Experimental and Mathematical Escherichia PlasmidTransfer ... · Experimental andMathematical Models ofEscherichia coli PlasmidTransferIn Vitro andIn Vivo ROLFFRETER,*ROLFR. FRETER,ANDHOWARDBRICKNER

INFECTION AND IMMUNITY, Jan. 1983, p. 60-84 Vol. 39, No. 10019-9567/83/010060-25$02.00/0Copyright 0 1983, American Society for Microbiology

Experimental and Mathematical Models of Escherichia coliPlasmid Transfer In Vitro and In VivoROLF FRETER,* ROLF R. FRETER, AND HOWARD BRICKNER

Department of Microbiology, The University of Michigan, Ann Arbor, Michigan 48109

Received 6 May 1982/Accepted 16 September 1982

Little is known about the factors that govern plasmid transfers in naturalecosystems such as the gut. The consistent finding by earlier workers that plasmidtransfer in the normal gut can be detected only at very low rates, if at all, has givenrise to numerous speculations concerning the presence in vivo of variousinhibitors of plasmid transfer. Plasmids Rl, Rldrd-19, and pBR322 were studied inEscherichia coli K-12 and wild-type E. coli hosts in two experimental systems: (i)gnotobiotic mice carrying a synthetic indigenous microflora (F-strains) whichresemble in their function the normal indigenous microflora of the mouse largeintestine, and (ii) anaerobic continuous-flow cultures of indigenous large intestinalmicroflora of the mouse, which can simulate bacterial interactions observed in themouse gut. Mathematical models were developed to estimate plasmid transferrates as a measure of the "fertility," i.e., of the intrinsic ability to transfer theplasmid under the environmental conditions of the gut. The models also evaluatethe effects of plasmid segregation, reduction of the growth rates of plasmid-bearing bacterial hosts, repression of transfer functions, competition for nutrients,and bacterial attachment to the wall of the gut or culture vessel. Some confidencein the validity of these mathematical models was gained because they were able toreproduce a number of known phenomena such as the repression of fertility of theRl plasmid, as well as known differences in the transmission and mobilization ofthe plasmids studied. Interpretation of the data obtained permitted a number ofconclusions, some of which were rather unexpected. (i) Fertility of plasmid-bearing E. coli in the normal intestine was not impaired. The observed low rates ofplasmid transfer in the normal gut can be explained on quantitative grounds aloneand do not require hypothetical inhibitory mechanisms. (ii) Conditions for long-term spread and maintenance throughout human or animal populations of adiversity of conjugative and nonconjugative plasmids may be optimal among E.coli strains of low fertility, as are found among wild-type strains. (iii) E. colistrains carrying plasmid pBR322 plus Rldrd-19 were impaired in their ability totransfer Rldrd-19, but strains carrying pBR322 were significantly better recipientsof Rldrd-19 than a plasmid-free recipient E. coli. (iv) Long-term coexistence of.plasmid-bearing and plasmid-free E. coli, in spite of undiminished fertility,appeared to be due to a detrimental effect of the plasmid on the growth rate of itshost bacterium, rather than due to high rates of plasmid segregation. (v)Mathematical analysis of experimental data published by earlier investigators isconsistent with the conclusion that plasmid transfer occurs consistently in thehuman gut, but that the resulting transconjugant E. coli populations are too smallto be detected regularly with the culture methods used by earlier investigators. Itis concluded that the long-term interactions observed were often the conse-quences of minor differences in parameters such as growth rates, fertility, rates ofsegregation, etc., which were too small to be detected except by precisemathematical analysis of long-term experiments, but which were neverthelessdecisive determinants of the ultimate fates of the plasmids and their hosts.

Genes for drug resistance and virulence fac- ment of their new host microorganisms. Where-tors of bacteria are often located on plasmids, as as much information is available concerning theare sequences inserted by recombinant DNA transfer of plasmids among pure cultures oftechniques. As such, they may transfer to other bacteria in vitro, we are largely ignorant of thebacteria, thereby increasing the genetic comple- factors which govern plasmid transfers in natu-

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MODELS OF E. COLI PLASMID TRANSFER IN VIVO 61

ral ecosystems such as soil, water, the gut, etc.Information obtained with pure cultures can beextended to a natural ecosystem only when thefollowing basic questions have been answered.

(i) Does a given ecosystem increase or reducethe fertility, i.e., increase or decrease the intrin-sic efficiency of bacteria to act as plasmid do-nors or recipients? It is known, for example, thatthe growth rate of bacteria affects their fertility.Fertility is certainly the most important parame-ter that must be determined to assess the likeli-hood of genetic transfer in various environ-ments. In the following mathematical analysisthe transfer rate constant (designated by theGreek letter gamma) will be used as a relativemeasure of fertility.

(ii) Given a certain intrinisic donor or recipientefficiency (fertility), in which way does the phys-ical structure of the ecosystem affect plasmidtransfer (that is, how does the rate of genetictransfer differ when the bacteria are spatiallyseparated in dilute suspension, when they are

present on surfaces as separate or mixed aggre-gates of potential donors and recipients, or whena "bolus" of potential donors passes over asurface on which potential recipients are aggre-gated)?We have analyzed some of the above ques-

tions for plasmid transfers in the gut by studyingthe fate of various plasmids after these are

ingested with an Escherichia coli donor strain.Their transfer to an E. coli recipient strain that isresident in the gut is then observed experimen-tally and analyzed mathematically. Gnotobioticmice or anaerobic continuous-flow (CF) culturesof natural or defined synthetic mouse intestinalflora were used in most of these studies. Asreported elsewhere, anaerobic CF cultures sim-ulate many of the interactions among indigenousand invading microorganisms that occur in themouse gut proper (R. Freter, H. Brickner, M.Botney, D. Cleven, and A. Aranki, Infect. Im-mun., in press; R. Freter, H. Brickner, J. Fe-kete, M. M. Vickerman, and K. E. Carey, In-fect. Immun., in press; R. Freter, E. Stauffer, D.Cleven, L. V. Holdeman, and W. E. C. Moore,Infect. Immun., in press).

It is generally agreed that plasmid transferoccurs readily in the gut of gnotobiotic animalswhich harbor only the donor and recipient bacte-ria involved in the reaction, but no indigenousmicroflora (9, 22, 34, 36, 37, 47), or in newbornor very young animals in which an indigenousmicroflora is either absent (46) or is not welldeveloped (40). In contrast, plasmid transfer inthe gut of weaned animals or of adult humanscould be demonstrated, if at all, only at extreme-ly low rates. For example, Smith (40) couldeasily demonstrate plasmid transfer in the intes-tines of 3-day-old chicks and 2-day-old calves,

but found such transfers much more difficult i

detect in 10- to 14-week-old pigs. Jarolmen andKemp (21) found R-factor transmission to be arare occurrence in weanling pigs. In humanfeeding experiments a number of authors (2, 4,41, 42, 47, 48) could not demonstrate any plas-mid transfer at all, whereas others (1, 39, 49)found only a low rate of transfer. This apparentinhibition of plasmid transfer in the adult intes-tine can easily be abolished by the administra-tion of antibiotics (e.g., 4, 10, 14, 19, 24, 25, 38,40). Starvation of the host can also promoteplasmid transfer in the gut (43), probably be-cause of the larger enterobacterial populationspresent in starving animals (reviewed in 16).As emphasized by Smith (40) and Sansonetti

et al. (37), among others, the natures of theplasmid and of the donor and recipient strainsare important determinants of in vitro as well asin vivo plasmid transfer. Nevertheless, interpre-tation of the literature clearly permits the follow-ing generalization to be made: very little or nodetectable plasmid transfer occurs in the normalgut populated by an undisturbed microflora. Incontrast, when the microflora is absent, as ingermfree or newborn animals, or when it isincomplete or disturbed, as in the very young orin antibiotic-treated individuals, then plasmidtransfer can be observed as readily as during invitro matings. Antibiotics, of course, can havethe additional effect of providing a selectiveecological advantage to transconjugants, there-by facilitating the detection of even a few conju-gative events that may have occurred in vivo.The reason why a normal intestine does not

readily permit plasmid transfer is not known.Anderson (3) reported studies implicating theBacteroides populations of the indigenous mi-croflora as inhibitors of in vivo plasmid transfer.In contrast, Duval-Iflah et al. (10) found no sucheffect of Bacteroides on plasmid transfer ingnotobiotic mice. Wiedemann (47) tested thefollowing mechanisms as potential inhibitors ofplasmid transfer in the gut: (i) the anaerobicconditions in the normal large intestine, (ii) theeffect of bile salts, (iii) metabolic products of theindigenous flora (especially acids), (iv) thegrowth phase of the donor and recipient bacte-ria, and (v) the efficiency with which potentialrecipients can accept a plasmid. He suggestedthat the inhibition observed in vivo may be aconsequence of the combined effects of all ofthese mechanisms. Falkow (13) concluded thatthe anaerobic conditions, low pH, and fattyacids produced by the indigenous microflora areprobably responsible for the diminished in vivotransfer of R factors. Burman (6) determinedthat anaerobiosis inhibited the conjugative trans-fer of some plasmids, but not that of others.These findings may explain contradictory state-

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62 FRETER, FRETER, AND BRICKNER

ments in the literature that anaerobiosis doesinhibit conjugation (15, 31) or that it does not (3,44).An obvious, but largely unknown, determi-

nant of plasmid ecology in the gut is the possiblenegative effect of plasmids on the in vivo growthrates of their host bacteria. Richmond (35) hasreviewed evidence suggesting that some plas-mids may reduce the in vivo growth of their hostbacteria, whereas others apparently do not. Du-val-Iflah et al. (11) found that some plasmidsmay indeed affect the fitness of their host bacte-ria in diassociated mice, but noted that other,difficult to define characteristics ("adaptation")of the bacteria were even more important deter-minants of successful colonization of the germ-free gut. There is suggestive evidence that thepresence of a plasmid may sometimes have theopposite effect as well, i.e., to increase the invivo fitness of a bacterium (e.g., 29, 49), even ifsuch a plasmid does not specify obvious viru-lence determinants (12). Another mechanismthat can contribute to the decline of transconju-gant populations is fragmentation or segregationof the plasmid, a phenomenon that has beennoted to occur in vivo (e.g., 48).

It is apparent, then, that the literature pro-vides no definitive answers to any of the above-mentioned basic questions. Specifically, we donot know whether in vivo-in vitro differences inthe rates of plasmid transfer are due to thephysiological characteristics of the gut environ-ment that may reduce the intrinsic efficiency ofplasmid transfer, whether they are due to thenature of the recipient strains present, whetherthey are due to adverse effects of plasmids onthe fitness of host bacteria, or whether thesequantitative effects result from patterns of spa-tial distribution of bacteria in the gut. Thesepatterns almost certainly differ from the densebacterial suspensions commonly studied in vi-tro. A resolution of these problems was difficultin the past, because no distinction could bemade in quantitative terms of the relative contri-butions (if any) of the various proposed mecha-nisms to the formation of transconjugant bacte-ria in the gut.Levin and co-workers (28) have reviewed the

scant literature of early attempts to put plasmidtransmission on a quantitative basis. They de-veloped a mathematical model, based on massaction kinetics, and determined the transfer rateconstants for various plasmids in mixed staticand CF cultures containing only the recipientand donor strains (28). Such cultures differ fromconventional mice or from CF cultures of indige-nous mouse flora in several ways: usually, in-oculation of the donor strain will not result in theformation of a constant donor population in themouse or CF culture, when an undisturbed

indigenous microflora is present. Rather, thedonors will be washed out rapidly at a rateequivalent to the flow rate of the CF culture(Freter, Brickner, Botney, et al.; and Freter,Stauffer, et al., Infect. Immun., in press). More-over, the population level of the recipient strainis considerably lower in the presence of indige-nous flora, such that plasmid transfer from do-nor to recipient is correspondingly slow. As aconsequence, plasmid transfer from transconju-gants to recipients (as opposed to transfer fromthe original donors to recipients) becomes animportant factor, whereas this process was oflittle importance in the cultures used by Levin etal. (28), such that these workers did not feel theneed to assign a distinct transfer rate constant tothe transconjugant-recipient reaction. Addition-al difficulties arise from the fact that E. coli K-12strains cannot be implanted into conventionalmice or CF cultures of indigenous mouse flora atpopulation levels that are sufficiently high forplasmid transfer to become apparent. To makematters worse, wild-type E. coli strains, whichwould implant somewhat better, are often ineffi-cient donors and recipients of plasmids.To overcome the above difficulties, we have

carried out our experiments in gnotobiotic miceor CF cultures which were first inoculated withthe E. coli K-12 recipient strain and subsequent-ly either with normal mouse gut contents or withthe 95 strictly anaerobic F-strains isolated fromthe cecal flora of normal mice (17). The F-strainsassume to a large extent the functions of acomplete indigenous gut flora (17), but permitsomewhat higher populations of E. coli K-12 toestablish in both gnotobiotic mice and CF cul-tures than would be the case in the presence of acomplete flora.

MATERIALS AND METHODSE. coli strains. The E. coli strains and plasmids used

in a number of the experiments described below wereobtained from Bruce Levin. They are the same asdescribed in Levin et al. (28). This allows a directcomparison between their experiments with plasmidtransfer in pure static or CF cultures and our dataobtained in the presence of indigenous flora. Thedonor strain was J53, and the recipient strain was analidixic-acid resistant mutant of CSH50, here desig-nated as strain 50N. In addition, strain chi 1776 wastested. This is the "subordinated" E. coli strain devel-oped for recombinant DNA work (7). It was obtainedfrom the Office of Recombinant DNA Activities, Na-tional Institutes of Health. Strain chi 1665 is the parentstrain of chi 1776, and chi 1666 is a chromosomalmutant of chi 1665 that is resistant to nalidixic acid.These strains were obtained through the courtesy ofRoy Curtiss. Strain 1665rs is a chromosomal strepto-mycin-resistant mutant of chi 1665, selected in thislaboratory by means of the gradient plate technique(without the use of mutagens). These strains will bereferred to below as "1665," "1666," "1665rs," and

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"1776," respectively. E. coli C25 is a streptomycin-resistant mutant of a wild-type strain originally isolat-ed from a food handler on routine stool culture. It hasbeen described in earlier publications (17).

Plasmids. Plasmids Rl and Rldrd-19 were obtainedfrom Bruce Levin. They are the same as used in hisstudy (28). In addition, pBR322 was studied as anexample of a nonconjugative plasmid that is used inrecombinant DNA work. This plasmid shows an un-usually low efficiency of mobilization by conjugativeplasmids (8). It was obtained from the Office ofRecombinant DNA Activities, National Institutes ofHealth.

Indigenous microflora. The 95 strictly anaerobic F-strains are described elsewhere (17). They have beenmaintained for several years in gnotobiotic micehoused in germfree isolators. Inoculation of culturesor experimental animals was made via cecal contentsfrom these stock mice (it would otherwise take about 6months to reliably implant an inoculum of 95 strictanaerobes).CF cultures. CF cultures were in enriched veal

infusion broth, as described elsewhere (Freter,Brickner, Botney, et al.; Freter, Brickner, Fekete, etal.; and Freter, Stauffer, et al., Infect. Immun., inpress). The entire apparatus was maintained in ananaerobic chamber (Freter, Brickner, Botney, et al.;Freter, Brickner, Fekete, et al.; and Freter, Stauffer,et al., Infect. Immun., in press). The flow rate was 1/6h-1 (i.e., fresh broth was added to the culture at a rateequivalent to displace 1/6 culture volume per hour).The culture of 7-ml volume was contained in a roundtube of approximately 12-mm inner diameter. It wasstirred at 800 rpm by means of a Teflon-coveredstirring bar of 7-mm length and 2-mm diameter. Be-cause of the small size of the stirring bar, this stirringaction was just sufficient to cause a barely noticeablemovement of the medium at its surface.

In a typical experiment, the recipient E. coli strainwas inoculated first. Two days later the culture wasinoculated with cecal contents of a conventionalmouse (BALB/cwm) or of a gnotobiotic mouse carry-ing the 95 F-strains. The culture was allowed toequilibrate for at least 2 weeks before inoculation ofthe donor strain. The purpose of introducing the E.coli strain first, before inoculating the indigenousflora, was to insure a stable E. coli population atequilibrium (18; Freter, Brickner, Fekete, et al., In-fect. Immun., in press). Donors, recipients, and trans-conjugants were differentiated on culture by plating onmedia containing the appropriate antibiotics. In a fewexperiments unusually high initial donor concentra-tions (>109/ml) had to be used to obtain quantifiablerates of conjugation (e.g., with strain 1776). In suchinstances, transconjugant colonies could sometimes bedetected during the first 1 to 3 h of the experiment onplates inoculated with undiluted CF culture, but neveron plates inoculated with a 1:10 dilution. Transconju-gant colonies would then be absent in later platecultures and reappear after 6 to 48 h. In these in-stances, the early transconjugant colonies were con-sidered to have originated from matings on the platesrather than in the CF culture. In most experiments, notransconjugants were found on plate cultures preparedimmedately after inoculation of the donors. In experi-ments involving cultures of conventional mouse flora,the population of indigenous E. coli strains was usually

suppressed by the initial presence of high numbers ofrecipient E. coli. For this reason, and because of thelow fertility of wild-type E. coli (cf. Results), these arenot likely to affect the rates of conjugation in ourexperiments.To avoid possible errors from insufficient mixing or

from delayed expression of antibiotic resistance innewly formed transconjugants, only transconjugantcounts obtained later than 1 h after inoculation of thedonor strain were considered in the modeling proce-dures.

Experiments involving pure CF cultures of donorand recipient E. coli strains (i.e., in the absence ofindigenous gut flora) were conducted in the samemanner in the anaerobic chamber, except that a 1:20dilution of the enriched veal infusion broth mediumwas used, to maintain the E. coli populations at similarlevels as in the presence of gut flora.Experiments in mice. The conventional mice were

strain BALB/cwm, maintained in this Department byW. Murphy. These have never been subjected togermfree conditions, unlike commercial mice whichare derived from germfree stock and therefore harboran entirely abnoral intestinal flora. Germfree micewere originally obtained from Charles River BreedingLaboratories, Inc (strain CD-1) and have been bredsubsequently in thks laboratory. They were maintainedin Trexler-type plastic isolators. Inoculation was di-rectly into the stomach of 0.5 ml of bacterial suspen-sion in 3.75% NaHCO3 buffer. The buffer was keptunder a CO2 atmosphere until the moment of inocula-tion.The "effluent" (= stools) from mice was collected

while the animals were housed in sterilized singlecages with wire mesh bottoms suspended over arefrigerated cage pan which contained 30 ml of sterilebroth at a temperature of approximately 4°C. E. coliK-12 remains viable under these conditions for at least24 h (18). Preliminary studies also established that noplasmid transfer takes place under these conditions.At the end of the sampling period the accumulatedstools were homogenized, and the total number oftransconjugants, etc., was determined by quantitativeculture.Agarose gel electrophoresis for the detection of plas-

mids. The method of Kado and Liu (23) was used fordetection of plasmids. The presence or absence ofplasmids inferred from the antibiotic susceptibilitydata was verified by this method on isolates fromrepresentative experiments of each kind described inthis paper.

RESULTSShort-term experiments in CF cultures. We

have extended the mathematical model of Levinet al. (28) to take into account the transientpresence of the donor population and the factthat significant transfer of plasmids to recipientstakes place from both the original donors andnewly formed transconjugants. Our initial modelmakes the following assumptions: (i) matingoccurs at random with a frequency that is pro-portional to the concentrations of plasmid-freeand plasmid-bearing cells, (ii) plasmid loss bysegregation occurs at a negligible rate, (iii) there

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is no significant delay between the time a trans-conjugant receives the plasmid and the timewhen it can begin to transfer it, (iv) the forma-tion of transconjugants does not significantlyreduce the number of recipients (i.e., this modelbecomes invalid and experiments have to beterminated when the concentration of transcon-jugants approaches that of the recipients), (v)the CF culture consists of a homogeneous sus-pension of the various bacteria, and (vi) recipi-ents and transconjugants multiply at the samerate (which equals the flow rate of the CFculture, resulting in constant E. coli populationsizes in the absence of conjugation).A CF culture containing a constant population

ofN potential recipients per milliliter is inoculat-ed to obtain an initial concentration of N+(o) permilliliter of the donor strain. The concentrationof the donor N+ decreases with the flow rateconstant p (hour-1) of the culture (this assumesno actual multiplication of the donor strain).Thus

= -p N+ (la)

where a dot indicates differentiation with respectto time, and

N+ = N+(o)e-PT (lb)

Formation of transconjugants occurs via twoconjugational reactions: recipient with donorand recipient with a transconjugant formed earli-er. If N* denotes the concentration of transcon-jugants per milliliter in the system, then

N = y1N+N + Y2N*N + (4i* - p)N* (2)where qj* is the rate constant of multiplication ofthe transconjugants. The term (4* - p) is as-sumed to be zero because the donors, like therecipients from which they derived, are assumedto multiply at a rate equal to the flow rate of theculture and therefore to have a zero net rate ofgrowth. The important parameters to be deter-mined are -Yi and Y2 (milliliters per cell per hour),the conjugational transfer rate constants of thereactions between donor-recipient and transcon-jugant-recipient, respectively. These constantsmay be regarded as measuring the relative effi-ciency of infectious transmission (i.e., the fertil-ity) of a given host-plasmid combination underthe conditions imposed by the habitat in whichthe experiment is conducted.An explicit solution for the concentration of

transconjugants in the CF culture has been ob-tained:

'yi N N+(o)x

[(qo - p) + yzN]T e(-p- ]

Since determination of this variable in vivowould require the sacrifice of an animal, deter-mination of the rise of transconjugants over timein a given animal would be impossible. We havetherefore integrated equation 3 to obtain thetotal number of transconjugants N.' in the efflu-ent accumulating between the times (1) of a andb:

T =aN*' ( To a = C [e(Y2Ib) - e(Y2Na)] +

'\T= bi Y2N

CA [e(-pb)_ e(-pa)]p

(4)

where A denotes the flow rate of the culture inmilliliters per hour and

C = yI N N+(o)Y2 N + p

Least-square, best-fit estimates for the trans-fer rate constants -Yl and Y2 were calculated fromexperimental data by a computer subroutinewhich calculates the differences between thelogarithms of the actual and calculated datapoints of N. (= transconjugant populations) andwhich varies the parameters of interest (e.g., thetransfer rate constants) until the sum of thesquares of these differences reaches a minimum.This subroutine ("Praxis") uses the principalaxis method (5).The experiment shown in Fig. 1 involved

sampling from the contents of the growth tube aswell as sampling of accumulated effluent. (Theeffluent was collected in a refrigerated beakersuch that no growth or death of bacteria and noconjugative events took place after the effluentleft the CF culture.) Each horizontal segment ofthe stepped curve, representing transconjugantsper milliliter in accumulated effluent, resultedfrom one sampling period. The transfer rateconstants calculated from the two sets of dataare shown in Table 1 for two experiments of thistype. As may be seen, the two sampling methodscorrelated well. This finding is important for theexperiments with mice described below, be-cause sampling from mice was done via accumu-lated stools.Table 2 gives the best-fit calculated values for

the transfer rate constants -Yl and Y2 for all CFculture experiments. All calculations in this ta-ble used equation 3. Because of the limitationsof this initial version of our mathematical model,only the early data from each experiment wereevaluated, i.e., until the transconjugant popula-tions reached 5% of the population size of thepotential recipients, an event which usually oc-

(*- P) + Y2N + p

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N* =


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-j

cr 8.0LLJa-

F 6.0

, 4.0

m

z, 2.00

k<DONORS.+ i

el RECIPIENTS

-~~~~~~~~~~~~~~~---~~~~~~~_TRANSCONJUGANTS

C)) IN SAMPLE FROM

)_

0*

1-_J

GROWTH TUBE

IN ACCUMULATEDEFFLUENT (ACTUAL)

IN ACCUMULATEDEFFLUENT (CALCULATED)

10 20 30 40 50 60

HOURS AFTER FEEDING THE DONOR STRAIN

FIG. 1. Transfer of plasmid Rl in CF culture ofmouse indigenous microflora (F-strains; experiment 3,Table 2). Symbols denote the experimental datapoints. The continuous (solid and dotted) lines arecalculated by equation 3 and represent best-fit instan-taneous concentrations of bacteria in the growth tube.The horizontal portions of the stepped lines indicatethe sampling periods; their position corresponds to theactual concentrations of transconjugants in the accu-mulated effluent (solid stepped line) or the correspond-ing best-fit values calculated by equation 4 (dashedstepped line).

curred after 30 to 100 h. As may be seen bycomparing experiments 3 to 14 with experiments19 to 23, the rate constant for transfer fromdonor to recipient (-Yl) was higher for the dere-pressed mutant plasmid Rldrd-19 than for theparent plasmid Rl, as would be expected (P =

0.0039 by Student's t test). It is noteworthy thatthe order of magnitude of the lYl transfer rateconstant for plasmid Rl is similar to that report-ed by Levin et al. (28) for this same plasmid inthose of their experiments where donor andrecipient were in the stationary phase. Thiswould be expected if the nonmultiplying state ofthe donor in our experiments were analogousphysiologically to that of the stationary phase ofa pure culture. In contrast, the best-fit calculat-ed values of Y2, the rate constant for transferfrom transconjugants to recipients, are of thesame order of magnitude (ca. 10-9) as calculatedby Levin et al. (28) for the permanently dere-pressed plasmid Rldrd-19 but are much higherthan those calculated by them for the Rl plasmid(ca. 10-12 in their exponentially growing staticcultures and ca. 10-14 in their CF cultures). Thisis not surprising since the transfer function ofthe Rl plasmid would be expected to be dere-pressed in the newly formed transconjugantspresent in our experiments (30) and therefore

should resemble that of plasmid Rldrd-19. OurY2 rate constants do indeed resemble those re-ported by Levin et al. (28) for plasmid Rldrd-19growing exponentially in static broth (ca. 10-9).The CF cultures used in this study were

stirred slowly to simulate the shear forces andmixing effects that presumably are generated invivo by the peristaltic movements of the gut.The influence of stirring on plasmid transfer wasevaluated in the absence of gut microflora. Asterile CF culture was first inoculated with therecipient E. coli strain and, 2 days later, with thedonor strain. Stirring was either absent or at theusual rate. Inspection of experiments 1 and 2and 15 to 18 (Table 2) shows that the rate ofstirring used in the present experiments slightlyreduced the Yi rate constant which describesplasmid transfer from the original donors, butthat it had no noticeable effect on the subsequentplasmid transfer from transconjugants (that isdescribed by the Y2 rate constant).Comparison of transfer rates in pure CF cul-

tures with those in CF cultures populated withindigenous microflora indicates that the latterhad no demonstrable effect on the transfer of theRl plasmid from donor J53 to recipient 5ON (cf.experiment 1 with experiments 3 to 14, Table 2)or on the transfer of the Rldrd-19 plasmid fromdonor 1776 to recipient 1665rs (cf. experiments31 and 32 with experiments 33 to 39, Table 2). Incontrast, the transfer of plasmid Rldrd-19 fromdonor J53 to recipient 5ON appeared to be some-what impaired by the presence of indigenousmicroflora, as evidenced by a yl rate constantthat is approximately 2 orders of magnitudelower in experiments 19 to 23 than in experi-ments 15 and 16 (Table 2). The latter -yl rateconstants (experiments 15 to 18, Table 2) are ofthe same order of magnitude as reported byLevin et al. (28) for the same plasmid, donor,and recipient combination in pure CF cultures ofminimal synthetic medium (ca. 10-11).The data in Table 2 further show that the

"safe" E. coli strain 1776 was considerably lessefficient as a donor of plasmid Rldrd-19 (experi-ments 31 to 39) than its precursor strain 1666(experiments 24 to 30), as evidenced by a -Yl rateconstant that is 2 to 3 orders of magnitude lower.

TABLE 1. Effectiveness of equations 3 and 4 forfitting transfer rate constants

mi/cells-hExpt Equation

'Yi 'Y2

3 3 6.3 x 10-16 8.4 x 10-94 6.1 x 10-16 6.8 x 10-9

4 3 7.9 x 10-15 3.1 x 10-94 1.1 x 10-14 1.6 x 10-9

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TABLE 2. Parameters of plasmid transfer in CF cultures (calculated by equation 3)

ml/cells-hExpt Culture

'Y I 'Y2DoorJ3(1)rciiet:_ODonor: J53(Rl); recipient: SON

1

2

34S67891011121314

Donor: J53(Rldrd-19);recipient: SON15

16

17

18

1920212223

Donor: 1666(Rldrd-19);recipient: 1665rs24252627282930

Donor: 1776(Rldrd-19);recipient: 1665rs31

32

33343536373839

Pure broth,stirred

Pure broth,not stirred

F-strainsF-strainsF-strainsF-strainsF-strainsF-strainsF-strainsF-strainsF-strainsCecal floraCecal floraCecal flora

Pure broth,stirred

Pure broth,stirred



F-strainsF-strainsCecal floraCecal floraCecal flora

F-strainsF-strainsF-strainsF-strainsF-strainsCecal floraCecal flora

Pure broth,stirred

Pure broth,stirred

F-strainsF-strainsF-strainsF-strainsCecal floraCecal floraCecal flora

1.2 x 10-14

9.2 x 10-14

6.3 x 10-167.9 x 10-"52.6 x 10-148.0 x 10-151.8 x 10-147.1 x 10-155.8 x 10-145.2 x 10-148.6 x 10-141.5 X 10-151.9 X 10-154.8 x 10-15

1.1 x 101-

1.2 x 10-11

1.0 x 101o

2.0 x 10`'

1.7 x 10-138.3 x 10-W34.1 x 10-142.8x10-x32.3 x 10-13

1.7 x1.8 x5.4 x1.1 x2.6 x1.1 x4.1 x

1.6 x 10-8

2.6 x 10-8

8.4 x 10-93.1 x 10-91.3 x 10-81.0 x 10-89.6 x 10-92.3 x 10-81.6 x 10-87.6 x 10-91.0 x 10-84.5 x 10-81.4 x 10-71.8x10-x

2.7 x 10-8

1.3 x 10-8

1.4 x 10-8

1.2x 10-9

7.4 x 10-91.3 x 10-81.7 x 1-76.8 x 10-84.8 x 10-7

8.4 x 10-81.4x 10-91.4 x 10-91.2x 10-98.3 x 10-91.1 X 10-84.4 x 10-7

1.4 x 10-8

4.6 x 10-9

2.8 x 10-88.2 x 10-81.9 X 10-81.3 x 10-81.2 x 10-72.6 x 10-73.5 x 10-7

10-130- 12

10-1310-13l- 12

l- 12

l- 13

6.3 x 10-16

2.0 x 10-16

9.1 X 10-161.3 x 10-141.8 x 10-165.1 x 10-171.1 X 10-142.6 x 10-152.3 x 10-15

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TABLE 2-Continuedml/cells-h

Expt CultureYi Y2

Donor:1665(Rldrd-19+pBR322);recipient: 50N (transfer ofplasmid Rldrd-19)40 F-strains 1.0 x 10's 6.7 x 10-941 F-strains 4.4 x 10-'3 5.0 x 10-9

Donor:1666(Rldrd-19+pBR322);recipient: 1665rs (transfer ofplasmid Rldrd-19)42 F-strains 7.0 x 1015 5.0 x 10`143 F-strains 6.2 x 10-17 4.3 x 10-9

Donor:1665(Rldrd-19+pBR322);recipient: 5ON (transfer ofplasmid pBR322)40 F-strains 2.9 x 10-17 Uncertain41 F-strains 4.1 x 10-17 Uncertain

Donor:1666(Rldrd-19+pBR322);recipient: 1665rs (transfer ofplasmid pBR322)42 F-strains 9.3 x 10-17 Uncertain43 F-strains 3.2 x 1018 Uncertain

Donor: 1666(Rldrd-19);recipient:1665rs(pBR322+R1-fragment)44 F-strains 1.5 x 10-1o 4.3 x 10-945 F-strains 2.0 x 10-10 Uncertain46 F-strains 6.6 x 10-11 Uncertain47 F-strains 1.5 x 10"1 7.0 x 10-948 F-strains 8.3 x 10-1i 6.8 x 10-9

Donor: J53(Rldrd-19);recipient: 50N(pBR322+Rl-fragment)49 F-strains 6.5 x 10-12 Uncertain50 F-strains 1.6 x 10-1 Uncertain

Donor: J53(Rldrd-19);recipient: 5ON(pBR322)51 F-strains 3.1 x 10-12 5.4 x 10-952 F-strains 2.6 x 10-11 5.5 x10-9

Donor:1666(Rldrd-19);recipient: C2553 Pure broth, 7.2 x 10-16 2.3 x 10-

stirred54 Pure broth, 8.9 x 10-17 4.1 x 10-11

stirred

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The Y2 rate constant is, of course, similar in bothsets of experiments, because it applies to trans-fers among transconjugants and potential recipi-ents of strain 1665rs.

Wild-type E. coli strain C25 was a very poorplasmid recipient (Table 2, experiments 53 and54), with transfer rate constants that were sever-al orders of magnitude below those of the K-12strains tested.The remaining experiments in Table 2 concern

the mobilization and cotransfer of the nonconju-gative plasmid pBR322. The rate constant yl fortransfer of plasmid Rldrd-19 from a donor whichalso harbors pBR322 was much lower than thatof the same donor harboring only plasmidRldrd-19 (cf. experiments 42 and 43 with experi-ments 24 to 30). Note that there are two entriesfor each of these experiments, one giving datafor transfer of plasmid Rldrd-19 and the othergiving data for transfer of both plasmids. The Y2constant in experiments 42 and 43 is not unusu-ally low, of course, because it refers almostexclusively to the transfer from strain1665rs(Rldrd-19) to strain 1665rs. Cotransfer ofpBR322 was still less efficient than transfer ofRldrd-19 from a donor harboring both plasmids(experiments 40 and 41, and 42 and 43, Table 2).No Y2 constant could be assigned to cotransferof pBR322, because no demonstrable plasmidtransfer occurred beyond the initial formation oftransconjugants during the first few hours ofthese experiments. There was no evidence fortransfer of the mobilized pBR322 alone, i.e.,without concomitant transfer of Rldrd-19.

In view of the fact just discussed, i.e., that thepresence of pBR322 in the donor strain reducedthe efficiency of plasmid transfer, it was surpris-ing to find enhanced transfer of plasmid Rldrd-19 when the recipient carried pBR322 plus afragment of Rldrd-19 (cf. the -lYj constants inexperiments 44 to 48, and 49 and 50 with those inexperiments 24 to 30 and 19 to 23, respectively,Table 2). Plasmid transfer was so rapid, in fact,that most recipients had received a Rldrd-19plasmid very early in the experiment, such thatthe calculations of the Y2 constants in experi-ments 44 to 50 are quite uncertain. The recipientstrains used in these experiments were segre-gants of 50N(pBR322+Rldrd-19) and1665rs(pBR322+Rldrd-19), which still carried afragment of the Rldrd-19 plasmid that had lostthe antibiotic resistance and incompatibilityfunctions, but could be detected by agaroseelectrophoresis. Further experiments weretherefore carried out to determine whether theincreased fertility could also be observed in thepresence of pBR322 alone. For this purposestrain 5ON(pBR322) was constructed by intro-ducing the plasmid via Ca2+ transformation bythe method of Olsen et al. (32). The data (Table

2, experiments 51 and 52) show similarly en-hanced fertility as in the preceding experiments.In contrast, E. coli 5ON carrying only the Rl drd-19 fragment was unchanged in its fertility (datanot shown). One must conclude, therefore, thatplasmid pBR322 significantly enhanced the abili-ty of this E. coli to receive plasmid Rldrd-19.

Long-term experiments in CF cultures. Aspointed out earlier, the mathematical model pre-sented above in equations 1 through 4 becomesinvalid when the population density of transcon-jugants approaches that of the potential recipi-ents. One intriguing feature of long-term experi-ments was the fact that plasmid transfer oftendid not go to completion. Rather, the popula-tions of transconjugants and potential recipientsremained at constant levels for several weeksuntil the end of an experiment. When potentialrecipients isolated late in those experimentswere grown in static broth culture and mixedwith a static broth culture of the donor strain,they were consistently found to be normallysusceptible to transfer of the plasmid, and noplasmids or plasmid fragments could be detectedby agarose gel electrophoresis. We have evalu-ated several possible explanations for this phe-nomenon by extending the equations describedabove to include the following variables.

(i) The first variable was loss of recipients, asthese were converted to transconjugants. Thenumber of transconjugants (N.) formed by con-jugation was therefore subtracted from the pop-ulation of potential recipients (N). The changesin the population sizes of donors (N+), potentialrecipients, and transconjugants are then ex-pressed as follows:

N = N(* - p) - y1N+N - y2N*N (5a)

N* = N*(4s* - p) + ,yN+N + Y2N*N

N+ = N+(*+ - p)

(Sb)

(5c)(ii) A further refinement was to model the loss

of transconjugants by segregation or by a de-crease in the growth rate of the transconjugants,relative to the recipients. The effect of plasmidsegregation was introduced into the equationsby a factor, "Z," that is applied to the growthrate J* of N. (i.e., of the transconjugants), withO < Z < 1. The corresponding increase in po-tential recipients due to this process is then:N.*+ (1 -Z).Use of a factor Z to model plasmid segregation

differs from the approach of Levin and Stewart(27) in that their segregation rate (Xr) is indepen-dent of bacterial multiplication (i.e., plasmids inresting bacteria would segregate at the same rateas those in multiplying hosts). In contrast, segre-gation described by the factor Z is a function ofthe growth rate constant tp*.

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Equations 5a and 5b become then:N = N(* - p) - yiN+N - y2N*N +

N4*. (1 - Z) (6a)

N*= N(*Z - p) + -y,N+N + y2N*N (6b)

Equations 6a and 6b indicate that the effect ofplasmid segregation is equivalent to a reducedgrowth rate of the transconjugants (N.) relativeto the plasmid-free host bacterium (N). In otherwords the introduction offactorZ into equations6a and 6b is equivalent to setting + > ip. in such away that the sum ofN + N* remains consistentwith the experimental data. For this reason, thismodel cannot distinguish between the effects ofplasmid segregation and a detrimental effect ofthe plasmid on the growth rate of its host bacte-rium.

(iii) Another variable was introduced on theassumption that some potential recipients mightnot be immediately accessible to potential do-nors by virtue of being sequestered in a refuge(e.g., in colonies among other bacteria that areadherent to the wall of the gut or the culturevessel). This was modeled by subtracting thesheltered population R (cells per milliliter) fromthe population of potential recipients (N) inequations 5a and Sb.

Equations 5a and 5b become then:

N= N(j - p) - -yN+(N - R) - y2N*(N - R)

(7a)= N*(l* - p) - ly N+(N - R) - Y2N*(N - R)

(7b)

It is assumed, further, that an increasing num-ber of the sheltered colonies of potential recipi-ents will in time become exposed at the mucosalsurface, where they may acquire a plasmid thatis subsequently transmitted to all members ofthe colony. This process reduces the number ofpotential recipients that remain in sheltered sta-tus. Consequently, the size of shelter R wasreduced at the rate D (hour-'), such that:

R=-DR (8)(iv) A final relation introduced into the equa-

tions was to model the repression of transferfunction that is known to occur with plasmid Rlbut not with plasmid Rldrd-19 (30). An approxi-mate simulation of this phenomenon was accom-plished by switching the transfer rate constant Y2to a lower value, designated Y(2 repr.), at a certainnumber of hours (i.e., at the time "SWITCH")after the start of the experiment.The differential equations involved in the

above models are too complex to yield explicitsolutions. Computer simulation was thereforeused to calculate best-fit transfer rate constants,

as well as estimates for the additional variablesfrom the experimental data. The computer pro-gram was given distinct values of the growth rateconstants 4, qI+, and i* for every time intervalbetween experimental data points. These valueswere chosen such that the calculated sum ofpotential recipients plus transconjugant popula-tions (N + N.) and the calculated populations ofdonor cells (N+) would precisely match thecorresponding experimental values. Calculatinggrowth rate constants for the sum of N + N*assumes, of course, that their respective growthrates (*, and +*) are equal. Formally, then, wehave assumed that:

4 =+* = 4'I (9)The growth rate constant *, was calculated

from the experimental data such that the ob-served dynamics of the total (recipient-plus-transconjugant) E. coli population N, were de-scribed by:

N, = (,t - p) N, (10)

This approach was followed in all instances,including with equations 6a and 6b. In the lattertwo equations, differences in growth rates oftransconjugants and recipients (or changes dueto segregation) are introduced by the factor Z.Consequently, in all equations except 6a and 6b,changes in the relative population sizes ofN andN* are assumed to be consequences of conjuga-tion, rather than due to differences in growthrates. As in the earlier model, the subroutinePraxis was used to calculate best-fit values forthe parameters to be estimated by minimizingthe sum of the squared differences between thelogarithms of the actual and calculated datapoints. The root mean square of these errorsums is shown as the "RMS-error" in Tables 3and 5. These errors were calculated and com-bined for both N (population of potential recipi-ents) and N. (transconjugant populations).A total of four computer programs, each in-

corporating one of the above parameters, werethus developed and were used to evaluate thedata from all long-term experiments. These pro-grams are the Basic Program in which only -yland Y2 are estimated, and no parameters areintroduced for segregation, shelter, or repressedfertility, (uses equations 5a, Sb, and Sc); Segre-gation Program, which incorporates a factor Z tomodel plasmid segregation or a reduced growthrate of plasmid-containing E. coli cells or both,and estimates Yl, Y2, and Z (uses equations 6a,6b, and Sc); Shelter Program, which incorporatesvariable R and parameter D (described above) forshelter size and shelter decrease, respectively,and estimates yl, Y2, D, and the initial size of R(uses equations 7a, 7b, 5c, and 8); and RepressedFertility Program, which switches to a lower

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z 7.0

w0-

J 5.0C)cnm

O 3.00z

0o 1.0Ij

0 240 480 720 960HOURS AFTER FEEDING THE DONOR STRAIN

FIG. 2. Transfer of plasmid Ri in a CF culture ofmouse indigenous microflora (F-strains). Calculationsare by the Basic Program (experiment 8, Table 3).

transfer rate constant, Y(2 repr.)9 at time SWITCHand estimates Yl, Y2, Y(2 repr.), and SWITCH (usesequations 5a, 5b, and 5c).

Figure 2 presents data from a typical experi-ment (no. 8). As may be seen, the basic mathe-matical model matched the observed populationlevels of transconjugants and potential recipi-ents reasonably well during the early stages ofthe experiment (i.e., up to 30 to 40 h). It wasunable, however, to accommodate the subse-quent persistence of potential recipients, suchthat the calculated population of potential recipi-ents actually disappeared. In other words, thebasic mathematical model predicted that all po-tential recipients should eventually receive aplasmid, whereas this was not the case experi-mentally (Fig. 2). When the same data weremodeled with the Repressed Fertility Program, amuch better match between experimental andcalculated data was obtained (Fig. 3).

Figure 4 shows the same data as Fig. 2 and 3,except that modeling was by the SegregationProgram. Here again there was an improvementin fit over the Basic Program (Fig. 2), but it wasinferior to that achieved with the RepressedFertility Program (Fig. 3). The data of thisparticular experiment were modeled only poorlyby the Shelter Program (not shown).Table 3 presents results obtained with all

experiments that were continued for at least 14days. The numbers of the experiments corre-spond to those shown in Table 2. The data wereevaluated with all four of the above programs.As may be seen, in the first six experimentslisted, which involved plasmid Rl, the best fitwas consistently obtained with the RepressedFertility Program. This is evident from the factthat the RMS-error of calculated versus experi-

mental data points, which was computed asdescribed above, was consistently lower, by afactor of approximately 2, than the RMS-error ofthe program giving the second best fit. Thisvalue is expressed in Table 3 by the ratio inwhich the RMS-error of the best-fitting programother than the Repressed Fertility Program isdivided by that of the Repressed Fertility Pro-gram. This finding does not rule out the possibil-ity that the other factors modeled here alsocontributed in some measure to the persistenceof plasmid-free potential recipients. It is obvi-ous, however, that our model points to repres-sion of fertility as being the most important one.The Repressed Fertility Program was not su-

perior in modeling the remainder of the experi-ments listed in Table 3, in which transfer of thepermanently derepressed plasmid Rldrd-19 wasevaluated. This is apparent from the fact that theorder of magnitude of the ratios of RMS-errorsin Table 3 was 1. This finding lends somecredence to the validity of the mathematicalmodeling procedures adopted in this study inthat these were clearly able to identify a feature,namely, the repression of fertility, that is knownto be a characteristic of plasmid Rl and not ofplasmid Rldrd-19.

It is not possible, on the basis of the aboveexperiments alone, to clearly identify the rea-son(s) why plasmid Rldrd-19 did not enter allpotential recipient E. coli in the long-term ex-periments, because no single one among theRepressed Fertility, Shelter, or Segregation Pro-grams was distinctly superior in modeling theexperimental observations, although each ofthese gave a better fit than the Basic Program(Table 3). Progress along these lines was made,however, by distinguishing the effect of plasmidsegregation from that of a retardation of the

-J

tr 7.0bJ0-

ui 5.00mLL-o 3.00zc-o 1.0IJ


FIG. 3. Same experiment as in Fig. 2. Calculationsare by the Repressed Fertility Program.

TRANSCONJ. (colc)

e-r44 A + '--1,.0+__

\eto O = ~~TRANSCONJ.(exptl.)

+ = RECIPIENTS(exptl.)

RECIPIENTS DONORS(calc.)

RECIPIENTS (colc.)

-, - ---

}\_ziW,,o2>-_, (c0 = TRANSCONJ., 0 TRANSCONJ. (exptl.)

(calc.)+ = RECIPIENTS

4 "I." (exptl.)

? DONORS

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CY 7.00-

LJ 5.0C-)

mLA-0 3.00z

CDo 1.0

Ij

RECIPIENTS (colc.)

+ - CI~~~~~~~~I

IO a)

\ a) 0c = TRANSCONJ.(exptl.)

+ = RECIPIENTS(exptl.)

TRANSCONJ.(colc.) CF~3-- DONORS


FIG. 4. Same experiment as in Fig. 2 and 3. Calcu-lations are by the Segregation Program.

bacterial growth rate by the presence of theplasmid (see below).

In the experiment illustrated in Fig. 5, strain5ON(Rldrd-19) was grown in a CF culture of F-strains. No other E. coli strains were intro-duced. Cultures were made periodically to enu-merate the E. coli resistant to kanamycin andchloramphenicol (resistances specified by theplasmid) and strains lacking these resistancemarkers. E. coli lacking resistance to theseantibiotics were first found at approximately 700h, but not before that time (Fig. 5). [There is noculture method that can select the plasmid-freestrain 5ON against 5ON(Rldrd-19). Consequent-ly, small populations of segregants which musthave been present in the early stages of theexperiment could not be enumerated experimen-tally.] The data show, therefore, that the popula-tion of sensitive bacteria must have been consid-erably below that of the resistant ones in theearly stages of the experiment. In modelingthese data, the assumption was made that allantibiotic-sensitive bacteria detected in this ex-periment were segregants that had lost all, or atleast portions, of the plasmid. (For reasons ofconvenience, these are designated as "plasmid-free bacteria" below and in the figures. Agarosegel electrophoresis revealed that most of thesestill retained a fragment of the original plasmidthat had lost the immunity functions, in additionto losing the antibiotic resistance markers.)A first attempt at modeling these data was

made by using the Segregation Program (i.e.,equations 6a, 6b, and 5c). When the segregationfactor Z is set sufficiently high to yield plasmid-free E. coli populations that approach the ex-perimental values, the curve of the calculatedpopulation of plasmid-bearing bacteria (dashedline in Fig. 5) considerably exceeds the late

experimental points. Significantly, the equilibriathus calculated are established very early, incontrast to the experimental observations. Forthese reasons, segregation alone cannot explainthe late appearance of antibiotic-sensitive segre-gants in this experiment.

Considerably improved results were achievedby modeling the growth rate of the plasmid-bearing and plasmid-free strains as a function ofthe concentration of a limiting substrate, usingthe classical Monod equation. We are assuminghere that a very small number of antibiotic-sensitive bacteria are formed continuously as aresult of segregation or fragmentation of theRldrd-19 plasmid. These segregants are as-sumed to have a slight advantage in growth rateover the parent strain by virtue of a slightlylower saturation constant. The equations aregiven by Powell (33; equation 9), whose notationwe have followed here. Accordingly, equations6a and 6b above were used with modifications asfollows.The value of *, in equation 6a is defined as

(11)in

= ln S][Ks(n) + s]-

and 4* in equation 6b is defined as

* = - +S_Ks(n*) + s_

(12)

with Any and pUt* being the maximum growth rates(at saturation level of the limiting substrate) ofthe plasmid-free (N) and the plasmid-bearing(N.) populations, respectively. The constantsKs(n) and Ks(n*) represent the saturation con-stants for N and N*, respectively (which arenumerically equal to the substrate concentrationwhere half the maximum growth rates are ob-tained). Since there are no plasmid donors,equations 6a and 6b become:

N = N(M - p) - 'y2N*N + N*%* (1 - Z) (6aa)N = N* (Z- p) + 'y2N*N (6bb)

The concentration of the limiting substrate isdesignated s. This concentration is described bythe following equation (33; equation 10):

s = P(Sr - s)- N N-y*'I - - (13)

with p being the flow rate of the culture, Y beingthe yield constant of the limiting substrate (i.e.,weight of bacteria formed divided by the weightof the limiting substrate used), and ij and 4*being as defined in equations 11 and 12, respec-tively. In Powell's classical treatment of contin-uous culture, the constant Sr represents the

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PLASMID-BERRING SEGR.FRCT=.950-8.0 + __

Cl- +~~~~_

L 6.0 ID 0DC: PLRSMIO-FREE BRCT.m SEGR.FRCT.=.950

CD

oD 4.0 ...............................................................................

(z) PLRSMIO-FREE BRCT. SEGR.FRCT.=.9999CD

0 400 800 1200 1600HOURS RFTER FEEDING THE PLRSMID-BERRING STRRIN

FIG. 5. Appearance of segregants in a CF culture of F-strains inoculated with E. coli 5ON(Rldrd-19).Modeled by the Segregation Program. The transfer rate constant Y2 was set to 10-9 mlIcells-h, and thesegregation factor Z = 0.9990. Data points: circles, plasmid-bearing bacteria; +, plasmid-free bacteria.

concentration of the limiting substrate in theculture reservoir. In our cultures and in the miceit is meant to represent that portion of thelimiting substrate which is not utilized by thepredominant anaerobic microflora (i.e., it repre-sents the concentration of this substrate thatwould establish itself in the absence of any E.coli populations).

Figure 6 shows the results obtained with thismodel. The numerical values used for IL,, R,*,K2, and Y are of the same order of magnitude asdetermined by Herbert et al. (20) for Aerobactercloacae in a chemically defined medium. Thetransfer rate constant Y2 was set to 10-9. Theremaining numerical values were estimated bytrial and error. As may be seen (Fig. 6), a goodfit of the experimental data can be obtained withthis model.

It appears, therefore, that the major factorresponsible for the observed late appearance ofsegregants was a difference in growth rates, withsegregation contributing only in a minor way byfurnishing a small initial population of plasmid-free bacteria. By analogy, one must consequent-ly conclude that in the experiments involvingplasmid Rldrd-19 in Table 3, the "segregationfactor" Z also represents for the most partdifferences in growth rates rather than segrega-tion.

Figure 7 shows a similar experiment in mice,where E. coli 5ON(Rldrd-19) was introducedinto animals harboring the F-strains as theirindigenous microflora. As may be seen, the dataobtained are similar to those shown for ananalogous CF culture in Fig. 6. The model usedwas the same as in the preceding experiment. It

L 8.0a-LC

> 6.0 - --

CDPLRSMI O-FREE PLRSM ID-BEAR ING

co 4.0 BRCTERI R BRCTERI R

I 0 400 800 1200 1600HOURS RFTER FEEDING THE PLRSMID-BERRING STRAIN

FIG. 6. Same experiment and same symbols as in Fig. 5. Modeled by assuming that the growth rates ofplasmid-free and plasmid-bearing bacteria are controlled by the same limiting nutrient. The following numericalvalueswere used: p = 0.16667 h-'; I2 = 10-9mI/cells-h;Z= 0.999990; Y= 0.5g(dryweight) ofbacteriapergofnutrient used; S, = 4.122 x 10-2 g/liter; Fn, and Fn, = 1.66 h-1; Kss5Y, = 7.5 x 10-4 g/liter; K.,(n, =1l0-3 g/liter.One gram (dry weight) of bacteria was assumed to contain 1.8 x 101 cells. The population of plasmid-bearing E.coli at time zero was assumed to be 3.7 x 107/ml, with no plasmid-free bacteria present initially.

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/ LASMID- (D '\\ CD/BEARING"BACTERIA NOWAL

2.0~~ ~ ~ ~~ GROWTH

0 400 800 1200 1600HOURS AFTER FEEDING THE PLASMID-BEARING STRAIN

FIG. 7. Appearance of segregants in gnotobiotic mice (harboring the F-strains), after inoculation with E. coli50N(Rldrd-19). Symbols as in Fig. 5. Each symbol represents 10 mice. The following numerical values wereused: p = 0.23 h-1; Y2 = 1i-0 ml/cells-h; Z = 0.999990; Y = 0.5 g (dry weight) of bacteria per g of nutrient used;S, = 2.08756 x 1i-0 g/liter; pU, and p.,,. = 1.66 h-1; KS(,,) = 9.1 x 10- g/liter; KS(,,.) = 10- g/liter; w = 250 (per mlof gut volume). One gram (dry weight) of bacteria was assumed to contain 1.8 x 1012 cells. The population ofplasmid-bearing E. coli at time zero was assumed to be 1.7 x 106/ml, with no plasmid-free bacteria presentinitially.

is apparent that the late appearance of plasmid-free bacteria is modeled satisfactorily. In con-trast, the late leveling off of the plasmid-bearingpopulation cannot be accounted for on the solebasis of a reduced growth rate (dashed linelabeled "no wall growth"). For this reason, afurther parameter, w, was introduced to repre-sent a small part of the initial population ofplasmid-bearing bacteria which is embedded inthe mucosa of the large intestine. It is assumedthat this mucosa-associated population re-mained constant throughout the experiment,that it multiplied at the same rate as the nonat-tached plasmid-bearing bacteria, and that alldaughter cells were shed into the lumen. Thus,whereas the term that expresses the multiplica-tion of the population N. of plasmid-bearingbacteria remains unchanged, the population thatis eliminated at the flow rate (p) of the gut isreduced by w bacteria. Equation 6bb becomesthen:

N. = Ni.Z - (N. - w)p + Y2N.N (6bbb)with *. as defined in equation 12. Since the rateof segregation assumed in this model is very low(Z = 0.99999) in relation to the number ofadherent bacteria (250 per ml of gut volume), theprobability for one segregant to appear amongthe few adherent bacteria is very low, and thispossibility has therefore been neglected in themodel. As is apparent, a good fit of the experi-mental points was achieved in this manner (Fig.7, line labeled "with wall growth").

In summary, then, the results presented inFig. 5 to 7 show that a low degree of plasmidsegregation must have been present to furnish asmall initial population of segregants. Neverthe-less, the dynamics of plasmid segregation aresuch that this mechanism can be ruled out as thesole cause for the replacement of plasmid-bear-ing bacteria by segregants. On the other hand, amodel assuming an inferior growth rate and asmall attached portion of the population of plas-mid-bearing bacteria is entirely consistent withthe experimental observations. The assumptionof an attached population was unnecessary inthe in vitro experiment shown in Fig. 6, proba-bly because the experiment had not been ex-tended long enough to reach a leveling off in thedecline of the plasmid-bearing population.The above-described findings of segregants in

cultures of strain 50N(Rldrd-19) were surprisingto us, because we had never observed segrega-tion of this plasmid when the E. coli hosts weregrown in the usual pure laboratory cultures.Apparently, the populations of plasmid-bearingE. coli in the usual laboratory media are suffi-ciently high to quickly redeem any loss of aplasmid by subsequent conjugational transfer ofa new plasmid (45). In contrast, the populationdensities of E. coli in most natural environmentsand in our laboratory models of the gut are toolow to permit rates of conjugation that aresufficiently rapid to offset plasmid loss by segre-gation or by a reduction in the growth rate ofplasmid-bearing E. coli. These phenomena

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TABLE 4. Rates of elimination via the feces of the nonabsorbable tracer 51CrC13 by various types of micea

Time (h) A = Fraction I = Fraction Cc=Animals ~~~No. of ~ trof tracer oinclmCalculatedAnimals animals after remaining in of inoculum rate of

infection gUtb recovered elimination(

Conventional mice 39 4.0 0.7318 0.8007 -0.23063 18.0 0.0293 0.7758

Mice with F-strains 10 4.o 0.820 0.9213 -0.21321.0 0.022

Gnotobiotic mice 10 4.0 0.905 0.6910 -0.078(experiment 5, Table 5) 18.5 0.292

Mice given morphine 10 5.75 0.999 1.046 -0.0283(experiment 2, Table 5) 24.5 0.587 -0.0932

47.0 0.721a Inocula ranged between 1 x 106 and 5 x 106 cpm of 51CrCI3 per mouse, given directly into the stomach.bI = [Inoculum x recovery rate (I) - actual cumulative recovery in stool]/[inoculum x recovery rate (I)].c p = [In (fraction remaining in gut at time b) - In (fraction remaining in gut at time a)]/(b - a).

therefore can be important factors in the naturalpopulation biology of plasmids, even when theyare not readily observed in the usual pure cul-tures commonly used for laboratory studies ofplasmid transfer.No segregation was noted when experiments

similar to those shown in Fig. 5 to 7 were carriedout with strain 5ON carrying plasmid Rl. Appar-ently, the rate of segregation was too low or thedetrimental effect of the plasmid on the growthrate of this E. coli strain was too minor, or both,to generate detectable populations of segre-

gants, even when the experiments were contin-ued for periods of up to 5 months.

Plasmid transfer in mice. The excretion rate ofstool by the mouse was determined as: A = pVml/h, where p (hour-1) is the flow rate constantfor intestinal contents through the gut and V isthe volume of intestinal space that is available toE. coli (which was arbitrarily set to 0.5 ml). Thevalues of p were calculated for the various kindsof mice on the basis of experiments involving theexcretion of nonabsorbable radioactive tracer(Cr51CI3) by the animals (Table 4). This calcula-tion implies, of course, a perfect mixing ofintestinal contents. Whereas this is probably notcompletely realized in vivo, preliminary studiesinvolving the mixing of 51CrCI3 into the hamstercecum show that such mixing may indeed bevery rapid (K. Wilson, personal communica-tion).Mice were first associated with the recipient

E. coli strain plus, where indicated, with the F-strains as a defined indigenous microflora. Asdescribed in Materials and Methods, feces werecollected during certain sampling periods. Eval-uation of these data was by equation 4. Inaddition, equations- 5a to 5c, i.e., the BasicProgram, and 6a, 6b, and 5c, i.e., the Segrega-

tion Program, were used. To apply the latter twosets of equations to data from mouse experi-ments, the simulation program was modified tocalculate the numbers of recipient and transcon-jugant bacteria excreted during each samplingperiod. Equation 4 and the Basic Program gavesimilar results, because the latter differs fromthe former mainly in reducing the populationsize of potential recipients by the number ofnewly formed transconjugants. This correctionbecomes significant only when the size of thetransconjugant population approaches that ofthe recipients, a condition which was neverrealized in our mouse experiments.The data obtained in the various experiments

are presented in Fig. 8 and in Table 5. Figure 8shows that the transconjugants originally formedslowly decreased in number. Consequently, theBasic Program was unable to match the experi-mental data points. In contrast, the SegregationProgram accomplished an excellent fit of theexperimental data, indicating that segregation ordifferences in growth rates between plasmid-bearing and plasmid-free bacteria were determi-nants of the size of the transconjugant popula-tions. As discussed earlier, the latter is the morelikely explanation. A superior fit by the Segrega-tion Program was observed in two of the experi-ments of this type (experiments 3 and 4, Table5), as evidenced by a significantly lower RMS-error. There was no difference in the fit achievedby the two programs in experiment 1 (Table 5)simply because that first experiment had notbeen continued long enough to show a decreasein the population of transconjugants. In experi-ments 3 and 4 the animals were kept in germfreeisolators to rule out the possibility that theobserved shifts in the E. coli populations mighthave been caused by antagonism from environ-

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MODELS OF E. COLI PLASMID TRANSFER IN VIVO

zLLJ

zC)

:DLO

FIG. 8. Plasmidrepresents 10 mice.

7.0

-j

r-5.0

CE

cr 3.0

m

z: 1.0

CD

0 400 800 1200 1600 2000HOURS RFTER FEEDING THE DONOR STRRIN

transfer in mice harboring F-strains. Data of experiment 3, Table 5. Each data point

mental bacteria that implanted in the animalsduring the course of the experiment.As discussed above, numerous laboratories

have reported that plasmid transfer occurs readi-ly in germfree animals associated only with thereacting E. coli strains. Experiment 5 (Table 5)was therefore designed to test this. Germfreemice were associated with the recipient strain50N. Only a small inoculum of the donor strainJ53(Rldrd-19) was subsequently introduced di-

rectly into the stomach. The donor strain multi-plied rapidly in these animals, rather than beingeliminated as in mice harboring an indigenousmicroflora. As reported in the literature, largepopulations of transconjugants (ca. 108 per ml ofgut content) appeared rapidly. However, the yland 'Y2 constants in this experiment were nodifferent from those calculated for the otherslisted in Table 5. One must conclude, therefore,that the intrinsic ability for plasmid transfer was

TABLE 5. Plasmid transfer in the gut of mice

Basic Program Segregation ProgramE No. of Length of Donor and Rmrs RMS-error RMS-error

Expt mice expt (h) recipient strains Remarro SegregationNoofLengthofDonor and Remarks ~ ~ ~ 'YifactorY2 Y

1 4 312 J53(Rldrd-19) F-strains 0.2855 0.28585ON 1.19 x 10-13 1.18 x 10-'3 0.9998

a 5.72 x 10-9

2 10 868 J53(Rldrd-19) F-strains, 0.5587 0.44065ON morphine at 7.45 x 10-15 1.75 x 10-14 0.9868

0.5 and 8 h 4.44 x 10-14

3 10 2,256 J53(Rldrd-19) F-strains, in 1.095 0.44225ON isolator 2.93 x 1i-13 7.99 x 10-13 0.9729

1.19 x 1o-9

4 10 504 1666(Rldrd-19) F-strains, in 0.6774 0.14821665rs isolator 1.87 x 10-13 1.15 x 10-13 0.8187

1.07 x 10-8

5 5 573 J53(Rldrd-19) Diassociated, 0.6249 0.21605ON in isolator 1.12 x 10-12 2.49 x 10-12 0.8350

1.42 x 10-8a ., Not determined.

E- DONORS RECIPIENTS

k(/TRRNSC. (BASIC PROGR.)

* $ _~~~~-- -- -- --_-_-_- _-_-_-_

*,-'TRRNSC..SEGR.PROG,.)

-3 TRANSCONJUGANTS, SAMPLING PERIOD

+- = RECIPIENTS. SAMPLING PERIOD

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not appreciably increased by the germfree statusand that the large populations of transconjugantsobserved in experiment 5 were simply a conse-quence of the large recipient and donor E. colipopulations present in animals that harbor nocompeting microflora.Experiment 2 (Table 5) was designed to deter-

mine whether a decrease in intestinal motilitythat would retard the elimination of transconju-gants would have a stimulating effect on plasmidtransfer. These animals were given intraperito-neal injections of morphine at 30 min and 8 hafter feeding the donor strain. This regimensignificantly reduced intestinal transit for at least2 days (Table 4). Contrary to expectation, trans-fer was severely impaired (as evidenced by thelow transfer rate constants; Table 5), which in alllikelihood was a consequence of the nearly staticgrowth conditions in the gut of the morphine-treated mice. Levin et al. (28) have shown thatplasmid transfer is impaired among E. coli instatic cultures.

It is interesting to note that the transfer rateconstants calculated for the mouse experimentswere of the same order of magnitude as those forthe analogous CF cultures. The mathematicalmodel which we use at present presumes thatthe interactions among the bacteria take place ina well-mixed suspension. This may be the caseto some extent (though not precisely) in our CFcultures, but probably less so in the animal, inwhich a significant fraction of E. coli presum-ably reside in microcolonies on the gut wall,where only one or a few cells at the surface ofthe colony can react when the plasmid donor E.coli pass through the lumen of the gut. This islikely to result in an underestimate of fertility inthe gut. (We have developed a mathematicalmodel which evaluates the resulting error morerigorously. This will be described in a futurepublication. Interested readers may obtain apreprint from the senior author.) If one thenconsiders (i) that the values for the yl and _)2constants calculated for mice are already of thesame order of magnitude as those for CF cul-tures and (ii) that the former values are not likelyto be underestimates (because that would implythat plasmid transfer in the gut is actually moreefficient than in broth culture, an unlikely possi-bility), one must conclude that the efficiency ofin vivo plasmid transfer did not differ from thatoccurring in vitro and that any observed pecu-liarities of in vivo plasmid transfer were refer-able to the population densities of E. coli in thegut.

Modeling of a triparental mating. One of thevalues of mathematical models lies in their abili-ty to predict with some accuracy the outcome ofinteractions that cannot be tested experimental-ly. The following discussion offers an example of

such predictions, namely, the probability oftransfer in the gut of nonconjugative plasmidsfrom one E. coli host to another. The major partof the experimental work described in this paperwas concerned with the transfer of a very effi-cient conjugative plasmid. Plasmids used inrecombinant DNA cloning experiments and,probably, many plasmids that carry determi-nants for virulence factors (12) are nonconjuga-tive; i.e., they cannot spontaneously promoteconjugation to permit their entry into a new hostbacterium. Such plasmids must first be mobi-lized by the entry of a conjugative plasmid intothe host bacterium, followed by conjugativetransfer to a new, plasmid-free recipient bacteri-um (triparental cross). Thus, two successiveconjugative events are required to bring aboutthe transfer of a nonconjugative plasmid to anew host bacterium. In view of the above data,one can assume with some certainty that theprobability for triparental transfer of a nonconju-gative plasmid in the normal gut will be smalland can be observed experimentally, if at all,only with great difficulty.We have studied experimentally and evaluat-

ed mathematically the two separate steps of atriparental cross, namely, (i) the transfer of aconjugative plasmid (Rldrd-19) from its host(which may enter the gut as a transient) to aresident bacterium carrying the nonconjugativeplasmid pBR322 (experiments 44 to 50, Table 2)and (ii) the transfer of pBR322 from the resultanttransconjugant (i.e., E. coli harboring both plas-mids) to a plasmid-free recipient E. coli that isalso resident in the gut (experiments 40 to 43,Table 2). The entire sequence of a triparentalcross has been modeled by computer simulation.As noted by Levin and Rice (26), the interac-tions among the potential plasmid donors andrecipients are by no means limited to thosetransfers that are commonly emphasized be-cause they lead directly to the eventual transferof the nonconjugative plasmid. On the contrary,numerous other plasmid transfers occur simulta-neously, and the effects of these may have aprofound influence on the final outcome. This isillustrated schematically in Fig. 9. The mathe-matical details of the model and the valuesassigned to the different parameters and varia-bles are given in the Appendix.

Figure 10 shows the results of modeling asituation where a donor strain is ingested by ahost that harbors an E. coli strain with thenonconjugative plasmid pBR322 plus another E.coli strain that is plasmid-free. As may be seen,the donor population (N+) disappears rapidly,because the model assumes that an invading E.coli is unable to colonize the normal gut. Most ofthe plasmid-free population (M) that conjugateat all receive only plasmid Rldrd-19 (to become

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FIG. 9. Reactions during tnparental matings. The various bacterial populations which participate in thereactions are shown within boxes. Arrows leading from one box to another indicate transfer of a recipientbacterium to a transconjugant population which takes place when a plasmid is acquired. For each such arrowthere is a second arrow which approximates the shaft of the first and which originates at the box showing therelevant plasmid donor population. The matings began when a mammalian host which harbored a population ofplasmid-free E. coli (M) and an E. coli population carrying plasmid pBR322 [N(pBR322)], ingested an E. colicarrying the conjugative plasmid Rldrd-19 [population N+(Rldrd-19)]. The other bacterial populations shownarose as a consequence of subsequent in vivo plasmid transfers. Symbols and transfer rates are explained in theAppendix.

-J

LUJ0La: 5.0cr

Cr- 3.0m

CD

Z 1.0CD

IJ

0 200 400 600 800HOURS RFTER FEEDING THE DONOR STRRIN

FIG. 10. Simulation of triparental transfer ofpBR322 mobilized by plasmid Rldrd-19. Assuminginteractions as observed with E. coli K-12 strains, i.e.,strains of high fertility. The values given the variousconstants are listed in the Appendix. The symbolsrefer to the following populations: N = resident E. coliwith pBR322; N, = donor E. coli with plasmid Rldrd-19 (strain is being ingested at time zero); N. = initialtransconjugant which carries both plasmids; M =

plasmid-free resident E. coli; M. = transconjugant,originally from M population, carries plasmid Rldrd-19; M.* = transconjugant, originally from M popula-tion, which carries both plasmids, the end result of thetriparental mating.

population M.). Likewise, most of the originalresident strain population with pBR322 (N) alsoreceive plasmid Rldrd-19 (to become populationN*, which carries both plasmids and whichconstitutes the secondary donor to transmit oneor both plasmids to members of population M).Plasmid transfer to population N is relativelyrapid, because the residence of pBR322 in theseE. coli is assumed to increase their efficiency asrecipients for Rldrd-19 (cf. experiments 44 to50, Table 2). Only very few of the plasmid-freepopulation M receive both plasmids, to becomepopulation M**, which is the ultimate product ofthe triparental cross. Extension of the model to100,000 h showed that this population will reacha level of 6.6/ml at about 1,000 h and will remainconstant thereafter (see also Table 6).

It should be noted that the actual valuesassigned to the variables in the above calcula-tions are typical of those obtained in the presentstudy with the partially rough E. coli K-12strains, which are exquisitely efficient donorsand recipients in contrast to the wild-type E. coliC25 tested in this study. Other smooth strains ofwild-type E. coli described in the literature werealso less efficient donors than their rough mu-tants (47). For this reason, the same simulation

I N

l*- ~~MN*

.

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TABLE 6. Effect of various parameters on efficiency of triparental transfersFinal recipients of pBR322 (populations M.. + G)

E. coli type Segregation Size atfactor 100,000 h Dynamics of the population(per ml)

Wild-type 1.Oa 14.9 Population rising continuouslyWild-type 0.9999 3.75 Population rising continuouslyWild-type 0.9990 0.190 Population constant after 20,000 hWild-type 0.9729 0.007 Population constant after 3,000 h

K-12 1.Oa 6.6 Population constant after 1,000 hK-12 0.9999 16.8 Population rising continuouslyK-12 0.9990 116 Population rising continuouslyK-12 0.9729 5.66 Population constant after 8,000 h

a A segregation factor of 1 specifies no segregation. Values of other parameters as in Fig. 13 or 14,respectively.

was repeated with values for the different pa-rameters as previously determined for E. coliC25. The calculated data show a steady rise inthe population (M..) of originally plasmid-freeE. coli that have acquired pBR322 (Fig. 11).Surprisingly, the population of wild-type E. coliwith newly acquired pBR322 (M..) will eventu-ally exceed that obtained in the analogous modelof the highly conjugation-efficient K-12-typestrain (note the different time scales in Fig. 10and 11).

This apparent paradox of a poorly conjugatingwild-type E. coli that can be more effective thanits highly fertile counterpart in acquiring a non-conjugative plasmid by triparental transfer iseasily understood by inspection of Fig. 10 and11. The low effectiveness of triparental transferamong K-12-type E. coli is a consequence of themodel's prediction that the conjugative plasmid

-J

cr 3.0L)CL

o 1.00

z

0-n -1.0

N * M RND N

,,-1

........i............. ....................

..K


FIG. 11. Simulation of triparental transfer ofpBR322 mobilized by plasmid Rldrd-19. This simula-tion differs from that in Fig. 10 in that the presence ofwild-type E. coli strains of low fertility (such as strainC25) is assumed. Symbols as in Fig. 10. The valuesgiven the various constants are listed in the Appendix.

Rldrd-19 will quickly enter most of the potentialrecipient E. coli populations (M and N), whichthereby become populations M. and N. that areassumed in our model to be immune to subse-quent cotransfers of pBR322 with Rldrd-19.Consequently, transfer of the latter plasmid isinefficient and ceases rapidly (Fig. 10). In con-trast, the model predicts that the low efficiencyof conjugation of the wild-type strain insures acontinuing large population of potential recipi-ents (M), such that transfer of pBR322 cancontinue for very long periods of time. In viewof this, one might expect that phenomena suchas segregation which increase the pool of plas-mid-free E. coli would tend to increase triparen-tal transfer of the nonconjugative plasmid amongK-12-type strains, but not among wild-type E.coli such as strain C25. Indeed, extension of oursimulations to include various rates of segrega-tion (Table 6) shows this to be the case. Con-trary to first intuition, the model therefore indi-cates that wild-type E. coli strains are actuallywell tuned to the task of promoting the spread ofcertain nonconjugative plasmids among theirpopulations, precisely because of their low in-trinsic efficiency for conjugative plasmid trans-fer. This is not only apparent for the spread ofnonconjugative plasmids, as discussed above,but must hold true also for at least some conju-gative plasmids, simply because a low intrinsicefficiency for plasmid transfer assures the pres-ence of a large plasmid-free population that canacquire additional plasmids. This conclusion isobvious for plasmids that are mutually exclu-sive, but it would also hold when E. coli hoststhat harbor more than one type of plasmid havedecreased fitness.

DISCUSSIONThe present study in mice was made possible

by our technique of implanting E. coli K-12strains of high fertility into gnotobiotic animals

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carrying a synthetic indigenous microflorawhich resembles in its function the indigenousflora of the mouse large intestine (17) and by theuse of anaerobic CF cultures of indigenous largeintestinal microflora which simulate the bacteri-al interactions observed in the mouse gut(Freter, Brickner, Botney, et al.; Freter,Brickner, Fekete, et al.; and Freter, Stauffer, etal., Infect. Immun., in press). It thereby becamepossible to conduct experiments under condi-tions closely resembling those found in nature,which nevertheless resulted in the generation ofa sufficiently large number of transconjugants topermit accurate quantitative measurements.The mathematical models described in this

paper inspire some confidence in their validitybecause they are able to reproduce a number ofknown phenomena. Among these are the repres-sion of fertility of the Rl plasmid and the ab-sence of this phenomenon in experiments withplasmid Rldrd-19. In addition, known differ-ences in the transmissibility and mobilization ofthe plasmids studied were reproduced correctly.The models also evaluated correctly the rela-tionship between growth rate of the donor bacte-rium and its fertility, which had been noted byseveral earlier workers (e.g., 28, 47). The pres-ent study has shown that this phenomenon is ofconsiderable importance to in vivo plasmidtransfer because it is responsible for the lowfertility of a potential donor strain which, afteringestion, passes through the intestine, but isprevented from multiplying by the antagonisticeffect of the indigenous microflora. Disregard ofthis and other quantitative aspects has causedsome earlier investigators to make erroneousextrapolations from in vitro data as to the num-ber of transconjugants that could be expected toresult from experimental in vivo matings. Forexample, a finding that "transfer occurred inbroth culture to a frequency of 2 x 1O-3 perdonor organism per hour" (4) does not justify anexpectation of finding the same ratio of donorsto transconjugants after in vivo matings, be-cause donors would be nonmultiplying andtherefore considerably less fertile in vivo. More-over, even iffertility were unimpaired, the lowerpopulation densities in vivo would result inconsiderably lower transfer frequencies per do-nor than could be expected in fully grown brothcultures.

Evaluation of the experimental data by themathematical models led to a number of conclu-sions which could not have been reached other-wise. Several of these were quite unexpected.Most surprising perhaps is the finding that plas-mid transfer in mice harboring an indigenousintestinal microflora (F-strains) had the samedegree of efficiency as in the CF cultures ofindigenous intestinal microflora. The same was

true in the gnotobiotic mice which harbored onlythe donor and recipient E. coli strains. Contraryto the intuitive assumptions of earlier workers,our results indicate that the low rates of in vivoplasmid transfer can be explained solely on thebasis of the quantitative aspects of in vivobacterial growth and do not necessarily implythe presence of direct inhibitors of conjugation.This finding does not rule out the possibility thatsome of the factors described by earlier workersmay indeed reduce the in vivo fertility of someplasmids or bacterial hosts. For example, plas-mid Rldrd-19 was somewhat less fertile in CFcultures of indigenous microflora than in cul-tures harboring only the donor and recipient E.coli (Table 2). In contrast, the Rl plasmid ap-peared to be unaffected in its fertility by thepresence or absence of other flora. Be this as itmay, our mathematical models show clearly thatthe low rates of in vivo plasmid transfer ob-served by numerous earlier workers must beexpected even in the absence of hypotheticalinhibitors of conjugation.

In speculating about the applicability of our invivo data to humans, one must consider that thebasic rules of microecology in the mouse largeintestine (16) are likely to hold for the largeintestine of humans as well. It is thus quite likelythat certain E. coli strains may be more efficientin colonizing one host species than another (e.g.,because of specific adhesive mechanisms). Nev-ertheless, beyond such specific differences, onemay expect that the principles that govern plas-mid transfer in the mouse intestine also havetheir counterparts in humans. This conclusioncan be substantiated by modeling plasmid trans-fer experiments in the human gut that have beenreported by others and testing the predictions ofour model against the published experimentaldata. Several of these earlier reports containsufficient quantitative details to permit suchreconstruction. Figure 12 reproduces the datapoints of a graph (Fig. 5 in reference 1) describ-ing a human volunteer experiment. The lines inFig. 12 were generated by our segregation pro-gram, using the parameters listed in the legend.The maximum sensitivity of this author's culturemethod was 10 transconjugant E. coli per g ofstool, and he therefore detected transconjugantsonly on the first day after feeding the donorstrain, when transconjugant populations reachedthat density. It should be noted that the Yl and'Y2 transfer rate constants used in modeling arethose found in our CF cultures for a wild-type E.coli recipient (Table 2, experiment 53) and thatfactor Z is in the order of magnitude calculatedfor our mice (Table 5) and CF cultures (Table 3).A similar reconstruction of the data for volun-teer "M" described by Wiedemann et al. (48)shows that less than one transconjugant E. coli

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7.0. '/

7.0 \ RECIPIENTSW5.0

'__ 3.0 DONORS

U-)

CD

CD 1.0

CD

-1.0 TRANS

CONJUGANTS

0 40 80 120 160

HOURS

FIG. 12. Modeling of plasmid transfer in the human

gut, using data published by E. S. Anderson (refer-

ence 1, Fig. 5). The Segregation Program was used in

this reconstruction with the following parameters: p =

-0.23 h-1 -yl- 7.2 x 10-16 ml/cells-h; _Y2 = 3.0 x

10-11 ml/cells-h; Z = 0.94; initial population of recipi-

ent E. coli = 4.0 x iO5 per ml of gut volume; initial

population of donor E. coli = 108 per ml of gut volume

(this assumes that the initial inoculum of donors was

diluted into 100 ml of gut volume).

per ml of stool could be expected at 24 h and a

rapid decline of the transconjugant populationthereafter (not shown). The maximum sensitiv-

ity of the authors' culture method was 10 bacte-

ria per g of feces, and none of the transconju-

gants were therefore detected in their study.

Volunteer "M" was chosen for this reconstruc-

tion because he had the highest level of potentialE. ccli recipients in the stools and therefore had

the highest predictable rate of in vivo conjuga-tion. The quantitative aspects of the data pub-

lished by Anderson et al. (4) were similar to

those of Wiedemann and co-workers, and recon-

struction by our model also predicted low levels,of transconjugants. These authors stated that the

sensitivity of their culture method was lo to 10

organisms per g of feces.

In summary, the reconstructions just de-

scribed are consistent with the conclusion thatthe quantitative aspects of fertility and plasmidtransfer in the human gut are similar to those inour mice and CF cultures. It appears that, evenin the absence of selection, plasmid transferoccurs consistently in the human gut, but thatthe resulting transconjugant populations are toosmall to be detected regularly with the culturemethods used by earlier investigators.Other somewhat unexpected findings made

during the present study include the higher fertil-ity of recipients carrying plasmid pBR322. Themathematical model also made it possible todistinguish between the effects of the plasmid onthe growth rate of its host bacterium versusplasmid segregation as possible factors responsi-ble for the continued presence of plasmid-freebacteria among populations of highly fertile plas-mid-bearing potential donors.

Quite surprising also were the predictions ofthe mathematical model for triparental matings,which showed that there are situations in naturewhere the artificially high fertility of rough labo-ratory strains, such as E. coli K-12, wouldactually be detrimental to long-term mainte-nance and spread of plasmids. This effect mayseem small when calculated for a single humanbeing. However, if one views the entire commu-nity of mammalian hosts, the relatively low levelof fertility of wild-type E. coli strains may actu-ally be optimal for maintaining an ecologicalbalance among numerous plasmids, which canbe called upon to supply specialized functions ifand when changing environmental conditions sodemand. The calculated data also show thatplasmid segregation, a phenomenon that onetends to regard generally as detrimental to thesurvival of plasmids in nature, may in certaincircumstances do just the opposite, namely,promote the spread and persistence of a varietyof plasmids in bacterial hosts of relatively highfertility.Most of the interactions described in this

paper were consequences of relatively smalldifferences in fertility, growth rates, rates ofsegregation, etc., among the interacting bacteriaand their plasmids. Many of these could nothave been detected by merely observing theresults of short-term experiments. Nevertheless,these small differences often were decisive de-terminants of the ultimate fate of plasmids andtheir hosts. Consequently, most of the moreinteresting conclusions drawn from the presentstudy could not have been reached withoutprecise mathematical interpretation of long-termexperiments. It seems likely that similar ecologi-cal problems, such as the interactions among thevarious bacteria comprising the indigenous mi-croflora (16), may require a similar combinationof experimental and mathematical analysis

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(Freter, Brickner, Fekete, et al., Infect. Im-mun., in press).

ACKNOWLEDGMENTS

This study was supported by Public Health Service contractNO1-AI-62518 and grant 1 RO1 Al 17154 from the NationalInstitutes of Health.We are grateful to Douglas Rabert for carrying out most of

the agarose electrophoresis procedures and to Ronald H.Olsen for transformation of pBR322 into E. coli 50N. Wethank Garth W. Jones, Ronald H. Olsen, and Michael A.Savageau for advice and fruitful discussions.

APPENDIX

The following equations were simulated. They de-scribe each of the possible reactions that may occurduring a triparental mating, as illustrated schematical-ly in Fig. 9. These equations differ from those of Levinand co-workers (26, 27) in that they permit differenttransfer rate constants to be assigned to the variousconjugational reactions. Also, our treatment of segre-gation differs from theirs, as discussed above in rela-tion to equations 6a and 6b.

N = N(4i - p) - y1N+N - Y2N.N - Y3M.N -

'2M.-N + N.(1 - Z)qi Wi+ = -pN4 (ii)

N. = N.(qiZ - p) + y1N+N + Y2N.N +y3M*N + Y2M.*N (iii)

M= M(I - p) - -Y4N+M - y2N*M - y5M*M -

y6NVM - yWN"M - Y2M.*M + M*(1 - Z) (iv)

M = MO(.Z - p) + -Y4N+M + Y2N*M +

Y5M*M + Y2M**M (v)

M** = M**(qZ- p) +yN*M + yM.*M (vi)

G = G(4i- p) + M**(l - Z)qj (vii)The definitions of these variables are given below.

Appended in parentheses are the initial values (at timezero) assigned in calculating the data shown in Fig. 10,followed by the values used in calculating triparentaltransfer among wild-type recipients such as strain C25(Fig. 11), where these are different.

N = Resident E. coli with pBR322 (5 x 105/ml).N+ = Donor E. coli with plasmid Rldrd-19 (5 x 109/

ml). This strain is assumed to be ingested attime zero.

N. = Initial transconjugant, carries both plasmids(zero).

M = Resident E. coli, plasmid-free, final recipient (5x 105/ml).

M. = Transconjugant, originally from "M" popula-tion, carries plasmid Rldrd-19 (zero).

M.* = Transconjugant, originally from "M" popula-tion, carries both plasmids (zero).

G = Segregant of M.. population, carries onlypBR322 (zero).

yl = Rate constant for the transfer of plasmidRldrd-19 from a transient (i.e., nonmultiplying)donor strain to a resident recipient strain thatcarries pBR322 (7.5 x 10-11 ml/cells-h), (7.5 x10- 14).

Y2 = Rate constant for the transfer of plasmidRldrd-19 from a resident donor that carriesboth plasmids to a resident plasmid-free recipi-

ent or to a recipient carrying pBR322 (5 x 10-9ml/cells-h), (3 x 10-11).

3 = Rate constant for the transfer of plasmidRldrd-19 from a resident donor that carriesonly this plasmid to a resident recipient thatcarries pBR322 (5 x 10' ml/cells-h), (3 x10-11).

4 = Rate constant for the transfer of plasmidRldrd-19 from a transient (nonmultiplying) do-nor to a resident recipient without a plasmid (5x 10"- ml/cells-h), (2.5 x 10-16).

_Y5 = Rate constant for the transfer of plasmidRldrd-19 from a resident donor that carriesonly this plasmid to a plasmid-free residentrecipient (5 x 10-9 ml/cells-h), (3 x 10-11).

_Y6 = Rate constant for the transfer of both plasmidsfrom a resident donor to a plasmid-free recipi-ent (1013 ml/cells-h), (1013).

= Growth rate constant for all resident E. coli(0.23 h-1).

p = Rate constant of flow through the gut (0.23h-1).

Z = Segregation factor for loss of plasmid Rldrd-19(1.0), (1.0). A value of 1 indicates no segrega-tion. Note that Table 6 shows data calculatedwith other values for Z.

The transient donor strain was assumed to be non-multiplying in the gut, and its population decayedtherefore with the flow rate of the gut contents.The values assigned to the above variables reflect

population levels typical of resident E. coli in humansand in experimental mice. The amount of donor strainingested (to make an initial population of 5 x 109/ml)represents a very large dose in humans. The transferrate constants assigned reflect the orders of magnitudedetermined for similar reactions in CF cultures andmice (Tables 2 and 3).

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Experimental and Mathematical Escherichia PlasmidTransfer ... · Experimental andMathematical Models ofEscherichia coli PlasmidTransferIn Vitro andIn Vivo ROLFFRETER,*ROLFR. FRETER,ANDHOWARDBRICKNER