Model of GTA evolution by infectious transfer

Here's the description of my model addressing Explanation 1 for GTA persistence.  For now I've just pasted in the text of a Word file I prepared about 10 days ago.

A constant-population-size model of large-head GTA transmission

(Based on Xin Chen’s model, but with stepwise generations and without logistic growth.)

Assumptions:

The population:

1.     Population size is constant.  Loss of GTA+ cells due to lysis during GTA production is made up by growth of all cells after the transduction step.

2.     Dense, well-mixed culture in liquid medium (so cells frequently encounter GTA particles)

GTA production:

3.     GTA particles come in two sizes.  Small particles contain 4 kb DNA fragments.  The hypothetical large particles contain fragments that must be at least 14 kb (the size of the GTA gene cluster) but could be as big as 50 kb. 

4.     The number of GTA particles a cell produces does not depend on the proportion of small and large particles.

5.     DNA packaging by GTA is random; all parts of the cell’s genome are equally represented.  But in this model we only consider the particles containing the full-length GTA cluster.

6.     This is the killer:  If the cell’s chromosome is 5 MB and the large-particle capacity is 15 kb, only 2x10^-4 of large particles will contain complete GTA gene clusters (will be G+ particles).  If we change the large-particle capacity to 20 kb, then about 1x10^-3 of large particles will contain a complete cluster.  A 50 kb capacity and a 3 MB chromosome would probably get it up to about 10^-2.  (And this ignores the recombination machinery’s need for homologous DNA flanking the GTA cluster to promote recombination.)

Transduction:

7.     GTA- cells completely lack the main GTA gene cluster.  They can only be converted to GTA+ by homologous recombination with GTA-containing DNA from G+ particles.

8.     GTA particles cannot tell the difference between GTA+ and GTA- recipients.  Particles capable of transducing GTA- cells to GTA+ can also ‘transduce’ GTA+ cells to GTA+.

9.     All GTA particles produced in one cycle are taken up by and transduce cells in that cycle.  (The efficiency of infection and recombination is 1.) 

10.  The model ignores large and small GTA particles that don’t transduce GTA+.

11.  Each cell takes up only one G+ particle (or none).  This is reasonable, since the number of G+ particles is always going to be much smaller than the number of cells.

Parameters:

F    Initial frequency of GTA+ cells (we want to consider a wide range)

c    Fraction of GTA+ cells producing GTA particles (and consequently lysing).  (In wildtype lab cultures this is <3 o:p="">

b    Number of GTA particles produced by each burst.  Default value is 100.  (We have no actual measurements.)

µ    Fraction of GTA particles that are large.  (We expect this fraction to be small, since large particles have not been observed.)

T    Fraction of large GTA particles that are G+ particles (able to transduce GTA).  (This is limited by genome size, GTA gene cluster size, and the DNA capacity of these hypothetical particles.  Plausible values are between 10^-2 and 10^-4.)

G   µ * T Fraction of GTA particles that contain complete GTA genes.

What happens in one generation:

GTA production and cell lysis:

N   Proportion of GTA particles to cells remaining in the medium after GTA+ cells have burst. 

      = (Fcb)/(1 – Fc)  (Note: Fcb is the GTA production per original cell.  1 – Fc normalizes this to the number of cells remaining after lysis.)

N⁺  Proportion of GTA particles, per remaining cell, that carry the complete GTA gene cluster (are ‘G+’ particles able to transduce the GTA-production genotype to GTA- cells). 

= NµT   =  NG

Fraction of surviving GTA+ cells per original cell (will be normalized to remaining cells later): = F(1 – c)

Transduction:

Fraction of GTA- cells transduced to GTA+: N⁺(1 – F).  {Note: the 1 – F corrects for the G+ particles that attach to and ‘transduce’ GTA+ cells.) 

Fraction of GTA+ cells (per original cell) after transduction:  F(1 – c) + N⁺(1 – F).  (Note: F(1 – c) removes cells killed by lysis, N⁺(1 – F) adds cells gained by transduction.)

Fraction of GTA- cells (per original cell) remaining after transduction:  (1 – F) – N⁺(1 – F).  (Note: 1 – F is the original fraction of GTA- cells, N⁺(1 – F) removes cells lost by transduction to GTA+.)

Cell growth:

Now we normalize the cell numbers to ‘per remaining cell’:

Total fraction of cells remaining after GTA production and transduction:

            1 – (Fc)  (Note: To normalize, divide the above cell fractions by this value.)

Fraction of GTA+ cells after one complete cycle:

F’ = F(1 – c) + N⁺(1 – F) / 1 – Fc

How to evaluate the change in the proportion of GTA+ cells?

We can expand N⁺ and pull out the F, then look at the before/after ratio:

F’   = F * (1 – c) + c * b * F * µ * T * (1 – F) / 1 – (F * c)

      = F * ((1 – c) + C * b * µ * T * (1 – F) / 1 – (F * c)

F’ / F = (1 – c) + c * b * µ * T * (1 – F) / 1 – (F * c)

When the value of this expression is greater than 1, GTA+ is increasing; when it is less than 1, GTA+ is decreasing.

For simplicity, below I combine b, µ & T as the compound variable W.

What happens if we vary F, holding everything else constant?

Increase of GTA+ depends only on W.  If W is >1, GTA+ increases.  If W is <1 decreases.="" gta="" o:p="">

The rate of change is very slow when F is close to 1 (when almost all cells are GTA+), and fast when F is close to 0 (when almost all cells are GTA-).

What happens if we vary c, holding everything else constant?

C affects how fast change happens, but not its direction.  If W>1, GTA+ still spreads; if W<1 decreases="" gta="" o:p="" still="">

What happens if we vary W, holding everything else constant?

If W<1 always="" be="" denominator.="" numerator="" o:p="" smaller="" than="" the="" will="">

If W>1, the numerator will always be smaller than the denominator.

In both cases., all the other parameters cancel out.  This confirms that the direction of selection o GTA+ depends only on whether W is higher or lower than 1.

Would the result change if the population were growing?

I don’t think so, since GTA+ and GTA- cells grow at the same rate.

Since plausible values of W are all much lower than 1, I conclude that GTA+ cells cannot increase by GTA-mediated transduction of GTA- cells to GTA+.

GTA could spread by transduction if it did preferentially package the GTA gene cluster into its particles.  Of course, then it would be a phage.

How the model’s assumptions affect this outcome:

Basically, all the assumptions are either neutral or increase the chance that GTA+ will spread. Making the simulation more realistic would just make things worse for GTA+, not better.

The population:

1.  Population size is constant.  Loss of GTA+ cells due to lysis during GTA production is made up by growth of all cells after the transduction step.

I don’t think adding growth would affect the outcome.

2.  Dense, well-mixed culture in liquid medium (so cells frequently encounter GTA particles).

If the culture were more dilute or poorly mixed, some GTA particles would not find new cells to attach to.  This would reduce the amount of transduction (effectively reducing W).

GTA production:

3.  GTA particles come in two sizes.  Small particles contain 4 kb DNA fragments.  The hypothetical large particles contain fragments that must be at least 14 kb (the size of the GTA gene cluster) but could be as big as 50 kb. 

This is the central assumption of the model.  The size of the small particles is known.  The hypothesized large particles could be as small as 15 kb (allows a bit of homologous sequence on each side of the cluster to promote recombination).  Phage capsids can in principle be very large, but it’s parsimonious to assume a modest size.

4.  The number of GTA particles a cell produces does not depend on the proportion of small and large particles.

Large capsids will require more capsid protein molecules.

5.  DNA packaging by GTA is random; all parts of the cell’s genome are equally represented.  But in this model we only consider the particles containing the full-length GTA cluster.

Experimental results show slightly less packaging of GTA sequences.  If this applies to the hypothetical large particles it would reduce production of G+ particles.  If particles preferentially package GTA, GTA would be a phage.

6.  This is the killer:  If the cell’s chromosome is 5 MB and the large-particle capacity is 15 kb, only 2x10^-4 of large particles will contain complete GTA gene clusters (will be G+ particles).  If we change the large-particle capacity to 20 kb, then about 1x10^-3 of large particles will contain a complete cluster.  A 50 kb capacity and a 3 MB chromosome would probably get it up to about 10^-2.  (And this ignores the recombination machinery’s need for homologous DNA flanking the GTA cluster to promote recombination.)

See point 3 above.

Transduction:

7.  GTA- cells completely lack the main GTA gene cluster.  They can only be converted to GTA+ by G+ particles.

Transduction depends on homologous recombination.  Small GTA particles can transduce functional alleles of individual GTA genes, replacing versions that became mutated or even deleted in an ancestor of the recipient cell.  But they cannot introduce GTA genes into cells that completely lack the GTA cluster, because there will be no homologous sequences to recombine with.

8.  GTA particles cannot tell the difference between GTA+ and GTA- recipients.  Particles capable of transducing GTA- cells to GTA+ can also ‘transduce’ GTA+ cells to GTA+.

I think some phages and conjugative plasmids may be able to detect whether potential hosts/recipients already have the element, but we have no evidence that transduction frequencies differ between GTA+ and GTA- recipients.  Wall et al (1975) surveyed 33 strains and found wide variation in both GA production and transduction, but no correlation between these abilities.

9.  All GTA particles produced in one cycle are taken up by and transduce cells in that cycle.  (The efficiency of infection and recombination is 1.) 

This is unlikely to be true, but assuming this increases the chance that each G+ particle successfully transduces a GTA- cell to GTA+.

If we were to relax this assumption the model would need to include an explicit uptake process and to specify what happens to particles that are not taken up.

10. The model ignores large and small GTA particles that don’t transduce GTA+. 

This should be OK, since these should not interfere with transduction by G+ particles, especially because their total number per cell will be small. Removing this assumption would make GTA + spread less likely.

11. Each cell takes up only one G+ particle (or none). 

This is a reasonable assumption, since the number of G+ particles is always going to be much smaller than the number of cells.  If the number of G+ particles were high, sometimes two G+ particles might inject their DNAs into the same s=cell, which would reduce the efficiency of transduction.

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