyla (see Table [2](#T2){ref-type="table"}). In all cases that included both *Bacillus subtilis* as a representative of the low GC Gram positives, and *Synechocystis* sp., the majority of QuartOPs supported the topologies that grouped these two organisms together. The alternative topologies were significantly supported by some QuartOPs, but the number of strongly supporting QuartOPs was lower than for the *B. subtilis* -- *Synechocystis* grouping. This also was true when one of the other two genomes was from a high GC Gram positive (genome quartets \#53, \#54, \#55 and \#68). Only when two low GC Gram positives were included in the same quartet, was the intra-phylum grouping of low GC Gram positives supported by many more QuartOPs than the grouping of *B. subtilis* with *Synechocystis* sp. (genome quartets \#51 and \#52).

::: {#T2 .table-wrap}
::: {.caption}
######

Summary of the genome quartets that include *Synechocystis* sp., *Bacillus subtilis*, and two bacterial genomes from other phyla.
:::

  **\#**   **Genome 1**      **Genome 2**       **Genome 3**       **Genome 4**        **((1,2),3,4)**   ** **   **((1,3),2,4)**   ** **   **((1,4),2,3)**
  -------- ----------------- ------------------ ------------------ ------------------- ----------------- ------- ----------------- ------- ----------------- ----- ------------ ------- ----- ----- ---------
  **9**    *Synechocystis*   *P. aeruginosa*    *D. radiodurans*   *B. subtilis*       94                76      63                ** **   101               73    *57*         ** **   186   158   **126**
  **10**   *Synechocystis*   *P. aeruginosa*    *T. pallidum*      *B. subtilis*       69                54      50                ** **   51                33    *28*         ** **   102   80    **67**
  **14**   *Synechocystis*   *R. sphaeroides*   *B. subtilis*      *E. coli*           65                53      44                ** **   286               263   **248**      ** **   28    25    17
  **15**   *Synechocystis*   *R. sphaeroides*   *B. subtilis*      *D. radiodurans*    95                72      60                ** **   201               173   **149**      ** **   73    60    47
  **16**   *Synechocystis*   *D. radiodurans*   *B. subtilis*      *T. pallidum*       63                50      *40*              ** **   94                74    **63**       ** **   60    46    *35*
  **17**   *Synechocystis*   *B. subtilis*      *T. pallidum*      *E. coli*           93                72      **66**            ** **   55                43    *34*         ** **   66    49    39
  **18**   *Synechocystis*   *D. radiodurans*   *T. pallidum*      *E. coli*           86                68      54                ** **   65                49    38           ** **   75    54    47
  **19**   *Synechocystis*   *D. radiodurans*   *B. subtilis*      *E. coli*           129               105     *86*              ** **   156               131   **104**      ** **   98    76    62
  **50**   *Synechocystis*   *B. subtilis*      *E. coli*          *M. loti*           276               255     **228**           ** **   44                31    21           ** **   54    43    38
  **53**   *Synechocystis*   *B. subtilis*      *E. coli*          *M. leprae*         125               104     **82**            ** **   101               84    65           ** **   119   19    66
  **54**   *Synechocystis*   *B. subtilis*      *E. coli*          *M. tuberculosis*   141               114     **97**            ** **   101               82    69           ** **   128   101   89
  **55**   *Synechocystis*   *B. subtilis*      *M. loti*          *M. tuberculosis*   189               164     **139**           ** **   92                74    59           ** **   80    58    44
  **64**   *Synechocystis*   *C. trachomatis*   *B. subtilis*      *T. pallidum*       47                31      24                ** **   108               96    **86**       ** **   36    25    18
  **67**   *Synechocystis*   *C. trachomatis*   *B. subtilis*      *M. loti*           64                48      32                ** **   116               104   **84**       ** **   72    51    44
  **68**   *Synechocystis*   *C. trachomatis*   *B. subtilis*      *M. tuberculosis*   77                55      45                ** **   94                80    **62**       ** **   68    52    38
  **51**   *Synechocystis*   *B. subtilis*      *E. coli*          *S. aureus*         33                19      15                ** **   361               349   [333]{.ul}   ** **   15    7     5
  **52**   *Synechocystis*   *B. subtilis*      *E. coli*          *S. pyogenes*       34                22      18                ** **   259               249   [227]{.ul}   ** **   24    17    9

The \# column refers to the unique number assigned to the genome quartets analyzed. Columns \"((1,2),3,4)\", \"((1,3),2,4)\" and \"((1,4),2,3)\" refer to the three possible tree topologies. Numbers in columns \"Tot\", \".90\" and \".99\" give the number of QuartOPs that support the indicated tree topology with a posterior probability higher than the other two posterior probabilities, or with 90% or 99% probability, respectively. The numbers in bold indicate the number of orthologs supporting the grouping of *Synechocystis* sp. and *Bacillus subtilis* in the absence of another low GC gram-positive in the genome quartets. Note that those numbers are the largest of the three numbers, a finding that supports the recent analyses by \[43\]. In the presence of another low GC Gram-positive in addition to *Bacillus subtilis*, the largest number of QuartOPs support grouping of low GC Gram-positives with each other (underlined). Other groupings that involve putative sister groups to the cyanobacteria (Deinococaceae and spirochetes) that had been suggested by others (e.g., \[40,41\]) are indicated in italics.
:::

Previous analyses based on a limited number of proteins and signature insertions and deletions had suggested different bacterial groups as closest relatives to cyanobacteria. Among the suggested sister groups were the Deinococcales \[[@B40]\] and spirochetes \[[@B41],[@B42]\]. Our analyses do not support these earlier claims, but are in agreement with the recent analyses of genes involved in chlorophyll biosynthesis \[[@B43]\], which indicated that the low GC Gram positive heliobacteria are closest to the last common ancestor of all oxygenic photosynthetic lineages. The analyses summarized in Table [2](#T2){ref-type="table"} also illustrate that interphylum HGT, while turning genomes into mosaics, has not eroded all associations between bacterial phyla. In the case of cyanobacteria, a close association between low GC Gram positives and the cyanobacteria is supported by the majority of conserved genes. Similar observations of reproducible associations between phyla based on genome wide comparisons were recently published \[[@B44]-[@B47]\]. However, at present it cannot be decided to what extent these closer associations reflects shared ancestry or are due to preferred HGT \[[@B19]\].

Interdomain genome quartets
---------------------------

In our search for the \"sister-phylum\" to the cyanobacteria we also analyzed a few quartets including Archaea. One noteworthy finding was that in the genome quartet including *Synechocystis* sp., *Halobacterium* sp., *Aquifex aeolicus* and *Thermotoga maritima* the grouping of *Halobacterium* sp. with *Synechocystis* sp. was recovered by many more QuartOPs (56 with p \> .99) than the grouping that would be expected following 16S rRNA phylogeny (12 QuartOPs with p \> .99; see Table [3](#T3){ref-type="table"}). To test if this association was specific for *Synechocystis* sp., we repeated the analyses replacing *Synechocystis* sp. with *Bacillus subtilis*. The result was qualitatively the same: at the p \> .99 level 53 QuartOPs supported grouping *Bacillus subtilis* with *Halobacterium* sp., and only 27 supported grouping *Aquifex aeolicus* with *Halobacterium* sp. (Fig. [7](#F7){ref-type="fig"}).

![ML map of the quartet representing *Bacillus subtilis*, the deep branching bacteria *T. maritima* and *A. aeolicus*, and the salt-loving archaeon *Halobacterium* sp.. The majority of the orthologous datasets support the grouping of the *Halobacterium* with *Bacillus subtilis*. The topology that corresponds to the 16S rRNA topology (lower left vertex) is supported by the least number of orthologous datasets. The result stayed qualitatively the same when *B. subtilis* was replacedwith the cyanobacterium *Synechocystis* sp. (see results for quartet \#11 in Table [3](#T3){ref-type="table"}). For details on the figure notations see legend for Figure [4](#F4){ref-type="fig"}. **A.** Probabilities calculated according to Strimmer and von Haeseler \[[@B24]\]. **B.** Probabilities calculated with the MrBayes program \[[@B31]\].](1471-2164-3-4-7){#F7}

::: {#T3 .table-wrap}
::: {.caption}
######

Summary of the genome quartets that include the mesophilic archaeon *Halobacterium* sp. or the thermophilic archaeon *Archaeoglobus fulgidus*, deep-branching bacteria *Thermotoga maritima* and *Aquifex aeolicus*, and bacteria *Synechocystis* sp. or *Bacillus subtilis*.
:::

  -------- --------------------- ----------------------- -------------------- -------------------------- ------------------- --------- ------------------- ------- ------------------- --------- ---------- ------- ---------- --------- ----------
  **\#**   **Genome 1**          **Genome 2**            **Genome 3**         **Genome 4**               ***((1,2),3,4)***   ** **     ***((1,3),2,4)***   ** **   ***((1,4),2,3)***

                                                                                **Tot.**            **0.9**   **0.99**            ** **   **Tot.**            **0.9**   **0.99**   ** **   **Tot.**   **0.9**   **0.99**

  **11**   *Synechocystis sp.*   *Thermotoga maritima*   *Aquifex aeolicus*   *Halobacterium sp.*        29                  20        12                  ** **   45                  34        27         ** **   86         69        56
  **13**   *Synechocystis sp.*   *Thermotoga maritima*   *Aquifex aeolicus*   *Archaeoglobus fulgidus*   47                  36        30                  ** **   63                  51        44         ** **   50         34        25
  **61**   *Bacillus subtilis*   *Thermotoga maritima*   *Aquifex aeolicus*   *Halobacterium sp.*        44                  32        27                  ** **   27                  19        11         ** **   83         62        53
  **62**   *Bacillus subtilis*   *Thermotoga maritima*   *Aquifex aeolicus*   *Archaeoglobus fulgidus*   64                  50        40                  ** **   50                  35        30         ** **   41         27        23
  -------- --------------------- ----------------------- -------------------- -------------------------- ------------------- --------- ------------------- ------- ------------------- --------- ---------- ------- ---------- --------- ----------

For table notations see legend for Table 2. Quartets \#11 and 61 indicate that the majority of the QuartOPs group *Halobacterium* sp. together with *Synechocystis* sp. and with *Bacillus subtilis* respectively, which is in disagreement with 16S rRNA topology. In two control quartets (\#13 and \#62) *Halobacterium* was substituted with *Archaeoglobus fulgidus*, and in these cases the majority of QuartOPs support the topology that is in agreement with SSU rRNA topology.
:::

Clearly, there are many artifacts possible in analyzing divergent sequences. For many QuartOPs the ortholog from *Halobacterium* sp. is expected to be the longest branch. To test for the possibility that long branch attraction \[[@B48]\] might be the reason for the strong support of *Halobacterium* sp. grouping with *Synechocystis* sp., we repeated the analysis replacing the *Halobacterium* sp. genome with that from *Archaeoglobus fulgidus*, another archaeon. Gratifyingly, many more QuartOPs supported the grouping of the thermophilic archaeon *Archaeoglobus* with the thermophilic bacteria *Aquifex* and *Thermotoga*. The different interdomain genome quartets that include a meso- or thermophilic archaeon are summarized in Table [3](#T3){ref-type="table"}.

An analysis of the putative functional assignments of the QuartOPs that grouped *Halobacterium* sp. with the mesophilic bacteria is given in Table [4](#T4){ref-type="table"}. To assess which of these categories have an increased percentage of QuartOPs as compared to distribution of ORFs within the genome, we also calculated the distributions of ORFs among functional categories in the *Halobacterium* sp. and *A. fulgidus* genomes. Open reading frames within a genome are distributed almost evenly among the four meta-categories (see columns labeled \"H\" and \"A\"). However, the QuartOPs that group the halobacterial orthologs with those from mesophilic Bacteria are distributed differentially among the meta-categories. Most of the QuartOPs are in the \"Metabolism\" and \"Information Storage and Processing\" meta-categories. These are also the categories in which *Halobacterium* sp. shows many more OuartOPs in support of topology 3 than *A. fulgidus*.

::: {#T4 .table-wrap}
::: {.caption}
######

Distribution of the datasets that strongly support (with 99% posterior probability) one of the three topologies among different functional categories.
:::

                                                                       **\#11**        **   **        **\#13**        **   **        **\#61**        **   **        **\#62**        **   **   **H**   **   **   **A**
  ---------------------------------------------------------------- --- ---------- ---- --------- ---- ---------- ---- --------- ---- ---------- ---- --------- ---- ---------- ---- --------- ------- --------- -------
  Information storage and processing                               5   7          20   **   **   7    7          10   **   **   11   4          17   **   **   12   4          0    **   **   24      **   **   17
  **J**     Translation, ribosomal structure and biogenesis        4   6          14   **   **   5    6          9    **   **   9    3          11   **   **   10   3          0    **   **   31      **   **   44
  **K**     Transcription                                          1   1          0    **   **   1    1          0    **   **   1    1          0    **   **   1    1          0    **   **   30      **   **   30
  **L**     DNA replication, recombination and repair              0   0          6    **   **   1    0          1    **   **   1    0          6    **   **   1    0          0    **   **   39      **   **   26

  Cellular processes                                               1   5          4    **   **   5    7          1    **   **   5    1          6    **   **   6    3          1    **   **   21      **   **   16
  **D**     Cell division and chromosome partitioning              0   0          1    **   **   2    0          1    **   **   3    0          0    **   **   3    0          0    **   **   7       **   **   5
  **O**     Posttranslational mod., protein turnover, chaperones   1   0          1    **   **   1    1          0    **   **   1    0          3    **   **   2    0          0    **   **   22      **   **   18
  **M**     Cell envelope biogenesis, outer membrane               0   3          0    **   **   1    1          0    **   **   0    1          0    **   **   1    0          0    **   **   13      **   **   14
  **N**     Cell motility and secretion                            0   0          1    **   **   1    1          0    **   **   1    0          0    **   **   0    0          0    **   **   15      **   **   10
  **P**     Inorganic ion transport and metabolism                 0   2          1    **   **   0    4          0    **   **   0    0          3    **   **   0    3          1    **   **   30      **   **   31
  **T**     Signal transduction mechanisms                         0   0          0    **   **   0    0          0    **   **   0    0          0    **   **   0    0          0    **   **   14      **   **   22

  Metabolism                                                       6   16         29   **   **   18   30         12   **   **   9    6          30   **   **   22   28         14   **   **   30      **   **   37
  **C**     Energy production and conversion                       2   3          1    **   **   1    4          1    **   **   0    0          0    **   **   3    0          0    **   **   23      **   **   31
  **G**     Carbohydrate transport and metabolism                  1   2          1    **   **   2    2          0    **   **   3    0          2    **   **   2    1          1    **   **   12      **   **   8
  **E**     Amino acid transport and metabolism                    1   5          16   **   **   12   10         7    **   **   2    2          14   **   **   10   12         6    **   **   27      **   **   24
  **F**     Nucleotide transport and metabolism                    2   4          10   **   **   1    9          4    **   **   2    2          11   **   **   4    10         7    **   **   10      **   **   7
  **H**     Coenzyme metabolism                                    0   2          1    **   **   2    5          0    **   **   2    2          3    **   **   3    5          0    **   **   19      **   **   16
  **I**     Lipid metabolism                                       0   0          0    **   **   0    0          0    **   **   0    0          0    **   **   0    0          0    **   **   9       **   **   14

  Poorly characterized                                             1   1          3    **   **   1    1          0    **   **   2    1          2    **   **   1    1          1    **   **   24      **   **   30
  **R**     General function prediction only                       1   1          3    **   **   1    1          0    **   **   2    1          2    **   **   1    1          1    **   **   64      **   **   58
  **S**     Function unknown                                       0   0          0    **   **   0    0          0    **   **   0    0          0    **   **   0    0          0    **   **   36      **   **   42

The distribution corresponds to the genome quartets listed in Table 3. Functional categories are as designated in Fig. 5. Columns 1, 2 and 3 correspond to the three possible unrooted topologies for each genome quartet (see Table 3). Column entries indicate the number of QuartOPs in each functional category. The last two columns represent the distribution of ORFs in *Halobacterium* sp. (H) and *Archaeoglobus fulgidus* (A) genomes among different functional categories. For these two columns, numbers in the rows corresponding to the meta-categories give the percentage of proteins in each meta category relative to the total number of classifiable proteins and numbers in the rows for each functional category indicate the percent distribution of the proteins within the corresponding meta-category.
:::

The analyses described in this section reconfirm that genes have been transferred across domain boundaries \[[@B6]-[@B12]\]. Not surprisingly, these transfers appear to occur preferentially between organisms living in the same or similar environment. The genome of the mesophilic *Halobacterium* sp. contains many genes that group with the orthologs from mesophilic bacteria, whereas the majority of genes from the thermophilic archaeon *Archaeoglobus fulgidus* group with the orthologs from the extremely thermophilic bacteria. The majority of QuartOPs that group the halobacterial orthologs with the ortholog from the mesophilic bactria belong to two of four meta-categories: \"Information Storage and Processing\" and \"Metabolism\". QuartOPs in Information Storage and Processing meta-category that support the grouping of *Halobacterium* sp. with *Synechocystis/Bacillus* are listed in the Table [5](#T5){ref-type="table"}. A complete listing is available in the supplementary material. As expected, this list includes several tRNA synthetases, which were previously found to be frequently transferred \[[@B6]-[@B8]\], and enzymes involved in DNA repair (*cf*. \[[@B9]\]). More surprisingly, this list also includes translation initiation factors and several ribosomal proteins. The latter were assumed to be infrequently transferred, but recent analyses reported them to be horizontally transferred among bacterial lineages \[[@B10],[@B11]\]. The initiation factor IF-2 in *Halobacterium* sp. was previously shown to have strong similarity to the initiation factor IF-2 from Bacteria \[[@B49]\]. Most of the genes that group *Halobacterium* with the mesophilic bacteria encode functions that were postulated to be frequently exchanged \[[@B50]\]. While no meta-category appears exempt from HGT, some functions appear to be more often transferred than others (cf. Table [4](#T4){ref-type="table"}).

::: {#T5 .table-wrap}
::: {.caption}
######

List of genes putatively horizontally transferred between *Halobacterium* sp. (*H.* sp.) and the mesophilic Bacteria *Synechocystis* sp. and *Bacillus subtilis* (\"Information Storage and Processing\" meta-category only).
:::

  ***Protein Name***                                                             ***H. sp. GI number***
  ------------------------------------------------------------------------------ ----------------------------------------------------------
  tRNA synthetases for serine, valine, methionine, cysteine, arginine, proline   10581491, 10581937,10579953, 10580644,10584349, 10580016
  phenylalanyl-tRNA synthetase subunit alpha                                     10581896
  Glu-tRNA amidotransferase subunits A, B                                        10580435, 10579969
  tRNA-pseudouridine synthase                                                    10581191
  dimethyladenosine transferase                                                  10580702
  DNA gyrase subunits A, B                                                       \[10580453, 10580452\]
  DNA helicase                                                                   10580995
  excision nuclease ABC chains A, B, C (involved in DNA repair)                  10582016, 10581796, 10581790
  endonuclease V (involved in DNA repair)                                        10579981
  DNA mismatch repair protein                                                    10579807
  Putative translation factor SUA5                                               10581723
  Translation initiation factor eIF-2B subunit alpha                             10581299
  Initiation factor IF2                                                          10581429
  ribosomal proteins L1, L11, L3, S4                                             \[10580652, 10580653\], 10581159, 10580672

GI numbers in brackets correspond to genes in operons. This list is derived analyses of genome quartets \#11 and \#61. A complete list of all GI numbers for each QuartOP as well as the four definition lines is available in the supplementary material.
:::

Conclusions
===========

Maximum likelihood mapping is a useful tool for analyzing and depicting the mosaic nature of genomes. ML-mapping is much less conservative than other approaches of estimating Bayesian posterior probabilities. If ML-mapping is used as the only probability mapping tool, the overestimation of supporting probabilities has to be taken into consideration. A posterior probability of .99 calculated with ML-mapping often corresponds to a posterior probability of only .90.

Many relationships among prokaryotes cannot be depicted by a tree-like pattern reflecting a core of rarely transferred genes. Rather prokaryotic genomes are mosaics where different parts have different evolutionary histories. However, HGT between divergent organisms has not erased all patterns of interphylum relationship. For example, the majority of QuartOPs group the cyanobacteria with the low GC Gram positives as sister phyla.

Due to horizontal gene transfer even organisms from different domains living in the same or similar environments share more genes with each other than organisms with a similar degree of divergence that live in different environments. These interdomain horizontal transfers mainly concern proteins involved in nucleotide, carbohydrate and amino acid transport and metabolism; however, proteins that are part of the translation machinery or are involved in DNA repair appear to be transferred across domain boundaries as well.

Materials and Methods
=====================

Genome Data
-----------

Completed genomes were retrieved from the NCBI\'s FTP site <> in the form of amino acid sequences encoded by open reading frames (ORFs) as identified in the annotated genomes. Mitochondrial genomes were obtained from the Organelle Genomes Page at NCBI <>. The genomes were formatted using the *formatdb* program from the stand-alone BLAST package, initially of version 2.0.11 and later of versions 2.1.2 and 2.2.1 as they were released \[[@B51]\]. All analyses were performed locally.

Data Flow in Quartet Analyses
-----------------------------

For each set of four genomes, BLAST \[[@B51],[@B52]\] searches of every ORF in one genome against the other three genomes were performed using the *blastp* program. The E-value cutoff for the BLAST searches was set to 10^-4^ (in one test case an E-value cutoff of 10^-20^ was used). For every BLAST search the GI number of the top hit (if it was below the cutoff) was saved along with the GI number of the query sequence forming a GI pair. This resulted in twelve lists of GI pairs for each of the twelve possible pairwise genome comparisons. This information was further used to identify quartets of orthologous proteins (QuartOPs). Following Tatusov et al. \[[@B27]\] we defined QuartOPs as those sets of genes that mutually pick each other as the top scoring hit in the BLAST comparisons. The detection of the QuartOPs was performed using the MySQL database software <>. The lists of GI pairs were entered into twelve tables of a database. The tables were joined into one table under conditions that satisfy the definition of the QuartOPs (see above). This resulted in a table with four columns of GI numbers for QuartOPs. The amino acid sequences for each QuartOP were retrieved from GenBank at NCBI and were aligned using ClustalW 1.8 \[[@B53]\]. QuartOPs were analyzed using the ML-mapping approach according to Strimmer and von Haeseler, Bayesian probabilities mapping and bootstrap support values mapping techniques (see details below).

Posterior probabilities according to Strimmer and von Haeseler
--------------------------------------------------------------

For all three possible unrooted tree topologies maximum-likelihood values and posterior probabilities were calculated using in-house JAVA programs that were written utilizing classes from the Phylogenetic Analysis Library version 1.0 \[[@B54]\] and parts of Vanilla package version 1.0 \[[@B54]\]. If not indicated otherwise, likelihood values were estimated using the automatically selected suitable substitution model (chosen from BLOSUM62, CPREV, Dayhoff, JTT, MTREV24, VT and WAG) with no ASRV. The maximum-likelihood mapping approach was further used to visualize support for each tree topology \[[@B24]\], i.e. the posterior probability vector for each QuartOP was plotted into an equilateral triangle. Maximum-likelihood maps were generated using GNUPlot v. 3.7 <>.

Posterior Probabilities calculated with MrBayes program
-------------------------------------------------------

Posterior probabilities were also calculated with MrBayes version 2.01 \[[@B31]\]. Each QuartOP was analyzed with two simultaneous Markov chains for 25,000 cycles under the JTT substitution model \[[@B55]\] without ASRV. One chain was heated with the temperature set equal to the default value of 0.2. Samples were taken at each cycle. The \"burn in\" option was set to 5,000 cycles. The remaining 20,000 cycles were used to calculate posterior probabilities for each of the three tree topologies. The posterior probabilities were plotted to equilateral triangle as described above. For discussion of the choice of the parameters see below.

Bootstrap Support Values
------------------------

As an alternative to posterior probability vectors, bootstrap support values were calculated and plotted. Each QuartOP was bootstrapped 100 times and the proportion of bootstrapped datasets supporting each tree topology was recorded as a bootstrap probability vector. The bootstrap probability vectors were plotted into an equilateral triangle with the zones changed to \"total\", \"70%\" and \"90%\" (see Fig. [2](#F2){ref-type="fig"}).

Empirical Search for Optimal MrBayes Parameters
-----------------------------------------------

To find parameters that will return consistent posterior probabilities within reasonable computation time, one QuartOP from mitochondrial genome quartet \#m1 was analyzed multiple times with different parameters. According to Strimmer and von Haeseler\'s approach \[[@B24]\] this QuartOP has posterior probabilities of 0.76, 0.10 and 0.13. In all runs samples were taken at each cycle; two chains and the JTT substitution model \[[@B55]\] without ASRV were used.

First, we analyzed the dataset with 250,000 cycles. We tried different \"burn in\" options in the range of 1,000--20,000. The posterior probability values changed by less than 0.3% from case to case. We selected a \"burn in\" of 5,000 cycles in further analyses. Second, we tried different numbers of cycles to calculate posterior probabilities. The probabilities were calculated using 10,000--240,000 cycles with increment of 10,000 cycles. Again, the posterior probability values did not change significantly from case to case. Third, we raised the \"temperature\" parameter *T* to 2.0 for the second, heated chain. This did not result in changes of the estimated posterior probabilities. Fourth, we used 25,000 cycles and repeated the analysis 10 times, calculating average and standard deviation of all runs. For all three probabilities the standard deviation was less than 0.01. Based on these analyses we selected 25,000 cycles with a \"burn in\" of 5,000 as a compromise between precision of probability estimation and computational time spent. As a final test, we performed the analysis of the quartet \#8 twice with selected parameters. This did not result in significantly different maps. Graphs and tables depicting the results of these analyses are given in the supplementary material.

Mapping taking ASRV into account
--------------------------------

For the genome quartet \#8 we calculated posterior probabilities under the model which takes ASRV into account with Strimmer and von Haeseler\'s \[[@B24]\] approach and with the MrBayes program version 2.01 \[[@B31]\]. TREE-PUZZLE 5.0 \[[@B56]\] was used to calculate posterior probabilities according to Strimmer and von Haeseler \[[@B24]\]. A discrete approximation of the gamma distribution \[[@B57]\] was used to describe ASRV. Eight rate categories were used in TREE-PUZZLE \[[@B56]\], and four rate categories were used in MrBayes \[[@B31]\]. The maps are available in the supplementary material. Due to the amount of time required for calculations, the analyses were not performed for other genome quartets.

Functional assignments using the COG database
---------------------------------------------

Datasets for QuartOPs with strong preference for a particular tree topology (i.e. with posterior probability above 99% for that particular topology, or in other words the QuartOPs located in the very corners of the equilateral triangle) were extracted. For each of those QuartOPs the COG functional category \[[@B27]\] was identified. In order to detect the functional categories, the COG database was downloaded from NCBI\'s FTP site (initially the year 2000 release and later the year 2001 release). The COG database was formatted using the *formatdb* program of BLAST package. Every QuartOP was compared to the COG database using the *blastp* program. The category of the each sequence in the QuartOP was assigned according to the category of the top hit of each BLAST search. The numbers of QuartOPs in each functional category were calculated for each of the three tree topologies.

Distribution of ORFs among COG categories for complete genomes of Halobacterium sp. and Archaeoglobus fulgidus
--------------------------------------------------------------------------------------------------------------

Every predicted ORF in a genome was compared to the COG database (release of year 2001) using the *blastp* program with E-value cutoff 10^-4^. The category of each ORF was set to be equal to the category of the top hit of the corresponding BLAST search. Category Q was dropped from the results, because the corresponding genome quartets were analyzed with the previous release of the COG database (release of year 2000) that did not contain the Q category.

Data Analysis Automation
------------------------

The repetitive tasks of analyses were automated using the SEALS package version 0.824 \[[@B25]\]. The tasks that were not available through SEALS package were programmed in PERL v. 5.005. The PERL scripts and JAVA programs are available upon request.

Mitochondrial Genome Quartets Analyses
--------------------------------------

Seven mitochondrial genome quartets were used as controls and were analyzed with the three approaches for genome quartet analysis described above. For calculation of posterior probabilities with MrBayes at least 25,000 cycles were used.

List of Abbreviations
=====================

**HGT** horizontal gene transfer

**COG** cluster of orthologous groups

**ML** maximum likelihood

**rRNA** ribosomal ribonucleic acid

**sp.** species

**BLAST** Basic Local Alignment Search Tool

**QuartOP** quartet of orthologous proteins

**SEALS** System for Easy Analysis of Lots of Sequences

**NCBI** National Center for Biotechnology Information

**ASRV** Among Site Rate Variation

Supplementary Material
======================

Supplementary material is located at the QuartOP web page <>. This web page includes the summary of all genome quartets analyzed (with maps), the results of control analyses, and a form to request the scripts described in this article. ML maps are available in postscript and PDF formats. An offline version of the QuartOP web page is available as a compressed archive named supp_material.zip and as a self-extracting archive supp_material.exe for Microsoft Windows users. The archive can be expanded using WinZip <> for Windows, StuffIt for Macintosh <>, or unzip utility for Unix. The uncompress utilities have to be run with the option to preserve the subdirectory structure inside the archive. To access the information in the archive, the file index.html has to be opened using an Internet browser. This index.html file is located in the root directory named \"offline_quartops\". All the files in the archive are hyperlinked and accessible through the index.html file.

Supplementary Material
======================

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###### Additional file 1
:::

::: {.caption}
######

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:::

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###### Additional file 2
:::

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Acknowledgements
================

We thank Paul Lewis and Lorraine Olendzenski for many stimulating discussions and for critically reading the manuscript. The work was supported through the NASA Exobiology Program and through the NASA Astrobiology Institute at Arizona State University.


Background
==========

Blood leukocytes are found in two sub-populations constituting the circulating and the marginating pools. The elements of these two sub-populations are interchangeable, i.e., marginated leukocytes return to bloodstream and vice-versa \[[@B1]-[@B3]\]. Therefore, a dynamical equilibrium situation occurs, and blood cell counts should reflect the rate constants of margination and return to circulation of those cells.

Several studies addressed the relationships between the sub-populations of white cells within the blood pool. Among other features, the ratio between the sub-populations under normal conditions \[[@B4]\], under altered states \[[@B4],[@B5]\] and the stability of the equilibrium situation (i.e., to resume a previous value after perturbations, \[e.g., \[[@B6]-[@B8]\]) were approached. It is presently accepted that the blood sub-pools play an important role as a white cell reservoir when increased demands supervene. Increased demands arise in both pathological and non-pathological situations, as during exercise \[[@B9],[@B10]\], burns \[[@B11],[@B12]\], infectious diseases \[[@B13]\], inflammatory processes \[[@B5],[@B14]\], etc.. It is also currently recognized that some hormones acutely alter the dynamics of the blood sub-pools of white cells \[[@B6]-[@B8]\]. In this sense, epinephrine is known to possess a demarginating effect that lasts for less than 1 hour, and such an effect is thought to be the result of changes in the rate constants of margination and return \[[@B2],[@B6]\]. Therefore, the interplay between marginating and circulating leukocytes is a relevant issue that should be taken into account in the interpretation of many results.

On the one hand, it is tacitly assumed that the rate constants of margination and return have higher values than other rate constants related to the dynamics of white blood cells (see below). On the other hand, there is no study addressing such an issue in a formal way in order to provide good estimation of these values. The aim of the present study is to provide the theoretical background to perform estimations of these rate constants, which may prove relevant for empirical studies on healthy normal situations as well as under altered states of the organisms.

Constructing the extensive model
--------------------------------

In this sub-section, we present a model based mainly on data from monocytes and neutrophils. The model describes the dynamics of three compartments of these cells and also contains the dynamics of specific growth factors. In the next sub-section we show a reduced model from this one, which will then be employed to obtain an improved estimation for the rate constants values. The extensive model is presented to assure consistence of the analysis.

Bone marrow total rate production apparently has three components: a fixed production rate (P~1~), a production rate dependent on self-regulatory factors (P~2~) and a production rate dependent on inflammatory/infectious factors (P(I)) \[[@B3]\]. Therefore, the production rate is not a time independent function, and we denote it by the sum of the three terms above. Once the newly produced cell gains the bloodstream, it may marginate (rate constant M). A marginated cell may then return to circulation (rate constant R). Cells in the blood may leave to surrounding solid tissues, and they do that according to a time-independent rate constant S~B~\[[@B1]\]. Exit to tissues is also a function of rate constants dependent on inflammatory/infectious factors \[[@B3]\]. Thus, let us denote such a total exit rate constant from the circulation by S~B~ + S(t,I). Tissue leukocytes may proliferate locally (rate constant D), produce self-regulatory factors and, eventually, die (rate constant Z) \[[@B15]\]. This set of empirical data suggests the scheme presented in Figure [1](#F1){ref-type="fig"} and the following equation system for describing the dynamics of these cells:

![Schematic representation of the dynamics of white cells in the organism. The self-reproduction in tissues is represented by \"D\", without arrows. Crosses denote death or clearance. See text for details.](1472-6793-2-3-1){#F1}

![](1472-6793-2-3-i1.gif)

Where α are self-regulatory factors (e.g., CSF-1) and φ~i~ refers to cells that take part in the circulating (c), marginal (m) and solid-tissues (~T~) pools. G is a constant related to the production of self-regulatory factors, and K is the rate constant of their clearance. Notice that the exit to tissues comes both from the circulating pool, φ~c~, and from the marginated pool, φ~m~. Such an exit has the same rate constant, S~B~, independently of the sub-pool. This partition of the total exit constant rate to tissues in two components is the basic reasoning that leads to the inference about the existence of two distinct sub-pools within the blood white cells \[[@B1]\]. Therefore, a cell that touches, attaches and passes through the vessel wall cannot return to the circulating pool, even though during a certain time interval this cell was marginated (semantically but not functionally). In this sense, these cells exit to the surrounding tissues coming directly from the circulating pool, and their dynamics is contemplated by the product φ~c~S~B~ of the differential equation. Those marginated cells that can potentially return to circulation have their dynamics accounted for by the rate constant R. On the other hand, part of these marginated cells migrate to surrounding tissues and the product φ~m~S~B~ contemplates this rate. This dynamics is also consistent to the empirical evidence that margination and diapedesis are distinct features arising from different signals and receptors \[[@B16]\].

Reducing to the compact model
-----------------------------

The injection of some substances, epinephrine in particular, changes the rate constants of margination and return to circulation (M and R) for very short periods of time (e.g., see \[[@B2],[@B6]-[@B8]\]). Under such a condition, the variable values change as the parameter values change. However, since the substance is rapidly metabolized, the parameters return to their previous value and then the variables would also return to some value maintained previously to the perturbation (in the parameters). Thereby, these substances act on the system described by equations 1a-d analogously to an impulsive function \[[@B17]\]. The returning phase is the relaxing period of the system, and it behaves as the perturbation had occurred in the variables instead of in the parameters. During the short time interval of the relaxing period, the production rate can be considered constant, P. Therefore, equations 1a and 1b maintain no loop-connection to the other two equations (1c and 1d), and a compact system can be written as:

![](1472-6793-2-3-i2.gif)

Results
=======

Analysis of the steady-state condition
--------------------------------------

The first step in the analysis was to verify whether the extensive model allows the existence of a stable equilibrium point. An equilibrium point means a set of values of the variables (in this case, φ~c~, φ~m~, φ~T~ and α) that does not vary with time (if the system is left externally undisturbed). Stability is related to the behavior of the system in face of perturbations in the variables that displace them to some vicinity of an equilibrium point. The equilibrium point is said asymptotically stable if the system returns to the equilibrium point attained previously. Otherwise, the equilibrium point may be neutrally stable, if the system does not return to the equilibrium point but remains somewhere around it, or unstable, if the system leaves that vicinity away (e.g., \[[@B18]\]).

We verified whether the extensive model posses one asymptotically stable equilibrium point as a first step because the biological system modeled seems to behave like this. Once disturbed, it returns to a previous value. The analysis revealed the existence of one equilibrium point, and such a point is asymptotically stable (see Methods). Let \* denote the value of the variable in the equilibrium point. Without loss of generality by considering the inflammatory/infectious factors absent and that φ~T~ and α are fixed (e.g., for a short time interval or changes in R or M, see Methods), the equilibrium values of the variables that we are interested in the present study are:

![](1472-6793-2-3-i3.gif)

As stated before, blood cell counts should reflect the rate constants of margination and return. Let f^\*^ denote the ratio φ~m~^\*^/φ~c~^\*^ and thus:

![](1472-6793-2-3-i4.gif)

Notice that one is not able to know, directly from the ratio φ~m~^\*^ /φ~c~^\*^, the values of R and M themselves.

The demarginated state and its return to the equilibrium condition
------------------------------------------------------------------

In a very short time interval succeeding a bolus injection of some substances (e.g., epinephrine) the parameters R and M change and return to their previous values. Therefore, the circulating and marginal pools attain different values from φ~c~^\*^ and φ~m~^\*^, respectively. Given the stability of the equilibrium point (see above) these variables return to φ~c~^\*^ and φ~m~^\*^. This returning is governed by a second-order differential equation (see Methods). Let ![](1472-6793-2-3-i5.gif) be the value attained above (or below) the equilibrium value φ~c~^\*^ by the circulating cells after the perturbation (see Methods). In short, ![](1472-6793-2-3-i6.gif) (see equation 9). Therefore, taking the equilibrium value of the circulating pool as a reference value, ![](1472-6793-2-3-i5.gif) is the difference between this value and the value found at a time t after the beginning of the returning to the steady-state equilibrium condition. This leads to the following equation governing the fast-phase of resuming the steady-state condition:

![](1472-6793-2-3-i7.gif)

where φ~c0~ is the peak (or nadir) value of the circulating pool attained after the perturbation, i.e., the initial value of this variable, and λ is the fast component of the process (λ~2~, see Methods equation 11b). Equation 5 can be linearized to simplify the estimation procedure. Finally, the λ of equation 5 is (see equation 11b):

λ = -(R + S~B~)(f^\*^ + 1)

This value of λ allows a good estimate of R during the fast phase of the returning to the equilibrium point (steady-state of the sub-pools). In the next section, we will work some examples of the application of the present model and its potential relevance.

Worked examples
---------------

The present study offers the means to compute the individual values of R and M. As stated before, the ratio f^\*^ is obtained by measuring the marginated and the circulating pools in steady-state conditions. This ratio is equal to M/(R+S~B~) (see equation 4). Therefore, its is important to note that the individual computation of R (and M) assumes that a series of other independent measurements were done: (a) φ~c~^\*^ (the steady-state value of the circulating pool); (b) φ~m~^\*^ (the steady-state value of the marginated pool); (c) S~B~ (the exit rate constant from the total blood pool to tissues). The knowledge of a general production rate value may also improve the picture, even though its is not a primary need.

The first example intends to compute the R value for mice monocytes under normal conditions. However, the paucity of adequate data prevents a true computation. Therefore, we will compute an approximate value. The other two examples are completely imaginary. They show the usefulness of the model to address changes in the rate constants that would otherwise pass unnoticed.

### I. The normal condition

The computations are based on data from van Furth and Sluiter \[[@B1]\]. The report give φ~c~^\*^ = 350 monocytes/mm^3^, φ~m~^\*^ = 500 monocytes/mm^3^ and S~B~ = 9.6 × 10^-4^ min^-1^. The f^\*^ value is 500/350 = 1.43. An intravenous bolus injection of epinephrine caused a two-fold increase in φ~c~ after 10 minutes of the injection. Therefore, ![](1472-6793-2-3-i8.gif) (or, φ~c~(0) = 700). After 60 minutes from the injection time (50 minutes after the monocytes peak), the variables resume their normal values. Notice the extremely limited temporal data. Let us suppose that we had a few more points, as illustrated in Figure [2A](#F2){ref-type="fig"}. From equation 5, the data could be linearized as:

![Simulated perturbation in the blood sub-pools of a white cell type (e.g., monocytes) under three different conditions: normal (blue), tumor carrying animal (red), long-term aerobic exercising animal (green); see text for details. (A) \"Raw experimental data\", the \"discrete\" time course of φ~c~ after a bolus injection of epinephrine under the normal condition. (B) Log-transformed (linearized) data, under the three different conditions. Notice the linear relationship obtained. (C) \"Raw \"continuous\" data\" of the φ~c~ time course. Data generated in MatLab 5.3 (The MathWorks, Natick, MA) by built-in numerical integration routines of the set of differential equations of the model.](1472-6793-2-3-2){#F2}

![](1472-6793-2-3-i9.gif)

Where T is the discrete sampling time. Figure [2B](#F2){ref-type="fig"} shows the result of the linearization procedure. A linear regression would then result in λ = -0.113 min^-1^. Finally, from equation 6, we know that this value must be decomposed as

-0.113 = -(R + S~B~)(1 + f^\*^)

Thus, R = 0.0456 min^-1^, a value 47-fold greater than S~B~. In the data simulation, R = 0.045 min^-1^, therefore the procedure seems quite adequate. This example highlights the use of the present model as well as the difference between the eigenvalue (λ, obtained directly from the data) and the constant R. Also, as far as we know, this is the first attempt to estimate the value of R (and M) under a complete and formal model.

### II. Tumoral effects

Consider, as another example, that mice carrying a certain tumor type have the exit constant rate equal to the normal value (S~B~ = 9.6 × 10^-4^ min^-1^). The f^\*^ value is 1.3 and these mice have φ~c~^\*^ = 490 monocytes/mm^3^, φ~m~^\*^ = 637 monocytes/mm^3^. Obviously, production rate have increased. Notice the potential confusing situation arising from these data. Does the tumor growth alter the dynamics between the blood pools? The epinephrine experiment (or some analogue) was conducted, and ![](1472-6793-2-3-i10.gif) (or, φ~c~(0) = 600). Figure [2C](#F2){ref-type="fig"} shows both the \"normal\" and the \"tumoral\" data. Linear regression of the log-transformed data computes λ = -0.065 min^-1^ (see Figure [2B](#F2){ref-type="fig"}) and thus R = 0.0273 min^-1^ since f^\*^ = 1.3. In data simulation, R = 0.027 min^-1^, again the results are in close agreement with the real parameter value. Therefore, in this constructed situation, one would be able to conclude that the rate constants of the transit between the blood pool were altered by the pathological condition. This could be an important result in tumor immunology, for example, allowing a better understanding of the pathology or devising new treatment strategies.

### III. Long Distance Running Effects

Our final imaginary example is related to the effects of long sustained aerobic exercise. Consider that mice trained in this type of exercise are known to have increased production rate of monocytes. During the steady-state condition of the exercise, φ~c~^\*^ = 350 monocytes/mm^3^, φ~m~^\*^ = 700 monocytes/mm^3^, thus f^\*^ = 2. S~B~ increased to 8.5 × 10^-3^ min^-1^. An epinephrine experiment is performed during the exercise section and samples are taken during the decaying phase after the injection (the exercise section proceeds during the sampling period). The \"continuos\" count profile is illustrated in Figure [2C](#F2){ref-type="fig"}. Linear regression of the log-transformed data (see Figure [2B](#F2){ref-type="fig"}) computes λ = -0.041 min^-1^ and thus R = 0.0052 min^-1^ (in data simulation, R = 0.005 min^-1^). This is a 9-fold decrease in the R value in relationship to the normal condition. On the other hand, the rate constant M decreased less than 2.5 times in relationship to its value in normal conditions (see above). This result might be important in the understanding of many immune suppression/enhancement phenomena related to certain exercise protocols. Notice that the f^\*^ value by itself do not tell anything about the change in the individual value of each rate constant.

Discussion
==========

The compartmentalization of blood leukocytes in two sub-pools is an important feature of these cells. For example, the marginal pool may acutely function as an extra source of cells at increased demand conditions. In this sense, knowledge about the rate constants governing the transit between the sub-pools may prove relevant in studies that approach both physiological and pathological situations. In the present manuscript we provide the theoretical background to support empirical studies related to the calculation of these rate constants.

From the general perspective of an extensive model, we first showed that this model corresponds to what is experimentally found in terms of existence and stability of an equilibrium point in the variables (the sub-pools). After that, we reduced the model to a compact one, concerning only the two blood sub-pools of interest here. Within the context of the compact model, it was shown how an impulsive function (e.g., an intravenous epinephrine bolus injection) perturbing the parameters (the rate constants of margination and return to circulation) can be translated to a perturbation in the variables. Then, the returning of the variables to their equilibrium values allows the estimation of the rate constants. This was done by the use of a second-order differential equation (see Methods). From that equation, the fast-decay component was identified and the corresponding eigenvalue contains the rate constants to be calculated.

The present study demonstrates, for the first time, how to adequately and completely estimate the rate constants of margination and return to circulation of white blood cells. Notice that first-order differential equations (as equation 5) were employed before in gross estimates of these rate constants (e.g., \[[@B19]\]). However, the constant then computed comprised the whole eigenvalue, without discriminating its parts. In other words, its was not recognized that the computed value should be partitioned into three components: R, M and S~B~.

The model also allows to establish the relationship between R and M. This relationship includes the S~B~ constant and the ratio between the marginal and the circulating sub-pools, f^\*^ (see equation 4). The need to take into account the f^\*^ value in the estimation of the constants is another important aspect for the first time presented. Therefore, the model leads to a more precise evaluation on how imposed disturbances interfere with the dynamics of the blood sub-compartments and thus increase our understanding of the physiology/pathophysiology of many conditions. Two imaginary situations of this kind were constructed and analyzed as examples of the use of the present model (see \"Worked Examples\" II and III).

Conclusions
===========

This study provides a complete model to approach the estimation of rate constants of margination and return to circulation of with blood cells. It shows, for the first time, how the value empirically found should be partitioned in order to adequately obtain the desired rate constants. In this sense, we were able to recognize that both an exit rate constant to tissues and the ratio between marginated and circulating cells should be taken into account in the computation procedure.

Methods
=======

1. Equilibrium point and its stability
--------------------------------------

The equilibrium point of the system described by equations 1a-d is found by setting all the derivatives to zero and computing the corresponding values of the variables that lead to such a condition. Without loss of generality, the inflammatory/infectious factors can be taken as zero. Therefore, the equilibrium point (denoted by \*) corresponds to the following values:

![](1472-6793-2-3-i11.gif)

Conditions of existence: (1) Z \> D; and (2) K(Z-D) \> P~2~G. The first condition reflects that tissue leukocytes must die at a rate higher than their own local replication otherwise their population would increase forever. The second condition is similar to the first in the sense that it reflects that the self-stimulatory loop must be lower than the loss loop of the system.

The stability of the equilibrium point is verified by the construction of the determinant of the matrix of the coefficients of the variables of the equations of the system \[[@B17],[@B18],[@B20]\]. In order to be asymptotically stable, all the eigenvalues (roots) of the characteristic polynomial of the matrix must have real parts lower than zero. Considering that the system has four equations, the characteristic polynomial is of the 4^th^-order with the following general formula:

λ^4^ + a~1~λ^3^ + a~2~λ^2^ + a~3~λ + a~4~ = 0

where λ is an eigenvalue of the equation (note that the equation has 4 roots). The coefficients (a~i~) come from the parameters of the system. To have all the eigenvalues with real part lower than zero, the Routh-Hurwitz conditions must verify. In short:

1A. All a~i~ must be greater than zero. This condition is true for the system.

1B. a~1~a~2~ \> a~3~. This condition is true for the system.

1C. a~3~(a~1~a~2~ - a~3~) \> a~4~a~1~^2^. This condition is not easily verified analytically. Therefore, we performed a numerical analysis. Random values were assigned to the parameters (conditions of existence of the equilibrium point verified, see above) and the condition 1C checked (pseudo-random number generator normally distributed built-in function of MatLab 5.3, The MathWorks, Natick MA). This procedure was taken 150,000 times by a routine specifically written for it and the condition always verified true. Therefore, the equilibrium point seems asymptotically stable.

The main point in this sub-section is to realize that both φ~T~^\*^ and α^\*^ are not affected by R and/or M. This is what assures the results concerning the compact model below.

2. The compact model and its second-order differential equation form
--------------------------------------------------------------------

Considering that production rate and inflammatory/infectious factors would not change in a short time interval, the extensive model allows the blood pool to be treated as having no loop connection to the rest of the system (see equations 2a and 2b). The determinant of the Jacobian matrix of the compact model (i.e., the sub-system represented by equations 2a and 2b) is to set to zero:

![](1472-6793-2-3-i12.gif)

Consider now a \"push\" in φ~c~^\*^ (and φ~m~^\*^) in such a way that:

![](1472-6793-2-3-i13.gif)

![](1472-6793-2-3-i5.gif) is, therefore, the time course of the perturbation around the equilibrium point of φ~c~. Equation 8 is homogeneous for ![](1472-6793-2-3-i5.gif):

![](1472-6793-2-3-i14.gif)

Taking into account that M = (R + S~B~)f^\*^ (see equation 4), the eigenvalues of equation 10 are:

![](1472-6793-2-3-i15.gif)

Notice that: (a) both eigenvalues are pure real (this means that the equilibrium point is attained without oscillations in the variables); and (b) both eigenvalues are negative (the equilibrium point is asymptotically stable). Given that λ~2~ \< λ~1~, λ~2~ is the fast component of the process of resuming the equilibrium value φ~c~^\*^ (see sub-section \"the demarginated state and its return to the equilibrium condition\"). The value of λ~2~ is -(R + S~B~)(f^\*^ + 1). This value is presented in equation 6.

List of abbreviations
=====================

φ~c~: the number of cells (or concentration) in the circulating pool;

φ~m~: the number of cells (or concentration) in the marginal pool;

φ~T~: the number of cells (or concentration) in solid tissues;

α: quantity (or concentration) of self-regulatory factors;

R: rate constant of return to circulation;

M: rate constant of margination;

S~B~: rate constant of exit to tissues from the blood pool;

P: bone marrow production rate of the cells;

f^\*^: the ratio between the marginal and the circulating pools at steady-state condition.

Acknowledgments
===============

This study was supported by a FAPESP (State of São Paulo Science Foundation) undergraduate fellowship to K.I. (process number 00/01112-7) and by a FAPESP research grant to J.G.C.B. (process number 00/02287-5).

This manuscript was significantly improved by the comments of two anonymous referees.