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Adsorption rate constants combine a number of concepts into a single number, as considered here along with more general phage adsorption rate theory.

Please cite as:**Stephen T. Abedon**

Phage Adsorption Theory.__adsorption.phage.org__

Adsorption rates are a function predominantly of virion diffusion rates, host target size, and the likelihood of virion irreversible attachment to the host given encounter. The term 'irreversible' is there solely to the extent that the endpoint of adsorption that is being considered is the point of virion irreversible attachment to a host. That is, reversible attachment could be included as an endpoint instead, or instead simply phage-bacterium encounter could be used as an endpoint. It really just depends on what the interest is. An implicit assumption in the basic theory is that rates of host motion or rates of virion motion that are a consequence of something other than diffusion (e.g., fluid flow) are small relative to rates of virion diffusion. Adsorption rates are expected to be greater given faster virion diffusion, larger hosts, relative rates of motion between phage and host which are fast in comparison to rates of virion diffusion, and with increases in the likelihood of whatever is the endpoint, given phage encounter with a bacterium, that is being employed. The essay consists of five sections titled Theory, Multiplicity of Infection, Collision Kernel, Conclusions, and References. For a multiplicity of infection calculator, see adsorption.phage.org. |

**Theory**

In deriving the theory, we start with the assumption that rates of phage adsorption to bacteria are a function of four parameters: phage density (*P*), bacterial density (*N*), an adsorption rate constant (*k*), and time (*t*).

The adsorption rate constant describes the likelihood of a single phage adsorbing to a single bacterium within some unit volume (generally 1 ml) over some unit of time. A typical unit of time is 1 min, but for many authors 1 hr is used instead.

It is really important to keep track of these units when comparing adsorption rate constants or, instead, when considering theory that is based on the magnitude of these constants.

We can start with the differential equation,

*dP*/*dt* = -*kNP*

This means that the instantaneous change in phage density as a function of time (*t*) is equal to the opposite of the product, in order, of the adsorption rate constant, the bacterial density, and phage density. The variable names are as employed by Stent (1963) .

Note from this equation that the instantaneous rates of phage adsorption will increase as a linear function of bacterial density and also as a linear function of the phage adsorption rate constant. Those issues are crucial to understanding phage biology/ecology since they determine rates of adsorption to bacteria by individual phages.

By contrast, the phage density, as indicated by *P* on the right side of the equation, is there because the left side of the equation refers to the rate of phage *population* adsorption rather than the adsorption of *individual* phages (yes, I know, the distinction is pretty subtle). The *population* rate of phage adsorption will be greater, in a linear fashion, the greater the current phage density, i.e., *P*, or which instead is indicated as *P*_{0}, below.

Alternatively, if we wanted explore the rate of adsorption of a single phage particle, then the equation would be presented instead as,

*dP*/*dt* = -*kN*

The rate at which individual phages find bacteria, in other words, is not a function of phage density. On the other hand, the rate at which individual bacteria are *found* by phages *is* a function of phage density,

*dN*/*dt* = -*kP*

The above equation and indeed all of these equations come with the caveat that bacteria tend to be limited in how many phages they can adsorb, i.e., their (so-called) adsorption capacity. The latter is being ignored here.

The above equations are easily solved, which for the first equation is as,

*P* = e^{-kNt}*P*_{0},

where
*P*_{0} is the phage density at the start of some interval, *t*, and *P* on the left side of the equation may more legitimately be labeled as *P*_{t}. Thus, we have,

*P*_{t} = e^{-kNt}*P*_{0}

**Multiplicity of infection**

As the above is the number of free phages remaining after an adsorption interval of time, *t*, the number of phages which adsorb over that interval is equal to *P*_{0} - *P*_{t}.

The value resulting from this subtraction divided by *N* is the multiplicity of infection (MOI) for that interval. More generally, multiplicity of infection (*M*) can be defined for various adsorption intervals, *t*, as,

*M* = (*P*_{0} - e^{-kNt}*P*_{0})/*N* or *M* = *P*_{0}(1 - e^{-kNt})/*N*

Note that the more phages you start with, then both the more and the faster phages will adsorb to target bacteria, and that this occurs as a linear function of the number of phages you start with.

The multiplicity you end up with is a more complicated function of bacterial density since though you will reach a given multiplicity of infection faster with greater numbers of bacteria (the first *N* in these equations), the overall multiplicity of infection will be smaller unless more phages are added as well. See Abedon (2016). for additional discussion of multiplicity of infection including consideration of its common abuse in the literature.

(Yes, *abuse*, though I probably should say, **ABUSE**!!!)

Keep in mind that *N* refers to the fraction of bacteria that phages are capable of encountering and further that an implicit assumption is being made that environments are homogeneous, and thus that all bacteria are equally likely to encounter a phage. That is, a given phage has an equal probability of encountering any given bacterium. At least some of these assumptions are likely violated when bacteria exist as biofilms, though those complications will not be our concern here.

**Collision Kernel**

All of the other subtleties associated with phage adsorption can be explored by considering the phage adsorption rate constant, *k*.

The phage adsorption rate constant is a special case of the more general variable that can be described as a "collision kernel" or a "collision frequency" as attributed to Smoluchowski (Meyer and Deglon, 2011) or, by Stent (1963) , to "von Schmoluchowski". The collision kernel describes the "number of collisions per unit volume and time" (Meyer and Deglon, 2011) , i.e., as defined for *k*, above.

There are three aspects to the collision kernel. These are particle motion, particle size, and what might be described as a collision efficiency (Meyer and Deglon, 2011) .

For phages there is a tendency to limit considerations (1) of motion to just that of virion diffusion (that is, assuming that hosts are stationary and viruses only move as a consequence of diffusion; here *C*, after Stent, for diffusion *C*onstant), (2) of size to just host size (assuming that virions are small relative to the size of hosts; here *S* for *S*ize), and (3) of collision efficiency to a description of the likelihood of virion attachment to host given collision (or "encounter"; here *f*, after Stent, for e*F*ficiency). The latter is rather than likelihoods of encounter, which would be relevant, for example, were phages attracted to bacteria at a distance, such as magnets are attracted to iron.

Thus,

*k* = *SCf*,

where, as noted, *S* is target *S*ize, *C* is the virion diffusion *C*onstant, and *f* refers to the e*F*ficiency of virion attachment to a host bacterium given encounter, i.e., collision.

From Stent (1963) , target size is considered to be a function of the radius of a spherical target bacterium, *R*, and thus equal to 4π*R*. *C* will vary with both virion properties, such as size (large particles tend to diffuse more slowly), and environment properties, such as the viscosity of the medium through which diffusion is occurring.

The efficiency parameter, *f*, will tend to vary as a function of both phage and bacterial properties. Stronger affinity of phages for receptor molecules found on the bacterial surface and greater numbers of the receptor molecules also as found on the bacterial surface, for example, will tend to increase *f* towards a maximum value of 1.0, which is where every phage encounter with a bacterium results in phage adsorption.

In general, then, we expect faster phage adsorption (1) given greater bacterial target size (including potentially as a consequence of differences in bacterial shape), (2) given faster virion movement, (3) given a *lack of inadequate* virion affinity for cell-surface receptor molecules, (4) given a *lack of inadequate* numbers of receptor molecules on bacterial surfaces, and, though not considered here, (5) also given substantial bacterium movement (Koch, 1960) (Murray and Jackson, 1992).

We can also consider the consequences of the impact of bacterial clumping, such as bacteria "clumping" into microcolonies (Abedon, 2012) .

**Conclusions**

All else held constant, phages should adsorb faster the more bacteria that are present to which they can adsorb.

Bacteria, in turn, should be adsorbed to faster the more free phages which can adsorb them that are present.

Both of these rates should increase given higher virion affinity for bacteria, faster virion diffusion (including in the immediate vicinity of target bacteria, e.g., as affected by bacterial glycocalyx), larger bacteria size, the faster as well as the higher the likelihood that virion encounter with a bacterium is translated into irreversible virion adsorption, and also given an initiation of a virion's search for a bacterium to adsorb that is spatially nearer to the bacterium rather than further away (for the latter, i.e., see Abedon, 2012).

**References**

**Abedon, S. T.** 2012. Spatial vulnerability: bacterial arrangements, microcolonies, and biofilms as responses to low rather than high phage densities. Viruses. **4**:663-687.

**Abedon, S. T.** 2016. Phage therapy dosing: The problem(s) with multiplicity of infection (MOI). Bacteriophage **6**:e1220348.

**Koch, A. L.** 1960. Encounter efficiency of coliphage-bacterium interaction. Biochim. Biophys. Acta **39**:311-318.

**Meyer, C. J., and D. A. Deglon.** 2011. Particle collision modeling – a review. Minerals Engineering **24**:719-730.

**Murray, A. G., and G. A. Jackson.** 1992. Viral dynamics: a model of the effects of size, shape, motion, and abundance of single-celled planktonic organisms and other particles. Mar. Ecol. Prog. Ser. **89**:103-116.

**Stent, G. S.** 1963. Molecular Biology of Bacterial Viruses. WH Freeman and Co., San Francisco, CA.