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Published online 17 May 2007
Published in Vadose Zone J 6:306-315 (2007)
DOI: 10.2136/vzj2006.0069
© 2007 Soil Science Society of America
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SPECIAL SECTION: SOIL BIOPHYSICS

Horizontal Gene Transfer on Surfaces in Natural Porous Media: Conjugation and Kinetics

A. Massoudieh, A. Mathew, E. Lambertini, K. E. Nelson and T. R. Ginn*

Department of Civil & Environmental Engineering, 1 Shields Ave., Univ. of California, Davis, CA 95616
* Corresponding author (trginn{at}ucdavis.edu).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.


Received 10 May 2006.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
 Appendix
 REFERENCES
 
The transfer of genetic material among bacteria in the environment has been linked to a range of important phenomena related to bacterial adaptation and evolution, including bioremediation capability, metal tolerance, antibiotic resistance by pathogens in the environment, and gene flow from genetically modified microorganisms. Transfer of genetic material can occur among microorganisms present in both the planktonic and attached states. Given the propensity of organisms to exist in sessile communities under oligotrophic conditions, and that such conditions typify the subsurface, study of subsurface gene transfer phenomena should include processes and kinetics for a range of sessile community structures, from reversibly attached single cells to mature biofilms, as well as planktonic communities. This study very briefly reviewed horizontal, primarily conjugative, gene transfer in natural porous media, and the kinetics used to date to describe conjugative gene transfer in both planktonic (aqueous suspension) and sessile (surface associated) communities. The mathematics so far used to describe the kinetics of conjugation have developed largely from experimental observations of planktonic gene transfer, and are absent of time lags that occur between gene transfer events or plasmid stability that appear experimentally. We develop a novel formulation of delay-difference equations for gene transfer for attached-state microbes using an exposure-time approach to account for lags.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
 Appendix
 REFERENCES
 
Horizontal transfer of genetic elements in nature has been linked to bacterial adaptation and evolution, thus giving rise to a range of important phenomena, in some cases beneficial, e.g., acquired or enhanced biodegradation capabilities by a bacterial strain, but in other instances potentially hazardous, for example, the development of antibiotic resistance by pathogens, and gene flow from genetically modified organisms to native soil microfauna (Miller and Levy, 1989). The main known transfer processes are transformation (uptake of DNA from the environment), transduction (DNA transfer mediated by a bacteriophage), and conjugation (cell-to-cell DNA transfer requiring contact). Conjugation is of particular interest, as it is possibly one of the major mechanisms of transfer of genetic information in the environment (Yin and Stotzky, 1997). Although the mechanisms at play have been elucidated, the processes controlling rates of conjugative gene transfer are currently incompletely identified, and associated expressions for kinetic rates of gene transfer have been established only for particular isolated microcosm studies, such as solutions or biofilms in batch. Because conjugation frequency depends on local cell number densities, when bacteria move through multiphase environments such as subsurface porous media, conjugative gene transfer rates will depend on bacterial transport and rates of attachment and detachment to surfaces.

Planktonic cells, biofilm pieces, and cell clumps in permeable porous media (aquifer materials) may undergo advection, dispersion, motile transport, attachment, adhesion, and detachment processes, which may or may not lead to biofilm formation depending on multiple abiotic and biotic factors, including aqueous chemistry and nutrient availability, physiological state of microbes, chemotactic activity, quorum sensing, and other community dynamics (e.g., Lappin-Scott and Bass, 2001; Ginn et al., 2005). Biofilm formation is a sequential process with five developmental stages: reversible attachment, irreversible attachment, sessile community, climax community, and mature biofilm (e.g., Jenkinson and Lappin-Scott, 2001). Biofilms are foci for the study of conjugative gene transfer (Beaudoin et al., 1998a; Lilley and Bailey, 2002), as microbes are in close proximity in the biofilm, arranged in a variety of architectural and functional phenotypes (Molin and Tolker-Nielsen, 2003). The focus of this study was on preliminary model development for conjugative gene transfer within the first three developmental stages of biofilms, i.e., before the formation of a climax community or a mature biofilm. Thus, our interest lies with the conjugative process that can occur on surfaces during the transport of individual microorganisms through porous media and their attachment to solid surfaces.

We begin with a summary review of gene transfer processes with primary emphasis on conjugation as performed by the F-plasmid in Escherichiae coli, the first and most studied mechanism (Birge, 1994). As we are interested in viewing conjugative gene transfer in the context of soil science and hydrogeological applications, we maintain a focus on kinetics of these processes in multiphase systems. Developing our conceptual model from the current knowledge reviewed, we introduce a novel modeling framework for kinetics of horizontal gene transfer on surfaces that honors lags observed in conjugation kinetics studies by use of the exposure-time approach (Ginn, 2000). We also describe how to integrate the novel kinetics of gene transfer by conjugation with bacterial fate and transport processes including advection, motility, and filtration.


   Horizontal Gene Transfer
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
 Appendix
 REFERENCES
 
The following review is limited to kinetics, with a general introduction to horizontal genetic transfer processes in prokaryotes, within a soil science context. Vertical gene transfer among unicellular organisms involves the inheritance of a copy of the parent cell genetic material by the "daughter" cell during replication. Horizontal gene transfer processes allow genetic material to move among cells and their environment, independent of reproduction. As horizontal transfer involves organisms of multiple species, it is of concern for (unintended) spread of antibiotic resistance, and transfer of engineered genes and virulence properties. Processes and rates of horizontal transfer also affect potential beneficial applications, such as conveyance of xenobiotic degradation ability in bioremediation of organic compounds or metal tolerance in engineered reductive precipitation. Understanding the kinetics of horizontal transfer is crucial to quantify these impacts and estimate related risks, and to provide insights into the evolution of microorganisms (Birge, 1994; Maloy et al., 1994; Davison, 1999).

Three main types of horizontal gene transfer mechanisms are known so far in bacteria. Transformation is the uptake of free DNA from the environment and was first observed in the 1920s in Streptococcus pneumoniae, during an experiment providing the first direct evidence that DNA is genetic material (Avery et al., 1944; Magasanik, 1999; Madigan et al., 2003). Transduction is the transfer of genetic material via a bacteriophage, i.e., a virus attacking a bacterial cell, and was discovered in 1952 in Salmonella typhimurium (Zinder and Lederberg, 1952; Lederberg, 1994; Magasanik 1999). Conjugation is the transfer of DNA, usually in the form of plasmids, via direct cell-to-cell contact (Salyers et al., 1998; Davison, 1999), and was reported as early as 1946 (Lederberg and Tatum, 1946). Gene transfer by these processes takes place despite the presence of what are effectively defense measures of the bacterial cell to prevent the entry of foreign DNA, such as endonucleases in the periplasm, hydrophobic membranes, and negatively charged lipopolysaccharides in the outer membrane of Gram-negative bacteria (Kado and Syvanen, personal communication, 2005). We discuss here only conjugative gene transfer mechanisms.

Conjugation
Conjugation is thought to be the most frequent and efficient of all mechanisms of horizontal gene transfer in the environment (Kado and Syvanen, personal communication, 2005). The process involves cell-to-cell contact and formation of a "mating bridge," followed by the transfer of horizontally mobile elements (HMEs) of genetic material, typically plasmids or transposons. Plasmids are double-stranded extrachromosomal genetic elements, mostly circular in shape and ranging from 300 to 2 400 000 base pairs long. They exist inside prokaryotic and also a few eukaryotic organisms, and are capable of replication independently from the main DNA filament, as well as of insertion into the main filament. They are present in a cell in specific copy numbers, varying from one to >100 depending on the plasmid, while different plasmid types may coexist in one cell (Madigan et al., 2003). Transposons, originally called "jumping genes," are mobile DNA segments that can frequently move from one location to another on the bacterial chromosome, and also to and from plasmids, but cannot replicate independently. They can be conjugative and, once in the cell genome, are reproduced by the normal cell duplication processes (Maloy et al., 1994; Roberts et al., 2001, Madigan et al., 2003). The distinguishing characteristic of HMEs is this peculiar ability of being transferred horizontally, in addition to vertically, while the chromosome can usually be transferred only from parent cell to daughter (Heinemann, 1998).

Plasmids and transposons can carry a variety of traits, including resistance to antibiotics or heavy metals, toxin production ability, virulence factors, ability to biodegrade a substrate, and plasmids encoding for the conjugative machinery itself (Maloy et al., 1994; Madigan et al., 2003). Nonconjugative plasmids can use the same mating bridge created by conjugative plasmids (Davison, 1999), but the resulting transfer is relatively rare. A distinction can be made between Gram-negative bacteria, which among other characteristics can express a mating pilus, and Gram-positive, which come into contact without a pilus. The specific conjugation mechanisms vary for each plasmid, but in most Gram-negative bacteria they follow that of the E. coli "fertility" (F) plasmid, while in Gram-positive bacteria they are more diverse (Birge, 1994; Madigan et al., 2003). The transfer of genetic material from the recipient back to the donor has also been observed, even though the frequencies of such retrotransfer are lower than "forward" conjugation (Heinemann and Ankenbauer, 1993; Sia et al., 1996; Blanco et al., 1991). Conjugative transposons follow a similar scheme of transfer, but with some important differences. Normally integrated into the host genome, a transposon can excise itself with great precision, and give origin to a covalently closed circular intermediate, which is then transferred to another cell by conjugation. The transposon also carries within itself integrase enzymes, which allow its integration into the genome of the new host. Due to the precision of the excision, the recipient cell receives all the genes needed for the transposon to transfer again, and is therefore fully functional as a donor if the transposon genes are expressed. Some other transposons are not self-transmissible, but can be mobilized by conjugative transposons (Salyers et al., 1998).

It has been observed that many plasmids and transposons have a wide host range, in that they can be transferred to and expressed in bacterial cells of different species and genera. Moreover, conjugative transfer can cross the border between kingdoms, i.e., the transfer can happen from bacteria to yeasts, or to plants (Heinemann and Sprague 1989; Davison, 1999; Roberts et al., 2001; Madigan et al., 2003). From a DNA-centric point of view, conjugation has also been viewed as an infectious process, through which a "selfish" DNA segment, such as a plasmid or a transposon, exploits a host cell and assures its survival and spread, while at the same time it may provide useful traits to the host (Doolittle and Sapienze, 1980; Orgel and Crick, 1980; Rensing et al., 2002). Similarly, plasmid behavior has also been interpreted as the migration of a "population" from a patch of resources (a bacterial cell) to another to ensure its survival and multiplication (Landis et al., 2000). Another aspect of plasmid interaction with the host cell and among themselves is that plasmids can affect the success of other plasmids in entering or residing in a host cell (Birge, 1994).

Conjugative transfer data arise from either the observations in cell culture (macroscale) or confocal microscopy (cytoscale). Conjugation begins when donor and recipient come in contact, either by collision or via a conjugative pilus that may sense recipient cells. In the case of E.coli, the ability of a potential donor cell to initiate conjugation is dependent on its possession of the F or other conjugative plasmid. A physiologically active state of donors and recipients is a prerequisite as well. The F-plasmid in the donor directs the expression of one to three pili, rigid pilin fibers sticking out from the cell wall. Cell contact is followed by pilus retraction (or not, Lawley et al., 2004), and the formation of a mating bridge by the fusion of a portion of cell membranes (Achtman et al., 1978). In the donor cell, DNA transfer is initiated at the oriT region of the plasmid, where one of the DNA strands is cleaved, and then gradually separated from the complementary strand while being transferred to the recipient through the mating bridge. When the single strand enters the recipient cell, a complementary strand is synthesized, and the DNA string reassumes the circular shape of a plasmid. While the single strand of DNA is being transferred to the recipient, this transferred strand is simultaneously replaced in the donor via a "rolling circle" mechanism. As the donor and recipient part, both cells own a complete copy of the plasmid (Madigan et al., 2003). The recipient develops thus into a potential donor, although the frequency of this happening is still debated (Sorensen et al., 2005) and is also dependent on inhibition mechanisms (Shi et al., 2005; Perez-Mendoza et al., 2005) or chromosomal integration of the plasmid (Birge, 1994; Madigan et al., 2003).

As the transfer process can be very efficient, the spread of a plasmid in a bacterial population can potentially convert all the cells to plasmid-positive. On the other hand, a plasmid can be lost by a cell population, especially when there is no selective pressure to justify its retention; this phenomenon is termed "curing," and consists in the progressive dilution of a plasmid in a population, as the plasmid doesn't replicate together with the cell (Madigan et al., 2003). Also, in some studies on biofilms it was observed that when transconjugants had competitive advantage in selective conditions, after some initial conjugative transfer they tended to transfer the acquired plasmid vertically as their population grew (Christensen et al., 1998; Haagensen et al., 2002).

Conjugation has been observed both in controlled and natural environments, including the rhizosphere, water, soil and subsurface ecosystems, the leaf surface, and the animal intestine, in many cases with different kinetics and efficiencies, so that extrapolation from one case to the other is not reliable (Dröge et al., 1999; Davison, 1999). In such natural environments, the frequencies of plasmid transfer seem to be dependent on the physiological status (e.g., growth rate) of the donor (Smets et al., 1993; Sudarshana and Knudsen, 1995), but not significantly on the status of the recipient (Arana et al., 1997; Muela et al., 1994). Nutrient-rich environments have been observed to lead to an increase in transfer frequency (Sandt and Herson, 1991; Clerc and Simonet, 1996; Götz and Smalla, 1997), even though gene transfer has been detected in oligotrophic environments, or after bacteria had undergone starvation (Goodman et al., 1993; Davison, 1999). On the contrary, Hausner and Wuertz (1999) reported transfer rates in biofilms to be independent of the concentration of nutrients; similar observations were made by Ehlers and Bouwer (1999). In biofilms, biomass surface has an influence on the efficiency of plasmid transfer, with high surface-to-volume ratios favoring the transfer (Molin and Tolker-Nielsen, 2003). There seems to be an optimal temperature for plasmid transfer, defined by the characteristics of both the specific plasmid and its host; out of the optimal region, the transfer still happens, but at lower frequencies (Dröge et al., 1999). In general, conjugation can occur under a vast range of conditions, and the effect of environmental variables on its efficiency and extent are under active investigation.

It has been demonstrated that cell culture and microscopy lead to substantially different conclusions about conjugation extent. The kinetics results presented below and elsewhere should be evaluated considering that the quantification methods used are, in general, different from one study to another, and most trials to date have used cell culture assays (Sorensen et al., 2005).


   Kinetics and Modeling of Conjugative Gene Transfer
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
Conjugation Kinetics with Lags...
 Conclusions
 Appendix
 REFERENCES
 
Different Approaches to the Quantification of Plasmid Transfer
Levin et al. (1979) measured F- and R-plasmid transfer rates among different strains of E. coli in batch cultures and in chemostat, and developed a model of transfer kinetics, with assumptions including: (i) mating is a random process and mating frequency is proportional to the concentrations of plasmid-free and plasmid-bearing cells; (ii) plasmid loss due to cell replication is negligible; (iii) transconjugants, cells originally free of a specific plasmid that have received it by conjugation, become potential donors without any delay; (iv) the plasmid transfer rate is the same for the original donors as well as transconjugants, and (v) all bacterial clones have the same growth rate. The result is a mass-action kinetics, with rate proportional to the product of the local number densities of donor and recipient cells:

Formula 1[1]
where k is the rate coefficient [in units of ([F]2)1/2 T–1], [F+] is the local number density of F-plasmid-bearing cells and [F–] is the local number density of F-plasmid-free cells, where local number density is defined as the cell number per local volume (Levin et al., 1979). This model performs well in the laboratory setting for active planktonic cells. In natural environments, however, donors, recipients, and transconjugants will not in general be homogeneously mixed and Eq. [1] should not work (Levin et al., 1979). Also, the form of Eq. [1] does not accommodate delays or lags observed in horizontal gene transfer. We address these two limitations below by structuring the model on space and including in the kinetics a measure of the mixing of donor and recipients on surfaces, and by accommodating lags. The mass action model does not perform well where recipients far outnumber donors ("donor saturation," Cullum et al., 1978), and it is not recommended for measuring relatively low transfer rates (Levin et al., 1979; Andrup et al., 1998; Simonsen et al., 1990). Fernandez-Astorga et al. (1992) found that cell number density and donor/recipient ratio had a significant effect on conjugation among E.coli strains. Simonsen et al. (1990) observed that the number of transconjugants formed on surfaces was strongly influenced by initial cell density, but the same did not hold for liquid cultures. Similar results are reported in Knudsen et al. (1988), studying conjugation on leaf surfaces.

The mass action model of Levin et al. (1979) has been used to compute conjugation rates, coupled with microbial growth and decay, in Knudsen et al. (1988) studying rhizosphere and phyllosphere environments, in Clewlow et al. (1990) and Simonsen et al. (1990) studying liquid cultures, and in Smets et al. (1994) and Beaudoin et al. (1998a, 1998b) studying biofilms. In particular, Beaudoin et al. (1998b) enrolled mass action kinetics with dynamic growth and decay in biofilm to quantify the transfer of mobilizable plasmid pDLB101 from Pseudomonas putida to a pure culture biofilm of Bacillus azotoformans. The biofilm model AQUASIM was modified to accommodate conjugation kinetics and plasmid loss and used to model data from a parallel study of Beaudoin et al. (1998a).

An alternative approach that treats bacterial conjugation as a classical enzyme catalysis type reaction was proposed in Andrup et al. (1998) and in Andrup and Andersen (1999), where donors and recipients are analogous to (catalyzing) enzymes and reactant, respectively. Thus a mating is viewed as an activated complex, and a transconjugant as a product, with a Michaelis–Menten kinetic governing the rate of bacterial conjugation. Assumptions include: (i) recipient cells vastly outnumber donors; (ii) transconjugants do not become donors with significant frequency; and (iii) plasmid loss is negligible. This Michaelis–Menten-analogous model predicts that, at low recipient concentrations, the conjugation rate is proportional to recipient concentration, whereas at high recipient concentrations the donors are saturated, thus a maximal conjugation rate is attained.

An ecological modeling approach was presented in Lagido et al. (2003), who described conjugative transfer on solid surfaces as a result of the contact between growing colonies of donors and of recipients. The model assumes that donor and recipient colonies grow exponentially until nutrient exhaustion and that, when donors and recipients come into contact, transconjugants are formed instantaneously, converting all the recipients to transconjugants at the same instant. Although this model incorporates the role of surfaces, it has limited applicability because of the required number of novel parameter values and because it does not account for any cell mobility in either aqueous or surface phases (Lagido et al., 2003). Another ecological approach, rooted in population dynamics, appeared in Landis et al. (2000), who simulated individual microbial generations as "resources" on which plasmid "infection" proceeds under a set of rules. While absent of a physical setting (neither liquid culture nor surfaces defined), the model includes selective pressure and an assumed Poisson distribution of conjugation rates.

Time Lags in the Plasmid Transfer Kinetics
Here we provide a brief summary of some relevant experimental observations of time lags and conjugation rates, considering in turn the conjugation event, post-conjugation recovery of the donor, and transconjugant maturation time. Although many researchers have modeled conjugative transfer, we are unaware of any attempt to date to include time lags in conjugation kinetics, despite the fact that such lags are consistently indicated.

Conjugation Event
In the study of plasmid transfer in E. coli by Andrup et al. (1998), the minimum time required for a single transfer event (including transconjugant formation evinced by gene expression) was 3.5 to 4.0 min. Similarly, Michel-Briand and Laporte (1985), while studying transfer of plasmid RP4 in E. coli, reported formation of transconjugant in ~5 min. These times certainly depend on the mating strains and experimental conditions.

Recovery Period
The recovery period for Bti encoded by the plasmid pXO16 was reported as approximately 10 min (Andrup et al., 1998). It was estimated that a transfer from a donor to recipient takes place every 15 min on average (calculated from the maximal conjugation rate) and, in view of the time required for a single transfer event, it was deduced that ~10 min is required for recovery and mating initiation (Andrup et al., 1998). In Cullum et al. (1978), however, the lag period between two rounds of transfer for the F-plasmid in E. coli was reported as 30 min, about three times as that of Bti. This disparity may be attributed to the difference in the ratio of donors to recipients at which the maximal rate of conjugation takes place, as well as the mating conditions (Andrup and Andersen, 1999).

Transconjugant Maturation Time
The maturation time of a transconjugant can be defined as the time it takes to achieve donor capability. Andrup and Andersen (1999) report maturation times for E. coli receiving the F-plasmid of ~80 min, and for Ent. fecalis receiving the pCF10 plasmid of ~100 min. These times were estimated by observing, in plots of transconjugants formed vs. time, dramatic increases in the number of transconjugants formed at 80 min for E. coli and at 100 min for Ent. fecalis. It was inferred that the increase results from the lag time needed for transconjugants to become donors. A similar maturation time of ~90 min was reported for the F-plasmid-mediated conjugation in E. coli by Cullum et al. (1978). The maturation time of Bti (pXO16) was studied by Andrup et al. (1998), who found that ~40 min are required for transfer from newly formed transconjugants.

The relevant data regarding kinetics and lag times in conjugative gene transfer is summarized in Table 1. The processes represented in this table include duration of: (i) conjugation events, (ii) donor resting (recovery) state after conjugation, (iii) time to transconjugant joining the donor pool, and (iv) maximum conjugation rate.


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  		TABLE 1. Compilation of mean times associated with conjugative gene transfer processes.

 
In summary, plasmid transfer rate was originally expressed in terms of cell densities and ratios of donor, recipient, or transconjugant populations, but later the transfer kinetics was cast as mass action and thus bilinear kinetics (e.g., Levin et al., 1979; Smets et al., 1994; Simonsen et al., 1990). Alternatively, conjugation has been represented by analogy to Michaelis–Menten enzyme catalysis (e.g., Cullum et al., 1978; Andrup et al., 1998; Andrup and Andersen, 1999). None of these models have accounted for the time lags, and most of the studies have not considered one or more of the following aspects of conjugative gene transfer: (i) spatial distribution and spatial separation of cells; (ii) population dynamics including growth and decay (with several important exceptions); or (iii) cell mobility in the case of conjugation on surfaces. We hypothesized that the ubiquitously poor fit of any kinetics to conjugative gene transfer data is due to these omissions. The lags especially often correspond to the lower bound on time intervals for a process to occur, thus limiting the applicability of bilinear and other forms that represent the transfer classically as a Markovian stochastic process with Poisson distribution. Thus we applied a non-Markovian approach that expresses rate coefficients as dependent on time in a given state (e.g., transconjugant).

Most efforts to measure conjugative kinetics were conducted for liquid cultures (Levin et al., 1979; Simonsen, 1990; Andrup and Andersen, 1999; Andrup et al., 1998), and few studies have taken place investigating plasmid transfer rates on surfaces (Simonsen, 1990; Lagido et al., 2003). Most of the environments where gene transfer takes place can be considered porous in nature; examples are human and animal tissues, the phyllosphere, all soil environments (e.g., the rhizosphere), and biofilms formed in all environments. Hence, most of the modeling approaches developed so far have only limited applications to natural environments. To overcome these constraints, our approach also accommodates processes of bacterial attachment–detachment between the aqueous solution and the solid surface of a porous medium, thus keeping track of the surface number density of cells on surfaces, and models conjugation within both planktonic and sessile communities.


   Conjugation Kinetics with Lags Modeled via the Exposure Time Concept
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
Conclusions
 Appendix
 REFERENCES
 
Building on the conceptual framework of conjugation processes outlined above, particularly of the E. coli F plasmid, we developed a mathematical model of bacterial transport and conjugation in a one-dimensional porous medium. The model takes into account the following processes: advective–dispersive transport of bacteria, reversible attachment–detachment of bacteria from bulk solution to the surface of the porous medium, and conjugation in the aqueous phase and on the surfaces. For simplicity, the model assumes that capture and release of bacteria to and from the solid surface of the porous medium can be explained using linear sorption kinetics. We also assumed a locally homogeneous concentration of bacteria on the solid surface. Although in reality the local concentration of captured bacteria is not uniform, due to differences in the geometrical (Nelson et al., 2007; Yoon et al., 2006) and mineralogical (Sun et al., 2001; Seeboonruang and Ginn, 2006) accessibility of various sites on the surface, and therefore bacterial populations may concentrate in some, here we assumed that the effect of such heterogeneities can be accounted for using modified conjugation rate coefficients for the attached bacteria. As explained above, during conjugation, genetic material is transferred from a donor (F+) to a recipient (F–); following that, the recipient becomes a transconjugant (T), and the donor becomes a temporarily "exhausted" bacterium (X) (Andrup and Andersen, 1999). Both T and X return to F+ donor pool after their lag times.

Using the above assumption, the transport–conjugation equation describing the behavior of mobile donors (F+) and recipients (F–) in the aqueous phase can be written as follows

Formula 2[2]

Formula 3[3]
and for the immobile (or captured) bacteria on the solid surface:

Formula 4[4]

Formula 5[5]
where F+ and F– are concentrations of donor and recipient bacteria, which are functions of space (x) and time (t); subscripts a and s indicates aqueous (in cells per aqueous volume) and attached (in cells per colonizeable surface area) phases, respectively; D is the diffusion coefficient for bacteria; S is the colonizable surface area of the solid phase per unit bulk volume of the porous medium; {theta} is the porosity; vx is the flow velocity; kf (in units of aqueous volume per colonizeable surface area per time) and kr (in units of 1/time) are attachment and detachment rates to the solid surface, respectively. The rates rac and rsc are conjugation rates in aqueous and attached phases respectively; rrT is the rate of transition from the transconjugant state T to a new donor state F+, and rrX is the rate of return from exhausted state X to donor state F+. The rate of conjugation is assumed to be proportional to the product of the concentrations of donors and recipients:

Formula 6[6]

Formula 7[7]
where kaz and ksz are conjugation rate coefficients in aqueous (in units of aqueous volume per cell per time) and attached (in units of colonizeable surface area per cell per time) phase, respectively.

The rate of conversion of exhausted and transconjugant bacteria into donor bacteria depends on the continuous time bacteria have spent in exhausted or transconjugant states, or "age" within the state. The distributions of transconjugant and exhausted bacteria concentrations within this age are denoted by T and X, respectively. Thus T = T(x,t,{tau}) and X = X(x,t,{tau}). For simplicity of notation these will be expressed without the x,t dependency. For example, the concentration of transconjugants between ages {tau}1 and {tau}2 will be equal to Formula 7T({tau})d{tau} . The distribution T at a certain point along the column changes due to fluxes in both the physical dimension and the aging dimension. Using the exposure-time approach of Ginn (2000), the mass balance of transconjugants T can be expressed with fluxes in both physical and age directions as (see Appendix):

Formula 7

Including attachment and detachment of bacteria to the surfaces, the governing equations for the aqueous and attached transconjugants and exhausted bacteria can be written as:

Formula 8[8]

Formula 9[9]

Formula 10[10]

Formula 11[11]

Here we assumed that all exhausted and transconjugant bacteria return to the donor state after lag time intervals {tau}T and {tau}X, respectively. Thus the rate of return to the donor state equals the flux in age dimension at age equal to the respective lag time, as

Formula 12A[12a]

Formula 12B[12b]

Formula 12C[12c]

Formula 12D[12d]

The flux of transconjugants at {tau} = 0 is equal to the rate of production of transconjugants, therefore the boundary condition for Eq. [8] to [11] can be written as

Formula 13A[13a]
and similarly:

Formula 13B[13b]

The set of Eq. [2] to [5] and [8] to [11], along with the boundary conditions in Eq. [13] and [14], is solved using a hybrid numerical method. The age distribution of transconjugants and exhausted bacteria is discretized into age intervals. The equations governing the fate and transport of F+, F–, T, and X bacteria at each time interval is solved using a Crank–Nicholson implicit finite difference scheme. To model the evolution of age distribution in time, the advective operator in age is imposed by shifting the concentration distribution over age to the next age step at each time step. To use such a shift operator, it is necessary that an age interval equal to the time step used in the finite difference scheme be chosen. This approach will reduce the computational effort significantly and also eliminate numerical diffusion error that a numerical solver can cause on the purely advective transport in the age direction.

Simplification for Aqueous Batch Systems
To interpret the results from typical horizontal gene transfer batch experiments (no transport and no solid phase present), it is worthwhile to study the form of governing equations describing the kinetics in such systems. For a batch system with no transport and no solid phase and with aging velocity v{tau} = 1, the concentration of transconjugants and exhausted cells can be expressed as

Formula 14[14]
and

Formula 15[15]

Substituting Eq. [14] and [15] (through Eq. [12]) into Eq. [2] and [3] and ignoring sorption leads to

Formula 16[16]

Considering the boundary conditions given in Eq. [13], the above equation can be written as

Formula 17[17]
and the governing equation for recipients can be obtained by eliminating transport and sorption terms from Eq. [3]:

Formula 18[18]

Equations [17] and [18] were used to obtain the parameters that best agree with kinetics results in the batch experiments by Andrup and Andersen (1999).

Demonstration Calculations
A demonstration computation is performed to present the performance of the model in a one-dimensional porous medium. First, data reported by Andrup and Andersen (1999) representing the kinetics of conjugation of E. coli were used to estimate the conjugation parameters, including the lags and the conjugation rate coefficient. In this study, E. coli strain MC1000 (F–, StrR) was used as recipient and strain XL-1 Blue (F+:Tn10, TetR NalR) as donor. Since the experiment was performed in a batch system, batch model Eq. [17] and [18] were used to estimate the parameters by minimizing the mean squared error between the transconjugant concentration calculated by the model and the transconjugant concentration measured by Andrup and Andersen. Figure 1 presents measured along with simulated transconjugant concentration vs. time. Rates kaz, {tau}T, and {tau}X were estimated to be 10–6 min–1 (cell concentration)–1, 34 min, and 10 min, respectively. The model matches reasonably well with the experimental data.


Figure 1
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  		FIG. 1. Observed vs. simulated transconjugant concentration in E. coli for donors initial concentration F+0 = 4.1 x 103 cell mL–1, recipient initial concentration F–0 = 7.6 x 106 cell mL–1, (lag) time spent in exhausted state ({tau}e) = 10 min, (lag) time spent in transconjugant state ({tau}T) = 34 min, and transconjugation rate kaz = 10–6 min–1 bacterium–1.

 
To perform the simulations of conjugative gene transfer among bacteria undergoing advective–dispersive transport in a one-dimensional porous medium, with kinetically controlled attachment–detachment, rate values of conjugation for the bacteria on the solid surface are also required. Under the assumption that this rate is intrinsically the same as the aqueous rate, but in units of surface area per cell per time, ksz, can be derived as:

Formula 18

This is a reasonable assumption considering the lack of experimental data on bacterial conjugation on surfaces without biofilms. This assumption needs to be refined, however, by further studies on bacterial conjugation on surfaces, both without and with biofilms, since the kinetics of conjugation on surfaces can be quite different from that in bulk water in many aspects.

Concentration profiles in a simulated column, with bacteria at various conjugative states both on solid and in aqueous phases at steady-state conditions, are presented in Fig. 2. Attachment and detachment rates were found by fitting a single rate model to the data presented by Li et al. (2004). Inflow concentrations adopted were from Andrup and Andersen (1999). Parameters used for this simulation are listed in Table 2.


Figure 2
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  		FIG. 2. Simulated (a) mobile and (b) immobile concentration profiles for donor (F+), recipient (F–), exhausted (E), and transconjugant (T) bacteria.

 

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  		TABLE 2. Parameter values used in the demonstration simulation.

 
In both phases, concentration profiles show that at the end of the 30-cm column, almost all F– bacteria are converted to transconjugants T. A fraction of transconjugants is still present at the end of the column, whereas almost all of the exhausted bacteria are converted back to F+. For mobile bacteria, conjugation rate is low at the beginning of the column due to the lower availability of donors, whereas in the middle of the column the rate increases due to the increase in the number of donors, which also produces a high concentration of exhausted and transconjugant bacteria. Further downstream in the column, transconjugant and exhausted bacteria gradually overcome their lag time and convert into donors. In the immobile phase, due to the smaller number of bacteria per surface area, the overall process takes place at a slower pace. Bacteria on the surfaces have a longer residence time, however, which affords more return to donor state of the transconjugant and exhausted cells. The low concentration of F– in the attached state at all locations along the column indicates that the rate of attachment of bacteria to the surface is significantly smaller than the rate of conjugation, which means that most recipients partake in conjugation soon after their attachment to the solid surface. Also, the decreasing concentration of exhausted and transconjugant bacteria with distance can be explained by the smaller number of F– bacteria remaining at the end of the column. These results are, of course, a consequence of the assumed experimental conditions; however, they demonstrate how transport, attachment–detachment, and conjugative gene transfer dynamics are coupled together.

Implications for the Vadose Zone
The foregoing conceptual and mathematical models focus on the conjugative gene transfer reactions among aqueous- and attached-phase microbes, and were developed mainly with saturated conditions in mind. There is some basis, however, for speculations about the behavior in the vadose zone. The gene transfer process in the aqueous phase requires bacterial transport across macroscopic scales and mixing of cells at the microscopic scale, and these processes slow down with decreasing water content. On the contrary, richer nutrient and electron acceptor (starting with O2) conditions in the shallow vadose zone and capillary fringe may lead more frequently to the occurrence of robust sessile communities, i.e., biofilms that could support gene transfer rates above that in the saturated zone. In either case, the results of gene spread from introduced species requires either transport of introduced species across macroscales, or positive selection for the plasmid conveyed. The preliminary modeling developed here shows the coupling of transport and conjugative gene transfer through the attachment–detachment processes controlling the locally mixed cell number densities on surfaces. The air–water interface may play a role as a surface supporting communities among which gene transfer can occur. These questions are ripe for subsequent research steps, and the model proposed here may serve as a starting point for quantitative analyses of such observations.


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
Appendix
 REFERENCES
 
In this study, we developed a novel modeling approach for quantifying the conjugative transfer rates on surfaces in porous media, taking into account some of the key aspects of conjugation in natural systems, including time lags and interactions of the bacteria in the bulk fluid with solid surfaces. Most of the transfer kinetics studies in the past did not account for important transfer-rate-determining factors such as time lags, spatial distribution of cell populations, etc., in their models. Moreover, most of the studies were performed in batch solution microcosms, which are not the ideal imitations of the natural environment in which conjugation generally occurs. The natural environment on which we focused is a porous medium, with bacteria undergoing transport and the initial development stages of biofilm formation, consisting of reversible attachment, detachment, and irreversible adherence that leads to colony formation on the surfaces. Hence, a one-dimensional bacterial transport–conjugation model was developed to estimate gene transfer rates of bacteria attaching to surfaces during their transport through porous media. We have tested an "aqueous batch" reduced model referring to the batch experiment data from Andrup and Andersen (1999), and obtained a reasonably good fit to their measured data. The obtained parameters were then used in an extrapolative modeling exercise to predict the distribution of donors and recipients in one-dimensional flow in porous media. This exercise is part of a premodeling effort to design small-scale experiments for observing the processes considered in the laboratory. As such, this model may be used to design flow rates and boundary concentrations for given porous medium physical and chemical characteristics and given microbial species. The model includes several far-reaching assumptions, one of which is the uniform distribution of attached microbes on surfaces. It is possible to relax this assumption by using individual-based simulation of bacterial transport in the Happel sphere-in-cell model of a porous medium, to obtain better approximations of how attached bacteria are distributed on surfaces of granular porous media (Nelson et al., 2007).


   Appendix
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
 Appendix
REFERENCES
 
Here we provide a brief description of the basis for the mass balance of species, such as transconjugants and exhausted species, across physical and age dimensions. The physical and age fluxes, Fx and F{tau}, respectively, are shown schematically in Fig. 3, and the corresponding mass balance of the distribution T at the finite volume can be written as

Formula A1[A1]
F{tau} is the flux in the age dimension, which is equal to the rate of change of age with respect to time v{tau}, termed the "aging velocity," multiplied by concentration at any point, or F{tau} = Tv{tau}. The physical flux Fx is defined using the conventional advection and diffusion: Fx = vxT + D{partial}T/{partial}x. Substituting F{tau} and Fx into Eq. [A1], dividing both sides by {Delta}x{Delta}{tau}{Delta}t and taking the limit when ({Delta}x,{Delta}{tau},{Delta}t) -> 0 yields

Formula A2[A2]


Figure 3
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  		FIG. 3. Mass balance diagram for a finite volume in x–{tau} space; shown is a two-dimensional control volume where one of the dimensions corresponds to a conventional spatial coordinate (x) and the other to exposure time ({tau}). Through each side of this control volume is shown the facewise specific fluxes in the directions of space and exposure time, respectively.

 

   ACKNOWLEDGMENTS
 
This project research was supported by NSF Project Award nos. 0620966 and 0416194, "Collaborative Research: CMG: Toward Understanding the Transfer of Genetic Information in Subsurface Hydrology," and NSF Project Award no. 0420374, "Biogeochemical Cycling of Heavy Metals in Lake Coeur d'Alene Sediments: The Role of Indigenous Microbial Communities," as well as the UC CARE Project entitled "Exposure-time Approach in Modeling Dynamics of Subsurface Microbial Communities in Contaminated Aquifers: Horizontal Gene Transfer in Adaptive Bioremediation."


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Horizontal Gene Transfer
 Kinetics and Modeling of...
 Conjugation Kinetics with Lags...
 Conclusions
 Appendix
 REFERENCES
 




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A. J.M. Smucker and J. W. Hopmans
Preface: Soil Biophysical Contributions to Hydrological Processes in the Vadose Zone
Vadose Zone J., May 17, 2007; 6(2): 267 - 268.
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