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Estimating the transfer rates of bacterial plasmids with an adapted Luria–Delbrück fluctuation analysis [1]

['Olivia Kosterlitz', 'Biology Department', 'University Of Washington', 'Seattle', 'Washington', 'United States Of America', 'Beacon Center For The Study Of Evolution In Action', 'East Lansing', 'Michigan', 'Adamaris Muñiz Tirado']

Date: 2022-08 

To increase our basic understanding of the ecology and evolution of conjugative plasmids, we need reliable estimates of their rate of transfer between bacterial cells. Current assays to measure transfer rate are based on deterministic modeling frameworks. However, some cell numbers in these assays can be very small, making estimates that rely on these numbers prone to noise. Here, we take a different approach to estimate plasmid transfer rate, which explicitly embraces this noise. Inspired by the classic fluctuation analysis of Luria and Delbrück, our method is grounded in a stochastic modeling framework. In addition to capturing the random nature of plasmid conjugation, our new methodology, the Luria–Delbrück method (“LDM”), can be used on a diverse set of bacterial systems, including cases for which current approaches are inaccurate. A notable example involves plasmid transfer between different strains or species where the rate that one type of cell donates the plasmid is not equal to the rate at which the other cell type donates. Asymmetry in these rates has the potential to bias or constrain current transfer estimates, thereby limiting our capabilities for estimating transfer in microbial communities. In contrast, the LDM overcomes obstacles of traditional methods by avoiding restrictive assumptions about growth and transfer rates for each population within the assay. Using stochastic simulations and experiments, we show that the LDM has high accuracy and precision for estimation of transfer rates compared to the most widely used methods, which can produce estimates that differ from the LDM estimate by orders of magnitude.

Funding: E.M.T. and B.K received support for this work from National Institute of Allergy and Infectious Diseases Extramural Activities grant no. R01 AI084918 from the National Institutes of Health. B.K received support for this work from Division of Environmental Biology grant no. 2142718 from the National Science Foundation. O.K. was supported by the National Science Foundation Graduate Research Fellowship grant no. DGE-1762114. C.E. was supported by the National Science Foundation Graduate Research Fellowship grant no. DGE-2019265372. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Here, we derive a novel estimate for conjugation rate by explicitly tracking transconjugant dynamics as a stochastic process (i.e., a continuous time branching process). This represents a notable deviation from previous approaches that are built upon deterministic frameworks. The random nature of conjugation can lead to substantial variation in the number of transconjugants at the end of a mating assay ( ) as this population will often be small. Prior deterministic frameworks rely on this number (e.g., Eq 4 ), such that transconjugant variation adds problematic noise to the estimate. In contrast, our stochastic approach leverages this noise to produce an estimate (akin to the way Luria and Delbrück estimated mutation rate [ 19 ]). In addition, our method allows for unrestricted heterogeneity in growth rates and conjugation rates. Thus, our method fills a gap in the methodological toolkit by allowing unbiased estimation of conjugation rates in a wide variety of strains and species. We used stochastic simulations to validate our estimate and compare its accuracy and precision to other estimates. We developed a protocol for the laboratory by using microtiter plates to rapidly screen many donor-recipient cocultures for the existence of transconjugants. In addition to its experimental tractability, our protocol circumvents problems that arise in the laboratory that can bias other approaches. Finally, we implemented our method in the laboratory and compared our estimate to the SIM estimate using a Klebsiella pneumoniae to Escherichia coli cross-species case study with an IncF conjugative plasmid.

The Achilles heel of this estimate, as with others, is found in violations of its assumptions. For example, we label Eq 4 as the “Simonsen and colleagues identicality method” estimate (SIM) for the donor conjugation rate because the underlying model assumes all strains are identical with regards to growth rates and conjugation rates. However, in natural microbial communities, this identicality assumption is misplaced, especially when the donors and recipients belong to different species. For example, suppose that the rate of plasmid transfer within a species (i.e., from transconjugants to recipients, which we abbreviate as the “transconjugant conjugation rate”) is much higher than between species (i.e., from donors to recipients), i.e., γ T ≫γ D ( Fig 1B ). This elevated within-species conjugation rate (γ T ) will increase the number of transconjugants and consequently inflate the SIM estimate for the cross-species conjugation rate (γ D ) compared to a case where the conjugation rates are equal (γ T = γ D , Fig 1C ). This Achilles heel is not specific to cross-species scenarios and can occur when estimating conjugation between any cells, including strains of the same species. One approach to minimize the resulting bias is to shorten the incubation time for the assay [ 12 ], as estimate bias tends to increase over time (e.g., Fig 1B ). However, new problems can arise when using this approach, such as the transconjugant numbers becoming exceedingly low and thus difficult to accurately assess [ 13 ]. Another approach was introduced by Huisman and colleagues [ 14 ], which squarely addressed the SIM identicality assumptions by developing a method to estimate donor conjugation rate when growth and transfer rates differ, thereby enlarging the set of systems amenable to estimation (see Section 1 in S1 File for full description of this and other approaches). Nonetheless, this new method can have difficulty with situations in which the donor conjugation rate (γ D ) is substantially lower than the transconjugant rate (γ T ), the example illustrated in Fig 1 . Such differences have been reported in multispecies systems [ 15 ] and recently several studies have recognized the importance of evaluating the biology of plasmids in microbial communities [ 7 , 16 – 18 ]. Therefore, a method that provides an accurate estimate despite substantial inequalities in rate parameters is desirable.

(a) In this schematic, the conjugative plasmid is a red circle, a donor is a red cell containing the plasmid, a recipient is a blue cell, and a transconjugant is indicated with a purple interior (a blue cell containing a red plasmid). The ψ D , ψ R , and ψ T parameters are donor, recipient, and transconjugant growth rates, respectively, illustrated by 1 cell dividing into 2. The γ D and γ T parameters are donor and transconjugant conjugation rates, respectively, shown by conjugation events transforming recipients into transconjugants. (b) When the transconjugant conjugation rate (γ T ) is higher than the donor conjugation rate (γ D ), transconjugants exhibit superexponential increase (purple curve) while donors and recipients increase exponentially (red and blue lines). The SIM estimate (orange line) increases over time, deviating from the actual donor conjugation rate (gray dashed line). (c) In contrast, when the conjugation rates are equal (γ T = γ D ), the transconjugant increase is muted relative to part b (purple line). The SIM assumptions are met, and the estimate is constant and accurate over time (orange line). Eqs 1 – 3 were used to produce the top graphs, with D 0 = R 0 = 10 5 , T 0 = 0, ψ D = ψ R = ψ T = 1, γ D = 10 −14 , and either γ T = 10 −8 (in part b) or γ T = 10 −14 (in part c). The donor and recipient trajectories overlapped but were staggered for visibility. Eq 4 was used to produce the bottom graphs. The code needed to generate this figure can be found at https://github.com/livkosterlitz/LDM or https://doi.org/10.5281/zenodo.6677158 .

The basic approach to measure conjugation involves mixing plasmid-containing bacteria, called “donors,” with plasmid-free bacteria, called “recipients.” As the coculture incubates, recipients acquire the plasmid from the donor through conjugation, and these transformed recipients are called “transconjugants.” Over the course of this “mating assay,” the densities of donors, recipients, and transconjugants are tracked over time (D t , R t , and T t , respectively) as the processes of population growth and plasmid transfer occur. To understand how such information is used to calculate the rate of conjugation, we consider an altered version of the foundational Levin and colleagues model [ 10 ]. In this framework, populations grow exponentially, and recipients become transconjugants via conjugation when they interact with plasmid-bearing cells (i.e., donors or transconjugants). The densities of the populations are described by the following differential equations (the t subscript is dropped from the variables for notational convenience): (1) (2) (3) In Eqs 1 – 3 , donors, recipients, and transconjugants divide at a per capita rate of ψ D , ψ R , and ψ T , respectively. The parameters γ D and γ T measure the rate at which a recipient acquires a plasmid per unit density of the donor and transconjugant, respectively. Thus, the ψ parameters are population growth rates, and the γ parameters are conjugation rates (see Fig 1A ). Assuming all the growth rates are equal (ψ D = ψ R = ψ T = ψ) and conjugation rates are equal (γ D = γ T = γ), Simonsen and colleagues [ 11 ] provided an elegant solution to Eqs 1 – 3 to produce the following estimate for the conjugation rate from donors to recipients (hereafter termed the “donor conjugation rate”): (4) For a mating assay incubated for a fixed period (hereafter ), the initial and final density of all bacteria (N 0 and , respectively), the final density of each cell population ( , , and ), and the population growth rate (ψ) are sufficient for an estimate of the conjugation rate.

This horizontal mode of inheritance makes it possible for nonrelated cells to exchange genetic material, which includes members of different species [ 4 ]. In fact, conjugation can occur across vast phylogenetic distances, such that the expansive gene repertoire in the “accessory” genome encoded on conjugative plasmids is shared among many microbial species [ 5 ]. This ubiquitous genetic exchange reinforces the central role of conjugation in shaping the ecology and evolution of microbial communities [ 1 , 3 , 6 ]. Notably, conjugation is a common mechanism facilitating the spread of antimicrobial resistance genes among bacteria and the emergence of multidrug resistance in clinical pathogens [ 7 – 9 ]. To understand how genes, including those of clinical relevance, move within complex bacterial communities, an accurate and precise measure of the rate of conjugation is of the utmost importance.

A fundamental rule of heredity involves the passage of genes from parents to their offspring. Bacteria violate this rule of strict vertical inheritance by shuttling DNA between cells through horizontal gene transfer [ 1 , 2 ]. Often the genetic elements being shuttled are plasmids, extrachromosomal DNA molecules that can encode the machinery for their transfer [ 3 ]. This plasmid transfer process is termed conjugation, in which a plasmid copy is moved from one cell to another upon direct contact. Additionally, plasmids replicate independently inside their host cell to produce multiple copies, which segregate into both offspring upon cell division. Therefore, conjugative plasmids are governed by 2 modes of inheritance: vertical and horizontal.

Conjugative plasmids play significant roles in the dynamics of microbial communities. Conjugation, the horizontal transfer of plasmids from one cell to another, is a common means of spread for genes of ecological significance, including those encoding antibiotic resistance. For both public health modeling and a basic understanding of microbial population biology, accurate estimates of the rate of plasmid transfer are of great consequence. Widely used methods assume the process of conjugation is deterministic and, under certain conditions, lead to biased estimates that deviate from true values by several orders of magnitude. Therefore, we developed a new approach, inspired by the classic fluctuation analysis of Luria and Delbrück, which treats plasmid transfer as a random process. Our Luria–Delbrück method is straightforward to implement in the laboratory and can accurately estimate the rate of plasmid conjugation for different bacterial systems under a wide variety of conditions.

Results

New laboratory protocol to implement the LDM We developed a general experimental procedure for estimating donor conjugation rate (γ D ) using the LDM approach in the laboratory. The LDM protocol is tractable and can accommodate a wide variety of microbial species and conjugative plasmids by allowing for distinct growth and conjugation rates among donors, recipients, and transconjugants. The basic approach is to inoculate many donor-recipient cocultures and then, at a time close to the average t*, add transconjugant-selecting medium (counterselection for donors and recipients) to determine the presence or absence of transconjugant cells in each coculture. In Section 1 in S1 File, we rearrange Eq 11 to provide an alternative form to highlight the quantities needed to conduct the LDM assay in the laboratory: (13) Similar to previous conjugation estimates, the LDM protocol requires measurement of initial and final densities of donors and recipients (D 0 , R 0 , , and ). In addition, the LDM approach requires a fraction of parallel donor-recipient cocultures to have no transconjugants at the specified incubation time ( ), which is the maximum likelihood estimate . Lastly, there is a correction factor when the coculture volume deviates from 1 ml; specifically, f is the reciprocal of the coculture volume in milliliters (e.g., for a coculture volume of 100 μl, f = 1/0.1 = 10, Section 5 in S1 File). Before executing the LDM conjugation assay, an incubation time and initial density for the donors (D 0 ) and the recipients (R 0 ) needs to be chosen so that the probability that transconjugants form ( ) is not close to 0 or 1. We developed a short assay (Section 6 in S1 File) for screening combinations of incubation time and initial densities to select a target incubation time ( ) as well as target initial densities ( and ) where . Note we add primes to indicate that these are “targets” to distinguish D 0 , R 0 , and in Eq 13, which will be gathered in the conjugation protocol itself. In addition, this pre-assay simultaneously verifies that the LDM modeling assumption of constant growth is satisfied. In our case, this pre-assay revealed several time-density combinations that could have been used. A useful pattern to note is that a higher donor conjugation rate will require shorter incubation times and lower initial densities compared to a lower rate. For the LDM conjugation assay, we mix exponentially growing populations of donors and recipients, inoculate many cocultures at the target initial densities in a 96 deep-well plate and incubate in nonselective growth medium with the specific experimental culture volume (1/f of 1 ml) for the target incubation time (Fig 5). To estimate the initial densities (D 0 and R 0 ), 3 cocultures at the start of the assay are diluted and plated on donor-selecting and recipient-selecting agar plates (Fig 5A). After the incubation time ( ), final densities ( and ) are also obtained by dilution plating from the same cocultures (Fig 5B). Liquid transconjugant-selecting medium is subsequently added to the remaining cocultures (Fig 5C). After a long incubation in the transconjugant-selecting medium, there should be a mixture of turbid and nonturbid wells. A turbid well results from one or more transconjugant cells being present at time (when transconjugant-selecting medium was added). Therefore, a nonturbid well indicates the absence of transconjugant cells at , since the first conjugation event had not yet occurred ( , Fig 3), although see Section 6 in S1 File for a caveat. The proportion of nonturbid cultures is (Fig 5C). Unlike the traditional Luria–Delbrück method, no plating is required to obtain . With the obtained densities (D 0 , R 0 , , and ), the incubation time ( ), the proportion of transconjugant-free cultures ( ), and the experimental culture volume correction (f), the LDM estimate for donor conjugation rate (γ D ) can be calculated via Eq 13. PPT PowerPoint slide

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TIFF original image Download: Fig 5. Overview for executing the LDM conjugation protocol. (a) The wells of a microtiter plate are inoculated with parallel cocultures (black-bordered circles) at the target initial densities ( and ). In addition, donor, recipient, and transconjugant monocultures serve as controls (red-, blue-, and purple-bordered wells, respectively). Three cocultures (top-right) are sampled to determine the actual initial densities (D 0 and R 0 ). Note empty wells (dash-bordered circles) are used later in the assay. (b) After the incubation time ( ), the same 3 cocultures are sampled for final densities ( and ). In addition, donor and recipient monocultures are mixed into the empty wells (indicated by gray arrows) to create coculture controls to verify that diluting with transconjugant-selecting medium effectively prevents conjugation. (c) Subsequently, transconjugant-selecting medium is added to the microtiter plate (indicated by the yellow background) and incubated for a long period. The transconjugant-selecting medium should inhibit donor and recipient growth, leading to nonturbid (gray-filled) donor and recipient control wells, but a turbid (purple-filled) transconjugant control well. In addition, the transconjugant-selecting medium should prevent new conjugation events leading to nonturbid coculture controls (gray-filled). Focusing on the wells inoculated with parallel cocultures, the proportion of transconjugant-free (i.e., nonturbid, gray-filled) cultures is Using this proportion, the actual incubation time ( ), initial densities (D 0 and R 0 ), final densities ( and ), and the experimental culture volume correction (f), the LDM estimate of the donor conjugation rate (γ D ) can be calculated. One microtiter plate yields 1 LDM estimate. https://doi.org/10.1371/journal.pbio.3001732.g005

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[1] Url: https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.3001732

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