Reviewer 1- Prof. Rob J de Boer
“The authors study the effects of coinfection of single cells with several copies of the same virus, considering scenarios where the burst size increases with the number of virus particles in the cell. Coinfection has been studied before, but this scenario with an increased burst size is new, interesting, and potentially true. The paper is well written. One the one hand one could argue that allowing for a positive feedback in the burst size leads to a faster than exponential growth is an expected result. On the other hand, I was not certain of some of the results, suggesting that it would indeed be worthwhile to study a model like this. One general question I asked myself is the case where having n viruses in one cell would just increase the burst size n-fold. My intuition tells me that one should then have no positive feedback, i.e., just exponential growth, because in such a system there would just be less target cell limitation as infected cells remain to be equally functional target cells. Given the rather complicated saturation functions in the model, which could even be wrong (see below), I find this simple case hard to find back in the results. I would recommend to help the reader to develop an intuition for the model by explicitly discussing this case.”
Author response: At the beginning of virus growth, when virus load is relatively low, the number of multiply infected cells is negligible compared to the number of singly infected cells. Mostly, singly infected cells are generated. Hence the growth rate of the virus population is determined by the rate of virus production in singly infected cells. When virus load becomes higher, multiply infected cells make up a higher fraction of the infected cell pool and their contribution will become visible. Because the rate of virus production in the multiply infected cells is higher than in singly infected cells, you will see the acceleration. This acceleration will be more pronounced in the quoted simple case (just increasing the burst size n-fold) compared to a lesser increase in the burst size, because the rate of viral replication in multiply infected cells is faster. We have tried to express this more clearly in the text. We have also added text discussion the conditions that promote acceleration in growth.
“In several places I find that the equations do not match with what is written in the text. This markedly complicated the reading and prevented me from fully understanding the paper.”
Author Reply: we have corrected mistakes that were present in the original version. Thank you for pointing them out! There was also a problem with PDF conversion that messed up some equations, although we are not sure whether this contributed to the problem or not.
“On page 4 you write "On the other hand, previous mathematical models", but where is the "First hand"?”
Author reply: corrected.
“On page 5 you write "can be addressed by specific experiments". Please elaborate: which experiments are you proposing?”
Author reply: we have now suggested experiments in the conclusion section.
“In Eq1 the saturation term is. In the text below the Eq you write that infection occurs at a rate. The latter I understand. The former would allow for infection in the absence of virus.” Author response:
First, the typo in the text has been corrected. It now reads
. Regarding the nature of this infection term, please note that the factor (1+ε) in the numerator is nothing but a constant, which does not change properties of the model. A simple rescaling of
brings the term to its more familiar form,
. The factor (1+ε) was added for convenience , such that the term has the following intuitive limiting cases: (i) when ε -> 0, we have βTv/T and (ii) when ε -> ∞, we have βT v (no saturation). Without (1+ε) in the numerator the latter limit would be zero and require a rescaling of β. Note, however, that the presence of (1+ε) in the numerator does not change biology: when v=0, the infection term is zero.
“A minor point: you write that cells die at a rate dT and aI, but each cell dies at rate d or a. Finally, why use a for a death rate, and why two variables in capitals and one in lower case?”
Author response: regarding first point, it was corrected. We used “a” for the infected cell death rate to differentiate it from the uninfected cell death rate, “d”. We have capitalized “V” throughout the paper for consistency.
“Eq2: please explain that the model is about identical viruses.”
Author response: done.
“On the top of page 6 you write that infected cells are infected at a rate βIiv, but in Eq2 infection of Ii cells happens with the strange saturation term.”
Author response: corrected.
“On page 7 you write that infected cells die at a rate aiIi whereas in Eq2 this is aIi.”
Author response: corrected.
“On page 8 you introduce an even more complicated saturation term, which could be suffering from the same problem with a (1 + η) in the numerator. Now it is more difficult to see whether this is a bug or a feature (I vote for a typo). Why not use a conventional Hill function, and write something likefor the production term? This would have a maximum production of k(1 + g) and be half maximal increaseswhen i-1 = η.”
Author response: Please note that the Hill function mentioned by the reviewer differs from our term by a constant factor (1+η). Therefore, there is no difference between our function and the one suggested. Please see our response about factor (1+ε), which explains the reasons behind using this constant.
“Page 9: Whether or not an infection gets established in the first phase should hardly depend on these effects because an infection is expected to start with a limited number of virus particles and probably hardly any coinfection. Shouldn'tcoinfection not be something happening at higher virus loads?”
Author response: The applicability of the initial condition dependence is now discussed. First, as mentioned in the text, it only occurs if the basic reproductive ratio of singly infected cells is less than one. If the replication rate is faster such that the infection could be established based on the kinetics in singly infected cells, establishment of infection is independent of initial conditions. Having said that, however, we think that the initial condition dependence is still an intriguing result that could have relevance to specific infections. HIV is discussed a lot in the paper. While there is no direct data that examines possible dependencies on initial conditions, it could be hypothesized that this contributes to the relative inefficiency of transmission of the virus to a new host. In order to ensure infection of monkeys with SIV in laboratory settings, a certain threshold infection dose has to be applied. Of course this does not show that the dynamics described here are at work, but it does point to a certain dependence on initial conditions. In addition, we recently examined very early events of viral spread using adenovirus infection of 293 cells in vitro (Hofacre et al. 2012, Virology 423, 89-96). The interesting finding was made that there is a high chance of virus extinction while only one or two infected cells were present, but that no extinction occurs once three infected cells exist. It appears that once thee infected cells exist, the MOI is sufficiently large such that multiple infection occurs (it is a spatially restricted setting), and that multiply infected cells show a significantly higher rate of virus production than singly infected cells. We are in the process of examining this in more detail. We have now brought this example into the discussion.
“Page 11: Do we have any evidence that the initial growth phase is faster than exponential?”
Author reply: No, so far there is no evidence, but it also has not been looked for. We have now extensively discussed this. In the context of HIV, it is possible that the acceleration occurs relatively early and has been missed in data, or that other factors come into play that modify the dynamics, specifics of which are discussed now. We are also in the process of examining this in the context of the adenovirus system mentioned above. The point of the model was to make the opposite assumption compared to previous work that assumed no change in burst size in multiply infected cells and to examine how this affects the dynamics. This information can then be used to gain a better understanding of specific infections. In the case of HIV, the dependence of burst size on the number of viruses in the cell can be measured directly. If the burst size does go up, the dynamics explored here can be investigated specifically. If aspects such as growth acceleration cannot be observed in specifically designed experiments, this means that other factors are important that are not taken into account in standard virus dynamics models, and it can help us identify these factors.
“Page 13: Discussing the effect of saturation, i.e., the value of ε, helps to gain better understanding of the results. Having 1000 target cells at the uninfected steady state and ε = 1, the (corrected?) saturationbasically remains one for a wide range of target cell levels. This implies that initially there is hardly any target cell limitation, and that all virus particles can easily find uninfected target cells. Nevertheless this is the regime where you claim to have have the strongest impact of coinfection. Can this be intuitively explained?”
Author response: We tried to do this. Essentially, the fraction of multiply infected cells is higher if ε is relatively small, so they contribute more strongly to the dynamics as the virus population grows. The fraction of multiply infected cells is very small if we have βTv (very larger ε in our model), and thus their contribution is less noticed and you only observe acceleration when addition of more viruses leads to a very large (unrealistic) increase in the viral replication rate (high g).
“Figures: what is the value of η ? Why not use parameter values that are somewhat realistic?”
Author response: The value of η has now been specified in all legends. Regarding parameter values, while we have discussed HIV a lot, this model is much more general and also is likely interesting with respect to other infections as well (perhaps more so), specifically the adenovirus example described above. HIV is discussed a lot because this is the system where multiple infection has been demonstrated experimentally in the most detailed way. However, it is also an infection that is characterized by many further complications that go beyond the basic virus dynamics model used here. We have added discussion to this effect. Using realistic parameter values will make most sense once a system is fully parameterized (and the model thus becomes truly predictive), as we are currently trying to do with the adenovirus system.
Reviewer 2 – Prof. Ruy M. Ribeiro
“This is a conceptual paper that proposes a simple model to raise interesting ideas about the effects of multiple infection of the same cell by HIV-1. The paper is well written and describes the research clearly. I have only a few comments.”
“In model 1 (and successive forms), it is not clear why the (1+epsilon) appears in the numerator. Or, saying it in another way, why was this form for the saturation of infection with T chosen? More usual forms would be epsilon/(T+epsilon) or even just 1/(epsilon+T), in which case than changing epsilon also implies changing beta. This probably does not make any difference for the analysis and results, but a short comment would be appreciated.”
Author response: We have added a short comment about this saturation term. Yes, a more usual form would be 1/(epsilon+T). The (1+epsilon) in the numerator was introduced for convenience such that increasing epsilon won’t significantly decrease the rate of infection.
“In the first line after model 2, you say that infected cells die with rate a_i I_i, but this is not reflected in the model nor is part of your analyses discussion. In fact, I think it would be good to say something more about the possibility of cells infected with more viruses, and thus producing more viruses, dying more rapidly.”
Author response: In this section, a_i I_i is a typo because we are discussing a model without increased viral replication in multiply infected cells. However, it should apply to the next section (Increased viral replication in multiply infected cells) and the reviewer correctly points out that this should be discussed. We have done this.
“In section 2.3(b), second paragraph, the authors discuss the initial growth rate of the virus and that under the current model it wouldn’t be a simple exponential. This is stressed again later in the Discussion. Something should be said that in general the exponential growth rate seems to decline at high viral loads, as the peak in primary infection is reached, and that there are not a lot of evidence to indicate that the exponential rate actually increases. Perhaps this should come together with the author’s argument that there not enough measurements in early infection. Another possibility is for the authors to give some more details regarding under what parameter regimens one could or could not see this increase in exponential growth rate.”
Author response: We agree. The acceleration should take place before growth slows down at the peak. However, there is so far no experimental documentation that the acceleration indeed takes place in HIV infection. We have now added more text to discuss this, to describe conditions that promote the occurrence of accelerated growth, and to describe possible experimental approaches to test the model.
“The last sentence before section 2.4 is not clear. Shouldn’t the infected cells accumulate in the I_n population? Why is it that in the figure all classes of infected cells go down with large beta?”
Author response: We do not plot I_n. In general, I_n is an artificial end of the multiple infection cascade that has to be assumed in an ODE framework like this. We have added a short explanation.
“InEffect of target cell saturation, when discussing the role of epsilon, it would be good to see if for large values of epsilon, viral load can continue to increase for a very long time and the set-point viral load is much higher, perhaps unrealistically so. Also, in this page, weren’t the co-infection experiments in vitro? If so, it is not clear that they can be used to justify the point that the authors are making regarding low epsilon in vivo.”
Author response: Yes, the experiments were in vitro, and we have now re-written it to reflect this and to be more cautious about what one can say about in vivo. Large values of epsilon do not lead to much higher virus loads in this model due to target cell limitation.
“In the figures for early infection, there doesn’t seem to be a well defined and pronounced peak; which is typically seen in the data. Is this because of the parameters chosen or the structure of the model? If the latter than this is an important issue.”
Author response: This is because of a combination of parameter choice and the lack of immune responses in the model. The model can show a more pronounced peak, but in the absence of immune responses, it can be difficult to account for a very pronounced decline in virus load following the peak, typically observed in in vivo data. We have added text to discuss this.
, where you show the “pro-virus” level, what does that correspond to in the model?”
Author response: It corresponds to the number of viral genome copies across all cells. We have now defined it more clearly.
“The authors study the effect of treatment during primary infection, and show that its success depends, in this model, on the viral load at the start of treatment. Is this also true after the set-point viral load has been achieved – i.e., therapy at different levels of set-point viral load (presumably different individuals) could have implications for treatment outcome?”
Author response: Yes, it would apply to this as well. We have added a figure to demonstrate this.
Reviewer 3 – Prof. Marek Kimmel
“The paper identifies an interesting model of viral infection depending on the initial viral load and draws conclusions concerning multistability of infection under such assumptions. The model includes infection classes defined by the number of viral particles per cell.
I find the paper methodology insufficient in that it neither provides a direct comparison to data based on a particular viral infection, nor does it provide a mathematical analysis, which might characterize general types of behavior. Instead, the authors only list several types of infection where the model might apply and, on the other hand, compute some straightforward mathematical characteristics such as equilibrium solutions.
However, the stability of the equilibria does not seem to have been studied, not mentioning an analysis of basins of attraction. Also, it seems that the same qualitative behavior of the model might be obtained by replacing the specific formula listed on p. 8, with a more general monotonous relationship. I think the paper has some potential for being publishable, but definitely needs more work. In additon I think that the authors might decide if they wish to fit the data or pursue a qualitative analysis of solutions. Both aims are worthwhile, but maybe too much for a single paper.”
Author response: We agree that a detailed stability analysis would be desirable. However, this is very complicated. It is impossible to even write down the equilibria of the system where multiple infection leads to an increased burst size of infected cells. In a follow-up study, we are considering simplified models in order to study these phenomena on a more mathematical level, but this is rather complicated, will take some time, and is beyond the scope of this paper. For this paper, we tried to tackle this numerically, but even this turned out to be characterized by complications. We thus did not want to present it in the current paper but decided to wait until our ongoing work has yielded a clearer picture. The current paper presents several results that are interesting from a biological and virus dynamics point of view and we think that it is valuable to get this out. A detailed mathematical analysis, aimed at a different readership, will follow. We have added some discussion to the manuscript.
The same applies to the expression describing the rate of viral replication as a function of the number of resident viruses in cells. We tried an alternative expression, and results did not change on a qualitative level. However, this again needs to be analyzed more rigorously and generally, using axiomatic modeling approaches, and we are currently working on that. It is premature to present at this stage, and we aim to include this in the same study that examines these phenomena from a more mathematical point of view. Again, we have added some sentences to discuss this.
The most interesting biological system to apply this model to would be HIV infection, because multiple infection is well documented, and we have discussed this case quite extensively in the manuscript However, currently there are no data that show an accelerated virus growth during primary infection. There could be a number of reasons for this, and they are discussed in the revision. The aim of this paper was to show how the very basic virus dynamics can change if you assume that multiply infected cells have a higher bust size in the context of standard virus dynamics models. In particular, it is important to contrast this with a recent study that assumed multiply infected cells to have the same burst size as singly infected cells. Understanding the properties of models with different assumptions is an important contribution such that we can distinguish between hypotheses, or identify further mechanisms that need to be taken into account to explain and reconcile experimental data.