Article CORE . ; ), 2006 . brought to you by by you to brought ), it can be ons, please Goethals et al RERO DOC Digital Library Digital DOC RERO by provided Young et al , though “small” plasmids ). ), bacteriocin production ), and antibiotic resistance Tazzyman and Bonhoeffer Anderson 1968 ; es and plasmids, and the 2009 Dowling and Broughton 1986 1975 has been suggested that there . . ), gall formation ( ), from hereon referred to as “es- y located on plasmids. We model Thomas 2000 e transfer, antibiotic resistance. nvestigate whether selection favors a Eberhard 1990 Evans et al ). Additionally, there are examples of plasmids Nogueira et al Eberhard 1990 € urich, Switzerland ; Jacob and Hobbs 1974 ), plant root nodulation ( ), all seem to be functions coded for by genes commonly There are several examples of such differences in genetic One particular element of plasmid genetics that has been nce Access publication December 23, 2014 3079 Gonzalez and Kunka 1987 Koch 1981 Eberhard 1990 ral, chromosomal locations for essential genes are indeed , it has been suggested that essential genes, such as those often lack this feature).exploit Because the there act are of phages conjugation that ( can 2001 composition. Naturally, there are athat suite plasmids of will phenotypic benefit traits fromto possessing, cause such conjugation as ( the ability virulence factors ( costly for a plasmid’s bacterialvious host, benefit without (as gaining the any donorchromosomes ob- receives are nothing). probably Consequently, lessgenes likely coding than for plasmids conjugation. to However,notypic bear there traits are other that phe- biased also genes. seem For to reasons(though that be have see coded not for been( fully by explained plasmid- genes ( observed is the factcode that for plasmids structural seldom( carry proteins genes or that basic metabolic functions bearing genescell that they are inhabit essential (either for all the the growth time; of the 2014 found on plasmids ( sential genes,” However,tween because of bacterial ratespopulation of sizes chromosom gene of bacterial flow populations,over be- evolutionary it time, seems most likely or that at all some bacterial time located genes on have both been ( plasmids and on chromosomes e evolution of prokaryotes. It wo degradation rates are equal, or if plasmid genes degrade mids can be a location for functioning essential genes is if etabolic functions, are rarel s and those in the “core” genome and that these differences € urich, CH 8092, Z ty for Molecular Biology and Evolution. All rights reserved. For permissi ). ,1 Jain ; 2010 . ntial genes will be found only on chromosomes. ile genetic elements, horizontal gen Goldenfeld ; Hacker and Hacker and Sørensen et al. Tazzyman and ; ; ng bacteria within a single population to i Rankin et al ; ). At the very least, the Ochman et al. 2000 be horizontally transferred ; e average genetic composi- Young et al. 2006 Integrative Biology (IBZ), ETH Z ; ). There are many mechanisms ssory” genome, which contains ories and interests of plasmids and Sebastian Bonhoeffer* 2010 ) and specifically in the case of plas- E-mail: [email protected]. . 1 Werren 2011 n, and transformation ), allowing for transfer of genes across Eberhard 1990 Koonin and Wolf 2008 ; Manolo Gouy ). A distinction has been drawn between 2009 . 32(12):3079–3088 doi:10.1093/molbev/msu293 Adva Rankin et al ), but conjugative plasmids have been cited as ; plasmids, genetic architecture, mob Bordenstein and Reznikoff 2005 Bergstrom et al. 2000 ;
Thomas and Nielsen 2005 ;
It has been suggested that there are functional differences The Author 2014. Published by Oxford University Press on behalf of the Socie Theoretical Biology, Institute of Norman et al Hacker and Carniel 2001 Bonhoeffer 2013 the “core” genome, consisting oftally genes transferred, and that the are “acce nothorizontally horizon- transferableCarniel 2001 genetic elements ( Plasmids replicate independently fromtheir the host and chromosome can of Consequently, be transferred their horizontally evolutionary andthose vertically: interests of may differ thegive from rise chromosome. to These conflictchromosomal or divergent to genes cooperation interests ( between can plasmid and large taxonomic distances. between genes that are likely to mid-biased genes ( implicated in horizontal transfer (broadlyjugation, grouped into transductio con- Carniel 2001 a particularly common( and successful vector of transfer and those that are( part of the core genome, both generally different evolutionary traject and chromosomes mean thattion th of each is likely to differ. and Woese 2007 *Corresponding author: Associate editor: ß Mol. Biol. Evol. et al. 2003 e-mail: [email protected] Mobile genetic elements play an importantevolution role in ( prokaryote Introduction Mobile genetic elements such as plasmids are important for th Abstract Samuel J. Tazzyman 1 Why There Are No Essential Genes on Plasmids competition between genotypically varyi 2005 are differences between functions codedcan for be by seen mobile between gene plasmidsinvolved and in chromosomes. the formation In of particular structural proteins or in basic m chromosomal location for essential genes.favored. We This find is that in becausedation” gene the rate inheritance as of the chromosomesessential is rate genes more at nonfunctioning. stable which The than chance only that genetic way for processes, in plasmids. for which We plas example, define mutation, the deletion, “degra- or translocation, render chromosomal genes degrade faster than plasmid genes. If the t faster than chromosomal genes, functioningKey esse words: View metadata, citation and similar papers at core.ac.uk at papers similar and citation metadata, View Tazzyman and Bonhoeffer . doi:10.1093/molbev/msu293 MBE
Dziewitetal. 2014; or under specific, regularly encountered on plasmids. In either setting, the gene can undergo “degra- environments, e.g., in the dark, Nagarajan et al. 2014,orunder dation.” By degradation, we mean any genetic process that anaerobic conditions, Ebertetal. 2013). Thus, because essen- makes the gene no longer functional, including, for example, tial genes are in fact sometimes found on plasmids (and must mutation, deletion, and translocation events, as long as they have been found there at times throughout evolutionary his- sufficiently disrupt the gene, so that it no longer works. For tory), why are they not found there more often? simplicity, we group all such events under degradation and The first possibility is that this is a semantic problem: assume that they occur at a rate c per individual for chro- Plasmids are often loosely defined as being replicons lacking mosomal locations and p per plasmid for plasmid locations. in essential genes, and consequently, no essential genes can be Thus, every generation functioning essential genes on chro- found on plasmids. As we are interested in what forces make mosomes and on plasmids lose their functionality at these essential genes less likely to be horizontally transferred, and rates. because (as noted above) there are in fact instances of essen- tial genes on plasmids, we can safely disregard this semantic Six Types of Individuals solution. Each individual in our population has one of two possible Another possibility is that there is only finite space on chromosome states: Either “wildtype” (i.e., with a functioning plasmids, and this space is taken up by the kinds of genes essential gene) or “mutant” (i.e., with a degraded essential that are favored on plasmids, as noted above. For example, gene). We denote these states as w and m, respectively. there have been various hypotheses to explain why plasmids Wild-type individuals have a functioning essential gene on would be an ideal locations for antibiotic resistance genes their chromosome, but mutant individuals do not. (Eberhard 1990; Rankin et al. 2010; Tazzyman and Therefore, to avoid death due to lacking the essential gene, Bonhoeffer 2014). We feel, however, that this explanation is a mutant individual must have a functioning essential gene begging the question somewhat: Even if these other kinds of on a plasmid. genes are favored in a plasmid context, we need to show that Each individual also has one of three possible plasmid essential genes are not also favored. Ideally, we would like to states: Either no plasmid (denoted with the “empty set" establish some sort of advantage accruing to essential genes symbol ;), a wild-type plasmid (i.e., with a functioning essen- from a chromosomal location. tial gene, denoted p), or a mutant plasmid (i.e., with a de- A final possibility is given by the “complexity hypothesis.” graded essential gene, denoted q). This suggests that the connectivity of a protein in the cellular There are thus six types of individual in total: Wild-type network, rather than the function of the protein, has a large chromosome with no plasmid (which we denote w;), wild- influence over how easily it can be horizontally transferred type chromosome with wild-type plasmid (wp), wild-type (Jain et al. 2003; Cohen et al. 2011). Genes whose products chromosome with mutant plasmid (w ), mutant chromo- have a high degree of protein–protein interaction will likely q some with no plasmid (m;), mutant chromosome with disrupt the metabolism of a cell into which they are trans- wild-type plasmid (mp), and mutant chromosome with ferred and consequently are unlikely candidates for horizontal mutant plasmid (m ). The relative frequency of a type x is transfer (Baltrus 2013). Essential genes by their very nature q y then denoted fðxyÞ, and the relative frequency after t gener- have been suggested to have high connectivity (Jeong et al. ations will be denoted ftðxyÞ. To simplify notation, we will 2001; Zotenko et al. 2008) and so by the complexity hypoth- denote the equilibrium relative frequency of a type xy as xy esis would be unlikely to be transferred horizontally. This is a rather than the more cumbersome f ðxyÞ. compelling explanation. However, our model below considers a much simpler case, attempting to explain the situation Fitness Scheme without reference to the complexities of protein–protein All fitnesses are defined relative to the fitness of type w;, interactions. which has a wild-type chromosome and no plasmid, and Understanding what forces bias essential gene location which we define to have fitness 1. There are then two ele- away from plasmids and on to chromosomes will help us ments to the fitness scheme. First, any individual lacking at to better understand the evolutionary processes shaping least one functioning copy of the essential gene dies and thus plasmid genomes. To investigate this, we here construct a has fitness zero. Therefore, the fitnesses of type m; and mq competition model, incorporating essential genes on both are zero, and no individuals of these types in fact exist in our plasmids and chromosomes and allowing for some degrada- population. tion of these genes. We can therefore see under which con- The second element in our scheme is the presence of ditions a chromosomal location is favored for such crucial the plasmid, which confers a small fitness cost, 0 < c 1. genes. The relative fitness of plasmid bearing types wp, wq,andmp are then all 1 c. The Model Interactions between the Types The Essential Gene In addition to the relative fitnesses, the frequencies of each Our model features a single, idealized essential gene. type are also determined by interactions between the types. Individuals require at least one working copy of this gene to There are three types of interactions: Mutation, conjugation, survive. The gene is found both on the chromosome and and segregation. We describe the details below; a diagram
3080 Plasmids and Essential Genes . doi:10.1093/molbev/msu293 MBE showing all interactions between the four surviving types is that our individuals are in a well-mixed, constant volume and shown in figure 1. that the process of horizontal transfer can therefore be rea- sonably well approximated by mass action-type equations Degradation. As mentioned above, degradation occurs on both chromosomes and plasmids, at rates and , respec- (Ross 1915). c p Conjugation can occur between any pair of individuals tively. Thus, in every generation, a proportion c of individ- uals with wild-type chromosomes become individuals with consisting of one plasmid bearer and one nonplasmid bearer. However, because in our system m; and mq individuals mutant chromosomes. In the case of w; and wq individuals, the chromosomal degradation immediately results in immediately die, we can ignore conjugation events featuring death, because it leaves the individual without a functioning these types. Conjugation therefore occurs between w; indi- viduals and wp, wq,ormq individuals. It transforms housekeeping gene. In the case of a wp individual, however, the nonplasmid-bearing w; into a plasmid-bearing transcon- the chromosomal degradation results in an mp individual, whichis“rescued”fromdeathbythefunctioning jugant. Thus, conjugation between w; and wp or mp individ- essential gene on the plasmid. Similarly, in every generation, uals results in the w; individual becoming the transconjugant wp, whereas conjugation between w; and wq results in the w; a proportion p of individuals with wild-type plasmids become individuals with mutant plasmids. In the case of individual becoming the transconjugant wq. Note that the an m individual, the plasmid degradation immediately chromosomal state is unchanged by conjugation. p Because we use mass action equations, the proportion of resultsindeath,asitmeansbecominganmq individual with no functioning essential gene. However, in the case of w; individuals that become transconjugants every generation is proportional to the relative frequencies of w; individuals, a wp individual, the plasmid degradation results in a wq indi- vidual, which still has a functioning essential gene on its and of the plasmid bearers, and is controlled by a parameter chromosome. . Given frequencies fðw;Þ; fðwpÞ; fðwqÞ,and fðmpÞ at a given generation, we then have fðw;Þ fðwpÞþfðmpÞ Conjugation. Conjugation is the process through which transconjugant wp individuals and fðw;ÞfðwqÞ transconju- plasmids are transferred horizontally in the population, gant wq individuals created, in that generation. from a plasmid bearer to a nonplasmid bearer. We assume Segregation. Segregation is the process by which plasmid- bearing individuals lose their plasmids, usually during replica- tion. It is generally believed to be rare because of a variety of phenotypic adaptations that plasmids have evolved to pre- vent it from occurring (e.g., postsegregational killing [Cooper and Heinemann 2000]; or methods to ensure even splitting of plasmids during cell division [Ebersbach and Gerdes 2005]). We model it by assuming that some small proportion of plasmid bearers lose their plasmids per generation. Thus, wp and wq individuals become w; individuals, whereas mp indi- viduals die because they become m; individuals and thus lack a functioning essential gene.
The Model Weusetheaboveelementstoconstructthemodelasfollows. We census the population at every generation. We concern FIG.1. Diagram of the interactions between the four surviving types. ourselves with relative frequencies and assume that the total Each box represents one of the six types, with the two nonsurviving population size is a (very large) constant. Therefore at any types marked with dashed lines and the label “(Dead).” Within each box, generation, the frequencies of each type add up to one: a cartoon shows the chromosomal status (squiggly line) and the plasmid fðw;ÞþfðwpÞþfðwqÞþfðmpÞ¼1. Given frequencies status (circles), marked with a tick for functioning essential gene and a ftðw;Þ; ftðwpÞ; ftðwqÞ,andftðmpÞ after t generations, we cross for degraded essential gene. Each arrow shows a transformation as wish to know the frequencies after t +1generations. a result of some interaction type. The labels for the arrows show the proportion of the type which the arrow leaves that are transformed into To simplify the mathematics, we first take fitness into ac- the type the arrow leads to. Some transformations result in the forma- count and then the interactions. The fitness of type w; is 1, tion of the nonviable types m; or mq; these transformations are shown whereas the fitness of types wp, wq,andmp is 1 c. here to lead to boxes marked “Dead.” The lines marked gwq and gwp gw; ¼ ftðw;Þ þ gmp are conjugation reactions, where is the conjugation rate, and the values gi are as given by equation (1). The lines marked c j gwp ¼ð1 cÞftðwpÞ are degradation events taking place on chromosomes, and the lines ð1Þ marked p are degradation events taking place on plasmids. The lines gwq ¼ð1 cÞftðwqÞ marked are segregation events. For full details of each reaction, see the ¼ð Þ ð Þ main text. gmp 1 c ft mp
3081 Tazzyman and Bonhoeffer . doi:10.1093/molbev/msu293 MBE
Then, the frequencies of each genotype in the next genera- Equilibrium with Only w; and wq Individuals tion are This equilibrium point corresponds to the case where all the essential genes on plasmids have degraded, whereas all those !ftþ1ðw;Þ¼gw; ðÞ 1 c gw; gw þ gw þ gm p q p on chromosomes are still functional. The equilibrium fre- þ gw þ gm ; quencies are p p
!ftþ1ðwpÞ¼gwp 1 c p þ gw; gwp þ gmp ; ð1 cÞ w ¼ ; ð5Þ ; ð1 cÞ cð1 Þ !ftþ1ðwqÞ¼gw ðÞþ1 c gw; gw þ pgw ; c q q p !ftþ1ðmpÞ¼gm 1 p þ cgw : p p and wq ¼ 1 w;. This solution is stable if c p,thatis,if ð2Þ plasmid genes degrade more rapidly than chromosomal genes, and unstable in the contrary case where 4 , where ! is the average fitness for generation t,definedas c p where plasmid genes degrade more slowly than chromosomal genes (Appendix). ! ¼ ftðÞw; ðÞþ1 ðÞ1 c ft wp c þ ðÞ1 c ft wq ðÞ1 ð3Þ c Equilibrium with All Four Types þ ðÞ1 c ft mp 1 p ; The exact location of the equilibrium is a lengthy expres- sion for each genotype frequency. We have reproduced so that the new genotype frequencies add to one. We can it in the Appendix, but here we instead make the substitu- combine the sets of equations (1) and (2) with equation (3) to tion ¼ , and note that is likely to be much find equilibrium frequencies w ; w ; w ,andm . p c c ; p q p smaller than the other expressions, and so we can let Results it tend to zero and see that the global equilibrium is approx- imately There are four different equilibria possible. Inspection of figure 1 reveals that equilibria are possible in which there ð1 cÞ w;& ; are only w; types, in which there are only mp types, and in ð1 cÞ c which there are only w; and wq types. There is also an equi- librium in which all four types are present. We now go þ cð1 þ Þ w &ð1 Þ ; through the stability of these equilibria in turn p ð1 cÞ c ð6Þ Equilibrium with Only w Individuals þ cð1 þ Þ ; w & ; This equilibrium point corresponds to the case where the q ð1 cÞ c plasmid is absent from the population. Under these circum- stances, because only chromosomal essential genes exist, all mp&0: degradations to these genes result in immediate death. The criterion for plasmids to be able to invade the population (for some parameter combinations this approximation (and thus for this equilibrium point to be unstable) is does not hold, but it is likely to be generally true for reasonable parameter values—see the Appendix for details). c < c ¼ ; ð4Þ We can see from this that it will only be a permitted solu- 1 þ c tion if <1, that is, if c4 p, since otherwise wp < 0 so there is an upper limit c on the cost that a plasmid can (we know that the numerator in the fraction found in the impose on its host and still be able to invade, depending upon expressions for wp and wq is positive because we assume the conjugation rate , the segregation rate ,andthedeg- c4c ). Thus, this equilibrium is stable for values c4 p radation rate for chromosomes c. As we are here interested (Appendix). in whether essential genes can be maintained on plasmids, we < assume from now on that c c (though note that the Growth Ratios reasons behind the maintenance of plasmids over evolution- ary timescales in the face of low conjugation rates and non- We further illustrate the system by calculating growth ratios. zero costs have been debated; Stewart and Levin 1977; The growth ratio p for the p plasmid (containing a function- Bergstrom et al. 2000; Lili et al. 2007). ing housekeeping gene) is p ¼ ftþ1ðwpÞþftþ1ðmpÞ = ftðwpÞþftðmpÞ , representing the frequency of the plasmid in generation t + 1 compared with its frequency in generation Equilibrium with Only Individuals mp t.If p41, then the p plasmid is growing in frequency, if This equilibrium point corresponds to the case where all p < 1, it is declining in frequency, and if p ¼ 1, its fre- functioning essential genes are plasmid borne. For reasonable quency is unchanging. Similar calculations can be made for parameter values, it is always unstable (see Appendix). the q plasmid, as well as for the chromosomes w and m.
3082 Plasmids and Essential Genes . doi:10.1093/molbev/msu293 MBE
To make analysis easier, we multiply all the growth ratios by the average fitness in generation t, !.
! p ¼ð1 cÞð1 þ ftðw;Þ pÞ ftðwpÞ ! q ¼ð1 cÞ 1 þ ftðw;Þ c þ p ; ftðwqÞ
cð1 cÞftðw;Þ ! w ¼ð1 cÞð1 cÞþ ftðw;ÞþftðwpÞþftðwqÞ ftðwpÞ ! m ¼ð1 cÞ 1 p þ c : ftðmpÞ We can now see more clearly why it is that the relative sizes of pqp q c and p have such an effect on the competition between p the p and q plasmids. If p4 c,then q4 p,andsothep plasmid will always be growing faster than the q plasmid. Thus, either plasmids go extinct entirely (if c is large enough relative to the other parameters, so that q < 1), or the q plasmid will fix. If, on the other hand, c p,thevaluesft FIG.2. Qualitative results of the model as a result of two parameters, , ðwpÞ and ftðwqÞ will determine which is the larger out of p ratio of the degradation rate of plasmids to that of chromosomes, and c, and q, with the two growth ratios being equal at the equi- the cost of bearing the plasmid. For c4c (eq. 4), no plasmids exist at librium with all four types present given above. equilibrium, and the equilibrium point is w; ¼ 1, with all other fre- We can rewrite the expression containing w in terms of quencies zero. For c < c , the results depend on ¼ p= c.For the expression containing m to give 8 1, there are only w; and wq types at equilibrium, and their fre- > quencies are given by equation (5). For <1, all four types are present > ! m > at equilibrium, and their frequencies are given by equation (6). > > þ <> ð1 cÞþð Þð1 cÞ ! ¼ p c ð7Þ small frequency, mp&0. A fraction 1 mp&1 of chromo- w > somes will bear functioning essential genes, whereas a fraction > þ > 1 of plasmids will bear functioning essential genes. > > ftðw;Þ ftðwpÞ > In particular, note that if c ¼ p, we will have only w; : cð1 cÞ cð1 cÞ 1 ftðmpÞ ftðmpÞ and wq types, and even if c& p (i.e., &1), we will be very close to this equilibrium. Thus, if the degradation rates on From this, we see that a higher degradation rate improves plasmids and chromosomes are similar, there will be very few the fitness of the w chromosome over the m chromosome, as essential genes on plasmids. does a greater positive difference p c. The effects of pa- rameters c and c will depend on the genotype frequencies in Discussion quite a complicated way. Our results theoretically validate the empirical observation that essential genes are mainly found on chromosomes, Overview of Results rather than on plasmids. At all possible equilibrium states, Our results are summarized in figure 2.Therearetwokey all (or nearly all) chromosomes in the population will bear elements, the cost c of bearing a plasmid, and the ratio of functioning essential genes. This will not necessarily be the the degradation rate of plasmids to that of chromosomes. case for plasmids. If the degradation rate on plasmids is equal If c4c , then plasmids are too costly to persist in the to or greater than the degradation rate for chromosomes (i.e., population and will go extinct. The only genotype in the 41), then no plasmids will have functioning housekeeping population is then w;, and consequently, all chromosomal genes. Alternatively, if the degradation rate on plasmids is less genes will bear functioning essential genes. If c < c ,then than the degradation rate on chromosomes, there will be a plasmids can persist in the population. Then, the equilibrium mix of plasmids in the population, some which have func- will depend on . tioning housekeeping genes and some which do not. The If 1, plasmid genes degrade more rapidly than chro- proportion of plasmids with degraded, nonfunctioning essen- mosomal genes. At equilibrium, there will be only w; and wq tial genes will be equal to the ratio of the degradation rate on types. This means that all chromosomes will bear functioning chromosomes with that on plasmids. essential genes, whereas no plasmids will. If <1, chromo- The intuition behind these results is that the key element is somal genes degrade at a higher rate than plasmid genes. the stable inheritance of essential genes. Offspring lacking Then, all four types will be present in the population at equi- essential genes die; consequently, the fitness of the different librium, although type mp will be present at a vanishingly genotypes in our model is determined to a large degree by
3083 Tazzyman and Bonhoeffer . doi:10.1093/molbev/msu293 MBE
0
−2
−4 log −6
−8
−10 −10 −8 −6 −4 −20 log 2 FIG.4. Log values of c required to give mp410 for given values of , ,where =0andc = 0. For example, for values of and along the FIG.3.Threshold value as a function of .Tohavemp4 ,we 4 2 require c4 (i.e., to be in the gray area). Increasing parameter line marked “ 4,” we would require c410 in order for mp410 . 5 values generally affects the threshold as shown by the arrows above. Looking at the plot, one can see that mutation rates below c ¼ 10 Here, the relationship is plotted on a log–log scale, and the parameter will require extremely small values of to lead to appreciable quantities values are ¼ 0:4; ¼ 0:5; ¼ 0:01, c = 0.05. of mp individuals at equilibrium, even where c =0and =0. how likely their offspring are to stably inherit the essential genes. The m chromosome relies on being paired with a p contingent on this. We simply aim to show the consequences plasmid to make up for its own lacking functioning essential of these rates, rather than claiming anything specific about gene. This plasmid can fail to be inherited due to segregation how they differ in reality. Our results show that if the two or due to mutation (rates and p, respectively). On the values are similar, then there will be almost no plasmids bear- other hand, the w chromosome will fail to pass a functioning ing functioning housekeeping genes (assuming plasmids are essential gene to its offspring only in the case where it mutates stably maintained in the population). The only way that plas- (rate c) and is not paired with a p plasmid. Thus, higher mids will bear functioning housekeeping genes in our model values of and p benefit the w chromosome, whereas is if they degrade less rapidly than chromosomes and are thus higher values of c benefit the m chromosome. The w chro- able to make up for their lack of stable inheritance. mosome also benefits from the extra fitness of the w; types. In addition to the effect of the relationship between p Although the exact relative fitnesses of each chromosome and c, there are also effects of the parameters , ,andc. depend upon the frequencies of the four types (eq. 7), in However, supposing that c and p are such that mp&0at general the w chromosome is usually fitter than the m chro- equilibrium (i.e., the two degradation parameters are suffi- mosome and rises to fixation (or near fixation) in all cases. ciently small and/or p4 c, see Appendix), we will be ap- Thefactthatthew chromosome is prevalent in turn affects proximately close to the equilibrium positions given by the competition between the p and q plasmids. The compe- equation (6), and these other parameters largely affect only tition between them is largely determined by the rates of the frequency of plasmid-bearing individuals in the popula- mutation. The q plasmid will lose out when the w chromo- tion, rather than the outcome of the competition between some with which it is associated mutates (at a rate c), be- the two types of plasmid. cause the resulting mq individual lacks a functioning essential There are some potential caveats to our conclusions. We gene and dies. The same chromosomal mutation does not have treated plasmids here as a single entity, with a binary immediately damage the p plasmid, as the mp individual cre- status of either “functioning” or “degraded.” Thus, our model ated survives. So high rates of c favor the p plasmid over the implicitly treats the situation in which there is only a single q plasmid by reducing the fitness of the latter. However, when plasmid location in each plasmid-bearing cell. In reality, how- the p plasmid mutates (at rate p), it becomes a q plasmid, ever, plasmids generally exist at multiple copy numbers, and meaning that higher values of p doubly favor the q plasmid there is the possibility that some of the plasmids within a cell over the p plasmid by increasing the fitness of the former and will carry degraded copies of the essential gene, whereas reducing that of the latter. Consequently, the result of the others will carry functioning copies. The inheritance, both competition between the two is largely determined by the vertically and by conjugation, of such a multiple copy relationship between c and p,withtheqplasmidhaving number system would require a model of greatly in- an inherent advantage, as described above. creased complexity. We expect, however, that our qualitative Note that in reality, p and c are not necessarily different. conclusions, that chromosomal locations are favored for Infact,weareunawareofanyapriorireasonwhythere essential genes, and that the frequency of these genes on should be a large difference between the two. Perhaps, they plasmids strongly depends on the degradation ratio , will do generally differ for some reason, but our work is not still hold.
3084 Plasmids and Essential Genes . doi:10.1093/molbev/msu293 MBE
Our work can be compared with the complexity hypoth- Appendix esis, which would also predict a lack of essential genes on plasmids but for different reasons; namely, that the high con- Stability of Equilibria nectivity of the essential proteins means that they are not We first note that, since the frequencies fðw;ÞþfðwpÞþ good candidates for horizontal transfer as they cause too fðwqÞþfðmpÞ¼1, there are only three independent vari- much disruption (Jain et al. 2003; Cohen et al. 2011; ables, so we can use the substitution mp ¼ w; þ wp þ wq.
Baltrus 2013). We could capture this disruption in our frame- For a given type xy,wedefineJxy ¼ ftþ1ðxyÞ ftðxyÞ,the work by introducing an additional cost k that represents the change in frequency per generation. Then, we define the problem caused by having two copies of the essential gene, so Jacobian matrix J, 0 1 that the fitness of type wp is 1 c k, whereas the fitnesses @Jw; @Jw; @Jw; of types mp and wq are both 1 c. Unfortunately, this addi- B C B @w; @wp @wq C tional parameter causes the complexity of the model to in- B C crease vastly, and so for space reasons, we have elected not to B @ @ @ C B Jwp Jwp Jwp C follow this line of research here but rather to leave it for a later J ¼ B C: B @w; @wp @wq C work. The complexity hypothesis is not mutually exclusive B C @ @ @ @ A with our hypothesis that the fidelity of inheritance of the Jwq Jwq Jwq genes is what is important; both could be contributory @w; @wp @wq factors. ð Þ Our findings are particularly interesting because they show The stability of an equilibrium point w;; wp; wq is then that, even at equilibrium, it is possible that essential genes determined by the three eigenvalues of J, denoted 1, 2, could be found on plasmids. This is the case if the degradation and 3, evaluated at this point, with the equilibrium being rate of these genes is higher on the chromosome than it is on stable if all eigenvalues have negative real part. a plasmid (fig. 2). This may help to explain the discovery of w; ¼ 1 plasmid-borne essential genes (e.g., Young et al. 2006; Ebert et al. 2013; Dziewit et al. 2014; Nagarajan et al. 2014); alterna- Evaluated at the equilibrium point for which there are just tively, such genes may be an example of a population that is w; individuals, the matrix J has eigenvalues yet to reach equilibrium. It might be that in the long term, cð1 þ Þ these genes will migrate to the chromosome. ¼ c ; 1 On a related note, one kind of mechanism that is essential 1 c under certain environmental conditions and is often encoded p cð1 þ c pÞ by genes found on plasmids is antibiotic resistance (Jacob and 2 ¼ ; 1 c Hobbs 1974; Eberhard 1990), for reasons that are imperfectly ¼ understood (Tazzyman and Bonhoeffer 2014). Our findings 3 1; here support the assertion that, over long timescales, antibi- and so the equilibrium will be unstable where otic resistance genes are likely to move to chromosomal lo- cð1 þ cÞ40. This gives the condition cations (Bergstrom et al. 2000), indirectly supporting the on c given in the text above. thesis that short-term effects are responsible for the presence of plasmid-borne antibiotic resistance (Bergstrom et al. 2000; mp ¼ 1 Tazzyman and Bonhoeffer 2014). In conclusion, we have shown that chromosomes are This equilibrium point corresponds to the case where all indeed favored locations for functioning essential genes, functioning essential genes are plasmid borne. Our eigenva- due to their stable inheritance. Whether plasmids will also lues are then bear such genes, however, will depend on the rate at which c 1 ¼ ; these genes degrade by chance events in each location. If 1 p plasmids very rarely bear functioning essential genes, it is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perhapsanindicationthattherateofdegradationinplasmid 2 þð1 cÞð c pÞ þð1 cÞð c pÞ þ locations is greater than that on the chromosome. 2 ¼ ; 2ð1 cÞð1 pÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Acknowledgments þð1 cÞð c pÞ þ þð1 cÞð c pÞ þ 3 ¼ ; The authors thank Daniel Angst, Alex Hall, and Gabriel 2ð1 cÞð1 pÞ Leventhal for helpful discussions on the topic, as well as the associate editor and three anonymous reviewers, through where whose efforts the manuscript was improved. This research was supported by the European Research Council under ¼ þ c p c 1þ p ; the 7th Framework Programme of the European Commission (PBDR: Grant Agreement Number 268540). ¼ 4ð1 cÞð1 cÞ ð c pÞ :
3085 Tazzyman and Bonhoeffer . doi:10.1093/molbev/msu293 MBE
We can have stability only where þð1 cÞð c pÞ 0 240, which is true if and only if c4 p,thatis,ifchromo- and <0, since otherwise 340. 40impliesthat somes degrade at a faster rate than plasmids. ð Þ 1 c All Four Types c p4 40; ð8Þ The equilibrium point is at and since 40, if c p40, we have that this equilibrium þ cð1þ Þ ð þ Þþc ð1 Þ ¼ c p p p c c ; is stable if and only if criterion (8) is true. As depends on c w; þ c p cð1þ pÞ and p, this is not a closed form solution, but we make the substitution p ¼ c, we can show that equation (8) will be ð1 cÞ ð c pÞ wp ¼ð c pÞ ; true where ð1 cÞ cð þ c p cð1þ pÞÞ ð1 cÞ ð Þ ð1 cÞ ¼ c p ð1 Þ 40: wq p ; c ð1 cÞ cð þ c p cð1þ pÞÞ þ c cð1þ Þ ð1 cÞ c ð Þ m ¼ c p ; This expression gives us a quadratic equation to solve for p ð1 cÞ , which shows that there is no solution in the region 0 < < 1 using our parameter ranges. So this solution is never where stable. ¼ þ c p c 1þ p :
Just w; and w Types q As wq ¼ wp p=ð c pÞ, the two will be of opposite sign This equilibrium point corresponds to the case where all if p4 c, and consequently, this equilibrium will be outside essential genes on plasmids are degraded, whereas all those the region we are interested in. Thus, this solution, in which all on chromosomes are still functional. The equilibrium fre- four types feature, will only occur if c4 p:Exactlythe quencies are wq ¼ 1 w; and region for which the above solution with only types w; and wq is unstable. In addition, as c ! p from above, the ð1 cÞ w; ¼ : solution with all four types tends to exactly the solution ð1 cÞ cð1 cÞ with only w; and wq given above. It remains to show that The corresponding eigenvalues of J evaluated at this this solution with four types is stable where c4 p. point are Unfortunately, the eigenvalues do not admit a simple analy- tical closed form solution, so we instead consider the char- ð cð1 þ ÞÞð cð1 þ ÞÞ ¼ c c ; acteristic equation of the matrix J, 1 ð1 cÞð1 Þð cð1 þ ÞÞ c c 3 2 x Tr½J x þjðÞJ11 jþjJ22 jþjJ33 j x jJj¼0; ðÞð cð1 þ cÞ c pÞ 2 ¼ ; where Tr½ is the trace of a matrix, j jis the determinant of a ð1 cÞðÞ cð1 þ cÞ matrix, and Jii is the two-by-two matrix formed from J by cð1 cÞ p ð1 cÞ ð þ pÞ deleting the ith row and column. Then, from the Routh– 3 ¼ ; ð1 cÞðÞ cð1 þ cÞ Hurwitz stability criteria, the solution is stable if, when J is evaluated at the appropriate fixed point, As c < c ,weknowthat cð1 þ Þ40, c j j < ; and so the denominator in all three cases is greater than 0. It J 0 follows that also cð1 þ cÞ4 Tr½J < 0; ð1 cÞ=ð1 þ cÞ40, so 1 < 0, and 2 has thesamesignas c p. Finally, after some rearrangement, we have that 3 < 0ifc < c^,where jJ11 jþjJ22 jþjJ33 j40; ð þ Þ and c^ ¼ p ; ð1 Þ þ ð þ Þ c p p Tr½J jðÞJ11 jþjJ22 jþjJ33 j 4 jJj but Define
ð1 Þð þ Þ þ c p c p c ¼ : c^ c ¼ 40; 1þ ðÞð1 þ c 1 cÞ p þ ð þ pÞ p Then if c < c ; 40. But we know c < c and so c^4c 4c,and 3 < 0. ð1 cÞð c pÞ Thus, for our region of parameter space, 1 < 0and c c ¼ ; ð1þ Þð1þ Þ 3 < 0, and so the equilibrium is unstable if and only if c p
3086 Plasmids and Essential Genes . doi:10.1093/molbev/msu293 MBE
so if c4 p,wehavec 4c 4c and so 40. After some To support our claim, note that we can rearrange equation calculation, it can be shown that (9)toderiveavalue , 0 1 ðð Þ ð1 cÞ Þ c ð Þð1 cÞ c p B C B 1 ð1 cÞ C B C jJj¼ 1B þ C ð Þ ð þð1 cÞ Þ2ðð1 cÞð þ Þþ Þ ¼ BsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC; c p c p : 2B C 2 3 3 @ ðÞc ð Þð1 cÞ 2 4ð1 cÞ A ð1 cÞ ð1 cÞ ð þ Þ þ ðÞ1 ð1 cÞ 2 ð1 Þð1 ð1 cÞ Þ
If c4 p, the fractional part of this expression has posi- tive denominator and numerator. Thus, jJj < 0if such that for any 0 < <1; mp4 if and only if c4 .The c p < ð1 cÞ = , which we showed above must be expression for is complicated, but we can investigate it the case for our parameter range. numerically, plotting as a function of " to see the required We can also show that mutation rate to sustain a proportion " of mp types at equili- brium (see fig. 3). Tr½J ¼ This investigation was carried out using Mathematica 0 1 (Wolfram Research, Inc. 2010), and the files are stored on ð þ þð1 cÞ cÞð þ Þ B C Dryad at http://dx.doi.org/10.5061/dryad.m7t63, allowing B ð1 cÞð1 cÞð þ Þ C B C interested readers to investigate the parameter space B C B þ C: themselves. B C B C To see that mp&0 for reasonable parameter values is to @ A c 1 2 4 c þ2 p þc 1þ2 þ2ð c pÞ investigate the case where c =0and =0,abest-casescenario ð1 cÞð1 Þð þ Þ in terms of increasing the size of mp. This corresponds to the c case where plasmid-borne essential genes do not degrade and plasmids cost nothing. Under these circumstances, we can As 40, the denominator of both expressions in brackets 2 determine the required value of c to give mp ¼ 10 .As on the right-hand side is positive, and the first expression is canbeseenfromfigure 4, even under these circumstances, also positive. Then, because we are assuming that , c, p,and which are very favorable to high mp values at equilibrium, we are all considerably less than 1, the second expression is c would require either (relatively) high values of c or (rela- also positive. So Tr½J < 0. 2 tively) low values of to get mp410 . The expression for j J11 jþjJ22 jþjJ33 j is considerably Finally, we numerically calculated values of mp across the less tractable. We were able to evaluate it numerically for a parameter space that we used to numerically investigate the wide range of parameter values (c varying between 0.01 and stability of the four-type equilibrium above and found that 0.05, varying between 0.01 and 0.1, varying between 10 4 across this range of parameter space, m < 0:09, and that 3 6 5 p and 10 , c varying between 10 and 10 ,and ¼ p= even this relatively high value is obtained at the edge of the 5 5 5 c varying between e and e ). For this region of parameter space with a combination of parameters c ¼10 , c = 0.01, 4 5 space, we found that as long as <1(i.e., c4 p), j J11 j ¼ 0:1; ¼ 10 , =0,and c ¼ 10 . þjJ22 jþjJ33 j 40 as required. In other words, in order for mp types to be frequent at We investigated the difference Tr½J jð J11 jþjJ22 jþ equilibrium, we would need an unrealistic combination of low j J33 jÞþ jJ j similarly and found that it was always positive, degradation rates on plasmids, high degradation rates on as required. The Mathematica (Wolfram Research, Inc. 2010) chromosomes, and/or low segregation rates. Under these cir- file in which these calculations were performed is available for cumstances, plasmids essentially become back-up chromo- download on Dryad at (insert URL). somes, as they are inherited almost as well as chromosomes. If they also degrade at a much slower rate, then we can see appreciable quantities of mp types. These Size of mp at the Equilibrium with All Four Types circumstances are somewhat peculiar, however, so that we Making the substitution p ¼ c,wehavefromabovethat feel justified in the assertion in the main text that mp&0for at the equilibrium with all four types, reasonable parameter values.
ð1 ÞðÞ þ ð1 Þ cðÞ1 þ References m ¼ c c c ; p ð1 cÞ Anderson E. 1968. The ecology of transferable drug resistance in the enterobacteria. Annu Rev Microbiol. 22(1):131–180. ð9Þ Baltrus DA. 2013. Exploring the costs of horizontal gene transfer. Trends Ecol Evol. 28(8):489–495. where 0 < <1 since otherwise the four type equilibrium is Bergstrom C, Lipsitch M, Levin B. 2000. Natural selection, infectious unstable, and we revert to the two-type equilibrium. We transfer and the existence conditions for bacterial plasmids. Genetics 155(4):1505–1519. claim that with reasonable parameter values, this equilibrium Bordenstein SR, Reznikoff WS. 2005. Mobile DNA in obligate intracellular value mp&0. bacteria. Nat Rev Microbiol. 3(9):688–699.
3087 Tazzyman and Bonhoeffer . doi:10.1093/molbev/msu293 MBE
Cohen O, Gophna U, Pupko T. 2011. The complexity hypothesis re- Koonin EV, Wolf YI. 2008. Genomics of bacteria and archaea: the emerg- visited: connectivity rather than function constitutes a barrier to ing dynamic view of the prokaryotic world. Nucleic Acids Res. horizontal gene transfer. MolBiolEvol.28(4):1481–1489. 36(21):6688–6719. Cooper TF, Heinemann JA. 2000. Postsegregational killing does Lili LN, Britton NF, Feil EJ. 2007. The persistence of parasitic plasmids. not increase plasmid stability but acts to mediate the exclusion Genetics 177(1):399–405. of competing plasmids. Proc Natl Acad Sci U S A. 97(23): Nagarajan S, Srivastava S, Sherman LA. 2014. Essential role of the plasmid 12643–12648. hik31 operon in regulating central metabolism in the dark in Dowling D, Broughton W. 1986. Competition for nodulation of legumes. Synechocystis sp. pcc. 6803. Mol Microbiol. 91(1):79–97. Annu Rev Microbiol. 40(1):131–157. Nogueira T, Rankin DJ, Touchon M, Taddei F, Brown SP, Rocha EP. 2009. Dziewit L, Czarnecki J, Wibberg D, Radlinska M, Mrozek P, Szymczak M, Horizontal gene transfer of the secretome drives the evolution of Schluter€ A, Puhler€ A, Bartosik D. 2014. Architecture and functions of bacterial cooperation and virulence. Curr Biol. 19(20):1683–1691. a multipartite genome of the methylotrophic bacterium Paracoccus Norman A, Hansen LH, Sørensen SJ. 2009. Conjugative plasmids: vessels aminophilus jcm 7686, containing primary and secondary chromids. ofthecommunalgenepool.Philos Trans R Soc B Biol Sci. BMC Genomics 15(1):124. 364(1527):2275–2289. Eberhard W. 1990. Evolution in bacterial plasmids and levels of selection. Ochman H, Lawrence JG, Groisman EA. 2000. Lateral gene transfer and QRevBiol.65(1):3–22. the nature of bacterial innovation. Nature 405(6784):299–304. Ebersbach G, Gerdes K. 2005. Plasmid segregation mechanisms. Annu Rankin D, Rocha E, Brown S. 2010. What traits are carried on mobile Rev Genet. 39:453–479. genetic elements, and why? Heredity 106(1):1–10. Ebert M, Laaß S, Burghartz M, Petersen J, Koßmehl S, Wohlbrand€ L, Ross R. 1915. Some a priori pathometric equations. Br Med J. Rabus R, Wittmann C, Tielen P, Jahn D. 2013. Transposon muta- 1(2830):546–547. genesis identified chromosomal and plasmid genes essential for ad- Sørensen SJ, Bailey M, Hansen LH, Kroer N, Wuertz S. 2005. Studying aptation of the marine bacterium Dinoroseobacter shibae to plasmid horizontal transfer in situ: a critical review. Nat Rev anaerobic conditions. J Bacteriol. 195(20):4769–4777. Microbiol. 3(9):700–710. Evans DG, Silver R, Evans D, Chase D, Gorbach S. 1975. Plasmid- Stewart FM, Levin BR. 1977. The population biology of bacterial plas- controlled colonization factor associated with virulence in mids: a priori conditions for the existence of conjugationally trans- Escherichia coli enterotoxigenic for humans. Infect Immun. 12(3): mitted factors. Genetics 87(2):209–228. 656–667. Tazzyman SJ, Bonhoeffer S. 2013. Fixation probability of mobile genetic Goethals K, Vereecke D, Jaziri M, Van Montagu M, Holsters M. 2001. elements such as plasmids. Theor Popul Biol. 90:49–55. Leafy gall formation by Rhodococcus fascians. Annu Rev Phytopathol. Tazzyman SJ, Bonhoeffer S. 2014. Plasmids and evolutionary rescue by 39(1):27–52. drug resistance. Evolution 68(7):2066–2078. Goldenfeld N, Woese C. 2007. Biology’s next revolution. Nature Thomas CM. 2000. Paradigms of plasmid organization. Mol Microbiol. 445(7126):369–369. 37(3):485–491. Gonzalez CF, Kunka BS. 1987. Plasmid-associated bacteriocin production Thomas CM, Nielsen KM. 2005. Mechanisms of, and barriers to, horizon- and sucrose fermentation in Pediococcus acidilactici. Appl Environ tal gene transfer between bacteria. Nat Rev Microbiol. 3(9):711–721. Microbiol. 53(10):2534–2538. Werren JH. 2011. Selfish genetic elements, genetic conflict, and evolu- Hacker J, Carniel E. 2001. Ecological fitness, genomic islands and bacterial tionary innovation. Proc Natl Acad Sci U S A. 108(Suppl 2), pathogenicity. EMBO Rep. 2(5):376–381. 10863–10870. Jacob AE, Hobbs SJ. 1974. Conjugal transfer of plasmid-borne multiple Wolfram Research, Inc. 2010. Mathematica, version 8.0 edition. antibiotic resistance in Streptococcus faecalis var. zymogenes. Champaign (IL): Wolfram Research, Inc. J Bacteriol. 117(2):360–372. YoungJP,CrossmanLC,JohnstonAW,ThomsonNR,GhazouiZF,Hull Jain R, Rivera MC, Moore JE, Lake JA. 2003. Horizontal gene transfer KH, Wexler M, Curson AR, Todd JD, Poole PS, et al. 2006. The accelerates genome innovation and evolution. MolBiolEvol. genome of Rhizobium leguminosarum has recognizable core and 20(10):1598–1602. accessory components. Genome Biol. 7(4):R34. Jeong H, Mason SP, Barabasi A-L, Oltvai ZN. 2001. Lethality and cen- Zotenko E, Mestre J, O’Leary DP, Przytycka TM. 2008. Why do hubs in trality in protein networks. Nature 411(6833):41–42. the yeast protein interaction network tend to be essential: reexa- Koch A. 1981. Evolution of antibiotic-resistance gene-function. Microbiol mining the connection between the network topology and essen- Rev. 45(2):355–378. tiality. PLoS Comput Biol. 4(8):e1000140.
3088