Sample preparation

The 180 bp rhodamine and fluorescein labelled dsDNA was prepared by initially annealing an internal rhodamine 90-nt ssDNA (O1) and a 5´end phosphorylated oligonucleotide (82 bases) (O2). Similarly, an 82-nt 5´-end phosphorylated oligonucleotide (O3) was annealed with a 98-nt oligonucleotide containing an internal fluorescein label (O4) to obtain another labelled dsDNA fragment. These two dsDNA labelled fragments were annealed and ligated overnight at 16°C with T4 DNA ligase in ligase reaction buffer (50 mM Tris, 10 mM MgCl₂, 1 mM ATP, and 10 mM dithiothreitol, pH 7.5 (New England Biolabs (NEB)). The 180 bp construct was purified3 by gel on 3% agarose in TBE (Tris/Borate/EDTA) buffer for 2 hours (6 V/cm). The 180 bp band was visualized with a midrange UV trans-illuminator. Finally, the 180 bp dsDNA band was cut out and was extracted from agarose using a Nucleospin kit (Machery and Nagel, Bethlehem, PA). The product was finally concentrated using 100 kDa Amicon filters (Millipore). The oligonucleotide sequences (Integrated DNA Technologies (Coralville, IA)) used to prepare the dsDNA construct are listed in S1 Table.

Strand exchange reactions

DNA strand exchange in vitro was achieved by mixing 0.06 μM ssDNA-RecA filaments with 0.06 μM labeled dsDNA. The sequences of the oligonucleotides used for these filaments are listed in S1 Table. The ssDNA-RecA filaments were prepared by mixing the ssDNA (final concentration 6 μM in bases) with 2 μM RecA (NEB) at 3:1 ratio (bases: protein), 1 mM ATP, a regeneration system containing 10 U/ml of pyruvate kinase and 3 mM phosphoenolpyruvate, and 0.2 μM E.coli single-stranded binding protein, SSB (Abcam) in RecA buffer (70 mM Tris-HCl, 10 mM MgCl₂, and 5 mM dithiothreitol, pH 7.6). Filament formation proceeded at 37°C for 10 minutes.

Fluorescence measurements

To measure the fluorescent signal of these strand exchange reactions, the ssDNA/RecA filament and dsDNA mixture was placed in a quartz cuvette. Initially, fluorescein emission was quenched by rhodamine because the outgoing and complementary strands were paired in the dsDNA. As strand exchange progresses, the outgoing and complementary strands separate and fluorescein emission increases. Using FluorEssence spectroscopy software and the automated FluoroMax spectrofluorometer (Horiba, Edison, NJ), the emission of the fluorescein label was read using 493 nm excitation wavelength and a 2 nm slit. The emission of the fluorescein fluorophore in counts per second (cps) was detected at 518 nm with a 2 nm slit. The reaction was run for 30 minutes, and emission measurements were collected every 1 second with an integration time of 0.5 s. Temperature was kept constant at 37°C.

Probabilities of incorrect pairings for a homology search following a DSB in a random genome: Analytical approach

For a random genome, the probability that one randomly chosen base will match another randomly chosen base is p = ¼. The probability of finding m or more correctly paired bases in a randomly chosen sample of n base pairs is then: (1)

For the 8 bp test, n = 8 and for the triplet tests n = 3. In either case m is the difference between n and the number of mismatches.

Thus, for an 8 bp test with one mismatch n = 8 and m = 7, and for a triplet test that accepts two mismatches n = 3 and m = 1.

The probability of passing a series of tests is just the product of the probabilities for all of the tests. Thus, for a given L_test, the probability of passing an 8 bp test followed by a series of triplet tests is the product of the probability of passing the first 8 bp test times the probability of passing (L_test-8)/3 triplet tests, where N_mismatch and N_mismatch8 are the number of mismatches accepted per triplet and 8 bp test, respectively.

(2)

The formula implies that the results for a series of triplet tests that accept only one mismatch per triplet are much more stringent than the results of a test that accepts 1/3 mismatches in L_test-8 bases.

For a genome that includes L_genome randomly chosen bases, and a randomly chosen sequence of length L_test, in the genome there are on average Pass(L_test, N_mismatch, N_mismatch8) L_genome matching sequences of length L_test.

In DSB repair for a random genome, we assume that the sequence of the searcher is always included in the target; therefore, there is also always one correct pairing between the searcher and the target, regardless of the probability that a randomly chosen searching sequence would find a match. Thus, for a random genome during the homology search that follows a DSB, the average number of pairings that would pass the RecA homology test is: (3)

Of those that pass, probability of being correct is then 1 out of the total number or (4)

The probability that a pairing passes and is incorrect is then: (5)

We note that all predictions of the analytical treatments are independent of the following: 1. The directionality of strand exchange; 2. whether strand exchange is unidirectional or bidirectional; 3. whether the bases are tested iteratively or all at once.

Probabilities of incorrect pairings for a bacterial genome using the simplified model that is applied sparsely to the sequences of bacterial genomes or a random genome: Simulation approach

A position in the given strand of genome was randomly chosen as the position of the 5′ end of the invading strand. A second position in the given strand of the genome was randomly chosen to represent the testing position in an unbroken chromosome that pairs with the 5′ end of the invading strand. The sequences of the 8 bp starting at the 5′ end and extending in the 3′ direction were then compared. If the number of mismatches was > N_mismatch8, homology testing at that position terminated and a new testing position was chosen. If the number of mismatches was ≤ N_mismatch8, then the next 3 bases on the 3′ side were tested for homology. Homology testing terminates, and a new testing position is chosen whenever the number of mismatches in a triplet > N_mismatch. If the length that has passed the homology tests reaches L_test, then the pairing is considered irreversible. If the test position and the search position are the same, the irreversible pairing is correct. If the test position and search position are different but there are no mismatches, then the irreversible pairing joins two copies of a repeat with length ≥ L_test. If the irreversible pairing contains mismatches, the pairing is an error. Typically, 100–100000 different invading strand sequences were chosen, and the results represent the averages of the results for the chosen invading strand sequences out of 4.6 Mbp possible invading strand sequences.

The computer simulations only included ∼ 100 to 10000 simulated DSB positions, so the genome was sparsely sampled; however, we increased the number of simulated breaks until results were insensitive to the number of simulated breaks. In addition, when no mismatches are accepted it is possible to sample the entire genome, and results for sampling the entire genome are similar to results for sparse sampling. We note that results for sampling the entire genome and rejecting all mismatches are independent of the directionality of strand exchange and are not affected by whether strand exchange is unidirectional or bidirectional. Finally, we also applied the computer simulation to random genomes, and the results of the computer simulations are in good agreement with the analytical results for sequences whose bases are chosen at random (random genomes).

We note that the results of these simulations are independent of whether the testing occurs simultaneously or iteratively. The model does assume that the triplet tests occur on the 3′ side of the 8 bp test, but results are insensitive to the positioning of the 8 bp test with respect to the triplet tests. Thus, bidirectional homology testing or 3′ to 5′ homology testing would give very similar results. Finally, exact homology testing that rejects all mismatches is independent of strand exchange directionality, and exact homology testing predicts the same stringency vs. L_test as homology testing in which the triplets are positioned to the 3′ side of the initial 8 bp test.

Exact probabilities of incorrect pairings when N_mismatch = 0 for entire bacterial genomes

Each possible 8 bp sequence was assigned a unique mapping number. The bacterial genome was divided into 8 bp sequences, each of which was assigned the corresponding mapping number. The 8 bp sequences were then sorted according to their mapping number, which grouped the entire genome into distinct 8 bp repeats. The total number of incorrect pairing locations for each member of the group is equal to the number of locations in the group -1. We repeated the same procedure for 11 bp sequences. To calculate exact repeats for 14, 17, 22, 33, 44, 55, and 99 bp we created a list of maps encoding a series of sequences with lengths ≤ 11 bp. For example, for L_test = 14 we used an 11 bp map and a 3 bp map and then grouped starting locations with the same values for both maps. For the E.coli MG1655 genome most 99 bp repeats occurred only twice, but some repeats occurred more than twice. The most frequent 99 bp repeat occurred 9 times. We obtained histograms of the long repeats by sorting the 99 bp sequences according to the starting position and counting the number of sequential starting positions.

Given the number of unique repeats with length L_test and the frequency of each repeat (frequency_repeat), we calculated the probability that a DSB in a bacterial genome would lead to an incorrect final pairing. For each repeat, the number of incorrect targets in the genome is given by (frequency_repeat -1), whereas the number of correct targets is 1. Thus, the probability that a sequence matched pairing of the repeat is incorrect is (frequency_repeat -1)/ (frequency_repeat). The probability that a DSB creates an invading strand terminating in this repeat is (frequency_repeat)/(genome length). Thus, the total probability that an invading strand consisting of the particular repeat leads to an incorrect pairing is (frequency_repeat -1)/(genome length). Summing over the result for all unique repeats gives the total probability that a DSB will lead to a sequence matched incorrect pairing with length L_test.

We note that all predictions of the exact probabilities for bacterial genomes are independent of the following: 1. The directionality of strand exchange. 2. Whether strand exchange is unidirectional or bidirectional. 3. Whether the bases are tested iteratively or all at once.

Calculation of the number of distinct long repeat pairs in bacterial genome

To determine the prevalence of longer repeats, each possible 12 bp sequence was assigned a unique mapping number. The bacterial genome was divided into 12 bp sequences, each of which was assigned the corresponding mapping number. The 12 bp sequences were then sorted according to their mapping number, which grouped the entire genome into distinct 12 bp repeats. Longer repeats were probed by extending the 12 bp sequences within each mapping group and counting only those pairs that had no mismatches over the length L_test. If the number is non-zero, that implies that there must be some repeats that have lengths ≥ L_test. If most repeats only occur twice, the ratio of the number of distinct pairs to the length of the genome gives the probability that a DSB will lead to an incorrect pairing of that length.

Probabilities of incorrect pairings for a bacterial genome using the simplified model with Chi sites that sparsely samples bacterial genomes

A DSB position in the genome was randomly chosen. The nearest Chi site on the 5′ side of the DSB was found. The 3′ end of the invading strand was positioned at the 3′ end of that Chi site. The homology test considers the L_test bases on the 5′ side of the Chi site.

Simplified probabilistic model of homology testing in triplets

A random number generator creates a number between 0 and 1. Tripletpass_M describes the probability that a triplet with M mismatches will pass the triplet homology test. A triplet passes the homology test if Tripletpass_M is greater than the random number. We first considered homology tests with Tripletpass_M = 1 when M = 0, which means that the triplet is completely sequence matched. We considered two fundamentally different homology testing models. In one model, Tripletpass_M is the same for all M > 0. This represents the case in which collective interactions within the triplet allow a single mismatch to destabilize the triplet. For this model in which all triplets containing mismatches have the same probability of passing. We considered two different passing probabilities: Tripletpass_M = 0.25 and Tripletpass_M = 0.5.

We also considered models in which each mismatch decreases the probability that a triplet will pass the homology test. In that model, a random number generator creates a number between 0 and 4. We ran a model in which a homology test of a triplet passes the triplet if the number of mismatches is less than or equal to the value determined by the random number generator. Since 0 mismatches is less than or equal to all those values, a triplet with 0 mismatches has a 100% chance of passing the homology test. Similarly, 1 mismatch, 2 mismatches, or 3 mismatches have a 75%, 50%, or 25% chance of passing the homology test, respectively. We ran a second more promiscuous homology test in which 1 mismatch, 2 mismatches, or 3 mismatches have a 75%, 50%, or 50% chance of passing the homology test. Thus, the second test is the same as the first except for the probability that a completely mismatched triplet will pass the test.

In vitro experiments probing RecA strand exchange in the presence of several periodically spaced single mismatches

Previous work has suggested that with ATP hydrolysis strand exchange can accept a single mismatch [17]. To gain some insight into the mismatch tolerance of triplet-based homology testing with ATP hydrolysis, we considered invading strands with periodically spaced single mismatches. How the invading strand is divided into triplet homology tests depends on the position at which strand exchange starts; however, periodically spaced single mismatches with 1 mismatch per 3 bases will always have exactly 1 mismatch/triplet. Similarly, periodically spaced single mismatches with 1 mismatch per 6 bases will have exactly 1 mismatch in every other triplet. The mismatch-containing triplets will be separated by single sequence-matched triplets.

We observed strand exchange using FRET due to fluorophores positioned in the homoduplex (Fig 1A). When the homoduplex is base paired, emission of the fluorescein is quenched by the nearby rhodamine fluorophore. When base pairing is disrupted and both fluorophores are separated, fluorescein emission increases. In particular, strand exchange is probed by measuring the interaction between unlabeled 98-nt ssDNA/RecA filaments and labelled 180 bp dsDNA. The same dsDNA was used in all the experiments. The mismatches between the invading and complementary strands were controlled by the sequence of the invading strand (S1 Table).

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Fig 1. Strand exchange across single mismatches monitored using FRET.

(A). Schematics of interactions of 180 bp dsDNA with 98-nt filaments containing different degrees of homology. The purple and light blue lines represent the complementary and outgoing strands, respectively. The gray rectangle indicates base paired regions. The dark blue ovals represent RecA. The red circle (rhodamine) and green star (fluorescein) show the locations of the fluorophores along the 180 bp dsDNA. All the invading strands contain 98 nt. The 16 heterologous bases nearest the 5′-end of the invading strand are shown in black. The green brackets indicate the region of the dsDNA that is partially homologous to the invading strand. The orange regions of the invading strand are partially homologous to the dsDNA. (B). Schematics of the experiments with mismatches periodically spaced 1 mismatch/3 bases. The orange regions of the invading strand include the periodically spaced mismatches, whereas the black and green regions are completely heterologous or completely homologous, respectively. The completely heterologous and homologous regions are also indicated by the black and green brackets. (C). Same as B but for invading strands with 1 mismatch every 6 bp. (D). Change in fluorescein emission (△F) vs. time curves. The dark red, bright red, and pink curves indicate results for 1/3 periodically spaced mismatches with a 20, 12, or 0 nt homologous tail on the 5′-side. The dark blue, medium blue, and light blue curves indicate results for 1/6 periodically spaced mismatches with a 20, 12, or 0 nt homologous tail on the 5′-side. Error bars represent root mean square deviation from two independent runs.

https://doi.org/10.1371/journal.pone.0288611.g001

The dsDNA had fluorophores positioned 90 bp (fluorescein, green star) and 89 bp (rhodamine, red circle) (Fig 1A) from the end of the dsDNA that is homologous to the 3′ end of the invading strand. The unlabeled ssDNA used for the fully homologous filament contained a 16-nt heterologous tail in order to have a total length of ∼ 100 bases (Fig 1A). The ∼ 100 nt length is long enough for ssDNA-RecA filaments to be stable with ATP hydrolysis. The ssDNA-RecA filament was mixed with the 180 bp dsDNA, and the resulting emission of the fluorescein label in the dsDNA was measured over time. To monitor strand exchange, we measure △F, the change in fluorescein emission, as a function of time.

In one set of experiments, the periodic mismatches were distributed over the entire invading strand (Fig 1B and 1C); therefore, strand exchange products with 1 mismatch/3 bases would be rejected by the initial 8 bp test [18], regardless of the tolerance of the triplet tests. To gain insight into the triplet tests we performed additional experiments in which the invading strands included homologous regions at the 5′ end that can pass the 8 bp test, as well as regions with periodically spaced mismatches that would be subject to triplet testing if the initial 8 bp test is passed (Fig 1B and 1C).

We considered interactions with 1 mismatch/3 bases (Fig 1B) and 1 mismatch/6 bases (Fig 1C). In the absence of a homologous 5′ tail, 1 mismatch/3 bases would always be rejected by the initial 8 bp test, and indeed we do not observe any significant fluorescence in that case (pink curve, Fig 1D). When a 12 or 20 nt homologous tail is present, 8 bp tests within the homologous tail can be passed. After that test is passed, strand exchange can progress through the remainder of the invading strand. Having a 12 nt homologous tail is insufficient to produce an increase in emission (Fig 1D, bright red curve) whereas a 20 nt tail might result in a very small increase in emission, but the increase is not statistically significant (dark red curve, Fig 1D). Thus, the results of invading strands with 1 mismatch/3 bases suggest that the triplet testing does not always pass triplets containing one mismatch.

Results of experiments with 1 mismatch/6 bases (Fig 1C) show that invading strands with 1 mismatch/6 bases can pass the initial 8 bp test (Fig 1D, blue curves). Thus, even without a homologous tail, invading strands with 1 mismatch/6 bases can provide information on the probability that a triplet with a single mismatch will be incorporated in a strand exchange product. Consistent with the results in yeast [12], interactions with 1 mismatch/6 bases show an increase in fluorescence due to formation of strand exchange products. The emission for 1 mismatch/6 bases was ∼ 25% of the emission for a perfectly matched invading strand. Thus, like Rad51 [12], RecA sometimes yields heteroduplex products from invading strands containing 1/6 = 16% mismatches. Unsurprisingly, the emission increases as the length of the homologous region at the 5′ end increases.

In these bulk experiments, we cannot determine the probability that a triplet containing a single mismatch will pass a homology test and be incorporated in a heteroduplex product; however, the probability that a triplet containing a single mismatch will pass a homology test is greater than 0, but less than 1.

Analytical treatment of the deterministic homology testing L_test base pairs in a random genome

The in vitro experiments presented above suggest that RecA can form readily observable heteroduplex products that contain 16% mismatches. This poor homology stringency might indicate that RecA mediated homologous recombination alone could not lead to accurate DSB repair; however, statistical considerations show that highly mismatch tolerant homology testing of a large number of base pairs can provide extremely accurate homology recognition. Thus, we studied whether there are L_test values that could allow highly mismatch tolerant homology testing to almost always lead to correct DSB repair.

The in vitro results in this paper indicate the probability that a triplet containing a single mismatch will pass a homology test is greater than 0, but less than 1; however, we first consider simple deterministic homology testing models in which the probability of passing a test is always 1 for interactions that meet a homology threshold and always 0 for interactions that fail to meet that threshold. Such models can provide insight into how the accuracy of mismatch tolerant homology testing is influenced by L_test, even if the models do not accurately capture the behavior of RecA mediated homologous recombination. Table 1 summarizes the terms used in the homology testing models.

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Table 1. List of terms used in the homology testing models.

https://doi.org/10.1371/journal.pone.0288611.t001

We begin by applying the deterministic models to “genomes” consisting of randomly chosen bases. We refer to these sequences as “random genomes”. We begin with random genomes for two reasons: 1. Simple analytical formulas describe the results of applying deterministic models to random genomes (Materials and Methods); and 2. Comparing results for random genomes to results for actual bacterial genomes allows us to highlight how the non-random nature of bacterial genome sequences could affect homologous recombination.

To highlight the influence of testing in triplets [22] and using an initial 8 bp test [16, 18–21], we consider three simple deterministic models (Fig 2). The first model considers the total number of mismatches in all L_test base pairs (Figs 2Bi, S1Ai, S1Bi and S1Ci). If the number of mismatches in L_test base pairs is > α L_test, then the homology test always rejects the attempted base pairing and no stable heteroduplex product is formed. Otherwise, the homology test is passed, and a stable heteroduplex product is formed. Mismatch tolerance increases with α. Importantly, the result of the homology test is completely insensitive to the distribution of the mismatches within the L_test base pairs. We present this simple and very incorrect model to highlight how homology recognition is improved by dividing homology testing into base pair triplets and incorporating an initial 8-bp test.

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Fig 2. Illustrations of simple homology testing models that compare L_test base pairs in the invading and complementary strands.

(A). RecA mediated homologous recombination forms a D-loop in which the invading (orange) and complementary (purple) strands bind displacing the outgoing strand (blue). The gray regions indicate base pairing. (B). Different homology testing models: i. the test is passed if the total number of mismatches in L_test bp is ≤ α L_test. The strictness of the test depends on the value of α If α= 0, no mismatches are accepted. ii. The L_test base pairs are divided into triplets. Six separate triplets are shown. Each triplet is tested separately. An individual triplet passes the homology test if it includes ≤ N_mismatch mismatches. The L_test bp pass the homology test if every individual triplet passes the triplet test. If N_mismatch = 0, no mismatches are accepted. iii. Triplet testing is the same as in ii, but testing also includes an 8 bp test. The 8 bp test is passed if the 8 bp include ≤ N_mismatch8 mismatches. The L_test bp pass the homology test if the 8 bp test is passed and every individual triplet passes the triplet test.

https://doi.org/10.1371/journal.pone.0288611.g002

The light blue lines with circular symbols in Fig 3 show the result for a homology test that does not accept any mismatches (α= 0). In the random genome, good stringency can be achieved by testing < 20 bp, which is why early work suggested that it was reasonable that RecA rejects homologous products that extend over less than 20 bp [11].

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Fig 3. Probability of an incorrect final pairing vs. L_test for a random genome.

In all panels, the light blue (circles) curve shows the result if no mismatches are accepted. (A). The red (triangles), gray (diamonds), and black (squares) curves show the result if the number of accepted mismatches is αL_test = (m/3) L_test where m = 1, 2, or 3, respectively. The results for m = 2 and m = 3 are indistinguishable. (B). The curves show the results in which the L_test base pairs are grouped into triplets. Each of the L_test/3 triplets is tested for homology. The homology test of an individual triplet is passed if the number of mismatches in the triplet is less than N_mismatch. All L_test base pairs pass the homology test if all L_test/3 triplet homology tests are passed. The red, gray, and black curves with black outlined triangles, diamonds, and squares show the result for N_mismatch = 1, 2, or 3, respectively. (C). The solid curves show results for homology testing that includes an 8-bp homology test that accepts one mismatch. (N_mismatch8 = 1) and homology testing of (L_test-8)/3 base pair triplets. The triangle, diamond, and square symbols indicate results for triplet tests that accept N_mismatch bp for N_mismatch = 1, 2, and 3, respectively.

https://doi.org/10.1371/journal.pone.0288611.g003

In Fig 3A, the curves indicated by the red triangles, gray diamonds, and black squares represent results for α= 1/3, 2/3, and 3/3, respectively. Fig 3A shows that homology testing of ∼ 50 bp provides good stringency for α= 1/3, but for α = 2/3, stringency is poor even if 120 bp are tested.

The second model divides the L_test base pairs into triplets, and then applies a homology test to each of the triplets (Figs 2Bii, S1Aii, S1Bii and S1Cii). Passing a homology test of L_test base pairs requires that no triplet includes more than N_mismatch mismatches. Otherwise, the homology test is always failed. The third model applies an 8 bp test that accepts one mismatch [18], with the remainder of the L_test -8 bases tested using the triplet test (S1Aiii, S1Biii and S1Ciii Figs). In both the second and third models, homology testing is sensitive to the distribution of mismatches within the L_test base pairs.

Fig 3B shows results for the model divides the L_test base pairs into triplets and then applies a homology test to each of the triplets. Again, there is a simple analytical formula that gives the stringency as a function of L_test. In Fig 3B, the curves with the red triangles, gray diamonds, and black squares correspond to results for triplet tests with N_mismatch = 1, 2, and 3. Thus, the total number of allowed mismatches in Fig 3B is (N_mismatch/3) L_test. Importantly, for a given α, the total number of mismatches allowed in Fig 3A is α L_test, and the symbolism for the curve in Fig 3A corresponding to α is the same as the symbolism in Fig 3B corresponding to N_mismatch/3 = α. Thus, one can determine whether triplet testing improves stringency by comparing a curve in Fig 3B to the curve in Fig 3A that is represented by the same symbols.

The curves represented by the blue circles are identical in Fig 3A and 3B. Thus, if all mismatches are rejected, dividing homology testing into triplets offers no advantage. In contrast, if some mismatches are accepted, testing in triplets can bring a significant advantage. The advantage of triplet testing can be seen by comparing the curves represented by the red triangles and the gray diamonds. In Fig 3A, the curve with the gray diamonds indicates that α= 2/3 provides poor stringency even after testing 120 bp; however, Fig 3B indicates that testing in triplets allows N_mismatch = 2 to provide good stringency by testing only ∼ 50 bp.

Finally, we considered the third testing model based on previous work suggesting that in an initial 8-bp test [16, 18, 20, 21] is followed by testing in base triplets [22] (Fig 2Biii and S1Aiii, S1Biii and S1Ciii Fig). Again, there is an analytical formula for the stringency, and that formula is independent of the order in which the tests are performed. Thus, the results do not depend on the directionality of strand exchange or the order in which the 8-bp test and the triplet tests are performed.

The results for testing that includes an 8-bp test are shown by the solid lines in Fig 3C. Comparison between the solid curves with the same symbols in Fig 3B and 3C demonstrates that for both for N_mismatch = 1 (red triangles) and for N_mismatch = 2 (gray diamonds), the initial 8-bp test decreases the L_test required to achieve a given stringency. The required L_test decreases because the 8 bp test is stricter than either triplet test. Interestingly, with an 8 bp test, even triplet tests that accept 2 mismatches can provide ∼99% stringency by testing ∼ 75 bp. For N_mismatch = 3 (black squares) the 8 bp test offers a negligible improvement because the triplet test never rejects any pairing. In sum, Fig 3 shows that dividing L_test into groups that are tested for homology separately can greatly improve the stringency of mismatch-tolerant homology testing; however, the stringency of mismatch intolerant testing is not affected by dividing L_test base pairs into groups. We note that all the results in Fig 3 are independent on whether homology testing occurs iteratively or simultaneously (Fig 3); however, iterative testing vastly improves searching speed. Furthermore, in iterative testing, searching speed improves if the 8 bp test occurs before the triplet tests.

Importantly, though different curves in Fig 3 represent various homology testing models, all the curves show that stringency as a function of L_test divides into two regimes. At low values of L_test, most sequences can pass a homology test at several positions in the genome; therefore, stringency is very poor and insensitive to L_test. At higher values of L_test, most sequences can only pass the homology test at the corresponding position in the genome, and stringency increases exponentially with L_test. Of course, the L_test value that divides the regime increases as homology testing becomes less strict.

Deterministic homology testing in bacterial genomes

Bacterial genomes do not consist of randomly chosen bases. Instead, they contain long repeats. S2A Fig shows histograms of the distribution of repeats in E.coli MG1655 that are longer than 99 bp. Importantly, there are 900 positions in a 1000-bp repeat at which a DSB could create an invading strand that includes 100 bp that occur at least one other position in the genome. Thus, S2B Fig shows a graph of the number of times that unique 100-bp sequences appear in the genome.

To determine the stringency for a homology test that considers L_test base pairs in the E. coli MG1655 genome and rejects all attempts to form a heteroduplex product that includes mismatches, we counted the number of 100-bp sequences that occur more than once in the given strand (Materials and Methods). Even a homology test that rejects all mismatches could not reject sequences that join different copies of these 100-bp repeats. S2C Fig indicates that there are more than 13,000 unique 100-bp sequences that occur exactly twice in the given strand of the E. coli MG1655, and one 100-bp sequence has 9 copies in that given strand.

For this work, we make the simplifying assumption that DSB are uniformly distributed across the genome. Furthermore, we assume that homology testing is also uniformly distributed across the genome. Given those assumptions, there is a >1% probability that a DSB at a random position in the E. coli MG1655 genome would lead to a sequence matched heteroduplex product that joins two different copies of a 100 bp sequence. The dark blue diamond in Fig 4B indicates the results of this calculation.

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Fig 4. Probability that after a DSB RecA will create a final incorrect pairing as a function of L_test for the random genome (hollow symbols) and the E. coli MG1655 genome (solid symbols).

(A). The probability that a final pairing will be incorrect as a function of L_test and N_mismatch. The blue line shows results for sparse homology testing that requires complete sequence matching. The red and gray lines with triangle and diamond symbols indicate results for sparse homology testing when the 8 bp test that accepts one mismatch is followed by triplet tests that accept 1and 2 mismatches, respectively. The dark blue vertical dashed line indicates L_test = 14. (B). Same as A with a logarithmic y-axis. The blue diamond highlighted by the blue arrow indicates the exact result for homology testing the entire genome when L_test = 98 and no mismatches are accepted. The yellow region indicates the stringencies that are achieved in the random genome but are not reached in the E. coli MG1655 genome.

https://doi.org/10.1371/journal.pone.0288611.g004

We now compare the results for the random genomes to the results for E. coli MG1655 (Fig 4A and 4B). The light blue lines in Fig 4B indicate that when L_test = 21, if no mismatches are accepted, the stringency for random genomes is ∼ 1-1x10^-6, which is much better than 99%. The stringency in the random genome is high because in a random genome the probability that two bases accidentally match is ¼ (S3 Fig). Thus, the probability that two 21-bp sequences in a random genome match is 1/4²¹∼2x10^-13. This low probability implies that random genomes do not contain long repeats. Additionally, the probability that a 21-bp sequence has a match in a 5x10⁶ nt sequence is 2x10^-13 x 5x10⁶ ∼ 10⁻⁶, consistent with the result shown by the dotted light blue line in Fig 4B. Therefore, Fig 4B indicates that if no mismatches are accepted the stringency achieved in a random genome when L_test = 21 is much better than the stringency achieved in a bacterial genome when L_test = 100 bp because when L_test > 20 bp most incorrect pairings in bacterial genomes join different copies of repeats. In sum, the long repeats in bacterial genomes limit stringency to ∼ 99% unless L_test is larger than 100 bp.

It is worth noting that stringency for bacterial genomes stringency cannot be expressed by a simple analytical formula, and the search technique that we used to find the exact matches does not extend to inexact matches. Thus, we used computer simulations to study how mismatch tolerance influences stringency in bacterial genomes (Materials and Methods). The solid curves with circular symbols in Fig 4 show results for computer simulations of homology testing using an 8-bp test that accepts one mismatch and triplet testing of L_test-8 base pairs for an E. coli MG1655 genome. For comparison, results for this test in a random genome are shown by dotted curves with hollow-square symbols. In both cases, the blue curves show results for tests that reject all mismatches. The remaining curves show results for an 8-bp test that accepts one mismatch, where the remainder of the L_test-8 bases are then tested in base pair triplets. The red and gray curves show results for triplet tests that accept 1 and 2 mismatches, respectively. The results for the random genome and the bacterial genome seem nearly identical (Fig 4A). Fig 4B shows the same results as Fig 4A, except that the y axis is logarithmic. In Fig 4B, it is clear that the results for the random genome and the results for the bacterial genome diverge in the region where stringency > 99% because at those stringencies most incorrect heteroduplex products join different copies of long repeats.

For L_test values that give poor stringency in the random genome, the E. coli results agree well with the results for the random genome (Fig 4) because most groups of L_test bp that pass the homology test in the E. coli genome do not join different copies of long repeats. In contrast, for L_test values that provide > 99% stringency in a random genome (yellow highlighted region in Fig 4B), the E. coli results show much poorer stringency than the random genome. Importantly, once the E. coli stringency reaches ∼ 99%, increasing L_test does not improve stringency. This saturation of stringency occurs for both sparse sampling and complete sampling (S4 Fig) because most incorrect groups of L_test base pairs that pass the homology test join different copies of long repeats (S5 Fig). Other bacterial genomes also show a similar saturation of stringency with L_test because they too contain long repeats (S6–S8 Figs). In sum, independent of any feature of RecA-mediated homologous recombination, if the 3′ end of the invading strand is located at a random position in the genome, then even if homology testing rejects all mismatches more than 1% of the 100 bp groups that pass the homology test would include complementary and invading strands from different copies of long repeats (Fig 4). Furthermore, if testing rejects all mismatches, then testing more than ∼ 14 bp does not significantly improve stringency.

After a DSB, if repair follows the RecBCD pathway the RecBCD protein interacts with the ends of the broken dsDNA [2, 26–28]. The function of RecBCD changes when it recognizes the 8-bp sequence GGCGGCGG, which is called the Chi site [28–30]. If there were no Chi sites in long repeats, then no heteroduplex product could join different copies of long repeats if the RecBCD pathway were followed; however, some Chi sites are positioned in long repeats [31]. Thus, to determine how terminating invading strands at or near Chi sites influences mismatch-tolerant homology recognition, we performed various simulations.

For Chi site simulations, the DSB occurs at a randomly chosen position in an E. coli MG1655 genome; however, the invading strand sequence used in the simulation terminates in the nearest Chi site on the 5′ side of the DSB. The homology test is then applied to the L_test base pairs on the 5′ side of the Chi site. We note that regardless of the direction of strand exchange, these are the L_test base pairs that are included in the heteroduplex that extends to the 3′ end of the invading strand.

There are ∼ 500 Chi sites in each strand of the 4.6 Mbp E. coli MG1655 genome [26]. Thus, the separation between Chi sites frequently extends over thousands of base pairs. As a result, thousands of different DSB positions will produce the same invading strand sequence. A detailed consideration of the distribution of Chi sites in genomes is required to determine whether the RecBCD pathway increases or decreases the probability that one side of a DSB will lead to a pairing between different copies of a long repeat. Interestingly, our simulations indicate that stringency as a function of L_test is not affected by whether or not the invading strand terminates in a Chi site (S8 Fig). These simulations only consider the invading strand formed by one side of the DSB. If the RecBCD pathway is followed, no single DSB could create two invading strands from the same long repeat [31].

Importantly, though different curves in Figs 4 and S6 represent various homology testing models, all the curves show that stringency as a function of L_test divides into three regimes. The first two regimes are shared with random genomes; however, for bacterial genomes an additional new regime begins when stringency as a function of L_test saturates. That asymptotic value occurs because pairings join different copies of long repeats. Finally, for bacterial genomes there is an eventual increase in stringency once L_test ∼ 1000 bp (S4 Fig), consistent with the histogram of repeat lengths shown in S2 Fig.

More realistic homology testing

The results of the deterministic homology recognition models provide insight into how L_test affects stringency. The results also highlight how long repeats in bacterial genomes limit the accuracy of homology testing, even if the testing does not accept any mismatches. Unfortunately, our in vitro results indicate that the deterministic models do not accurately capture homology testing by RecA; the in vitro results suggest that 1-bp mismatch is sometimes accepted and that 1-bp mismatch is sometimes rejected, whereas in the deterministic model the mismatch would either always be accepted or always be rejected. Therefore, we also considered models where the acceptance of mismatches is probabilistic. For example, in such models the probability of accepting a single mismatch in a triplet might be 75%, which implies that the probability that the mismatch would be rejected would be 25%.

Unfortunately, the bulk FRET measurements in Fig 1 do not allow us to accurately determine the probability that a triplet homology test will accept 1, 2, or 3 mismatches within that triplet. Thus, we considered various probabilistic models. In the models, the probability of passing a triplet test is characterized by tripletpass_M, where M represents the number of mismatches in the triplet. It is known that RecA family proteins sometimes reject perfectly homologous sequences [32]. Thus, one might think that it would be useful to run simulations in which perfectly homologous triplets are sometimes rejected (tripletpass₀ < 100%); however, if for all M tripletpassM is proportional to tripletpass₀, then the stringency vs L_test for a homology test with tripletpass₀ < 100% is same as the stringency vs L_test for tripletpass₀ = 100%. Reducing tripletpass₀ below 100% increases searching times; therefore, for all the results shown in Fig 4, we chose tripletpass_M = 100% for completely matched triplets. This is perfectly compatible with many sequence-matched products reversing before they extend over L_test base pairs. Indeed, simulations in which perfectly matched sequences could reverse before reaching L_test showed the same stringency vs. L_test as simulations in which sequence-matched regions never reverse.

The various models we considered may not exactly capture the details of RecA-mediated homologous recombination in vivo, but they allow us to test whether stringency as a function of L_test is very sensitive to the details of the probabilistic homology testing. If the results of our models are insensitive to details of the models, then the modeling results may provide insight to mismatch tolerant homology testing by RecA.

Results of probabilistic homology testing are shown in Fig 5. The triangle and diamond symbols in Fig 5 repeat the curves shown in Fig 4 that indicate results of deterministic homology testing of E. coli MG1655 genomes using an initial 8-bp test followed by triplet tests that reject any triplet with more than N_mismatch mismatches. The results of the deterministic tests can be compared with results of several different probabilistic testing models that include an initial 8-bp test that is followed by triplet tests that depend on the number of mismatches in each triplet.

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Fig 5. Probability that after a DSB RecA will create a final correct or incorrect pairing as a function of L_test for E. coli MG1655 genome for deterministic and probabilistic homology testing.

(A). Probability that a final pairing will be incorrect as a function of L_test and N_mismatch. The red and gray lines with triangle and diamond symbols indicate results for sparse homology testing when the 8-bp test that accepts one mismatch is followed by triplet tests that accept 1 or 2 mismatches, respectively. The orange and purple curves represent various probabilistic testing strategies. For both the orange and purple curves tripletpass_M = 100% for completely matched triplets. The orange and purple curves with the circular and square symbols represent results for homology tests in which tripletpass_M = 25% or 50%, respectively, for any triplet that contains at least one mismatch. (B) The curves with the triangular and diamond symbols are the same as in A. For both the orange and purple curves tripletpass_M = 100% for completely matched triplets. The orange curve shows results for tripletpass_M = 75%, 50%, and 25% for 1, 2, and 3 mismatches, respectively. The purple curve shows results when tripletpass_M = 75%, 50%, and 50% for 1, 2, and 3 mismatches, respectively.

https://doi.org/10.1371/journal.pone.0288611.g005

Previous work has indicated that collective interactions between bases might allow a single mismatch to destabilize a triplet [22, 33, 34]. In that case, tripletpass_M would be the same for all M > 0. In Fig 5A the orange and purple curves show results for tripletpass_M = 25% or 50%, respectively, for all M > 0. For the data shown in Fig 1, when there is a 20-nt homologous tail the invading strand with 1 mismatch per 6 bases includes 10 mismatched triplets. Thus, the probability of getting through all 10 mismatches would be .25¹⁰<10⁻⁶ or .5¹⁰<10⁻³. Of course, a FRET signal may not require progressing through all the mismatched triplets.

To further probe the robustness of the results, we considered probabilistic models in which each additional mismatch decreases the stability of the triplet. In this case tripletpass_M would decrease with M (orange and purple curves in Fig 5B). The orange curve shows results for tripletpass_M = 75%, 50%, and 25% for M = 1, 2, and 3 mismatches, respectively; therefore, for tripletpass_M = 75% the probability of getting through all 10 mismatched triplets would be .75¹⁰ ∼ 6%.

For bacterial genomes the predicted stringencies as a function of L_test fall in the range between the predictions of the simplistic deterministic model for N_mismatch = 1 (Fig 5, red curve) and N_mismatch = 2 (Fig 5, gray curve). We also considered other probability distributions that are compatible with the in vitro results shown in Fig 1, and those results also fall largely in the same range. In sum, for probabilistic models consistent with the in vitro results shown in Fig 1, the stringencies as a function of L_test are insensitive to the details of the models, including whether or not all mismatched triples have the same probability.