Estimation of hepatitis B virus cccDNA persistence in chronic infection using within-host evolutionary rates

Hepatitis B virus (HBV) infection is a major global health problem with over 240 million infected individuals at risk of developing progressive liver disease and hepatocellular carcinoma. HBV is an enveloped DNA virus that establishes its genome as an episomal, covalently closed circular DNA (cccDNA) in the nucleus of infected hepatocytes. Currently available standard-of-care treatments for chronic hepatitis B (CHB) include nucleos(t)ide analogues (NA) that suppress HBV replication but do not target the cccDNA and hence rarely cure infection. There is considerable interest in determining the lifespan of cccDNA molecules to design and evaluate new curative treatments. We took a novel approach to this problem by developing a new mathematical framework to model changes in evolutionary rates during infection which, combined with previously determined within-host evolutionary rates of HBV, we used to determine the lifespan of cccDNA. We estimate that during HBe-antigen positive (HBeAgPOS) infection the cccDNA lifespan is 61 (36-236) days, whereas during the HBeAgNEG phase of infection it is only 26 (16-81) days. We found that cccDNA replicative capacity declined by an order of magnitude between HBeAgPOS and HBeAgNEG phases of infection. Our estimated lifespan of cccDNA is too short to explain the long durations of chronic infection observed in patients on NA treatment, suggesting that either a sub-population of long-lived hepatocytes harbouring cccDNA molecules persists during therapy, or that NA therapy does not suppress all viral replication. These results provide a greater understanding of the biology of the cccDNA reservoir and can aid the development of new curative therapeutic strategies for treating CHB.


INTRODUCTION
We propose the within--host evolutionary rate of HBV can be used to estimate cccDNA lifespan. We developed a novel mathematical model to determine the relationship between HBV evolutionary rate and the lifespan of cccDNA, and combined with published mutation and evolutionary rates 10,14,16 , we inferred the lifespan of cccDNA during different phases of CHB. To the best of our knowledge, these are the first estimates of cccDNA lifespan in treatment naïve subjects and provide important insights into the HBV reservoir that will be valuable for the design and evaluation of future treatment interventions. A: Simplified HBV replication cycle. A virus particle containing relaxed circular DNA (rcDNA) enters a hepatocyte (blue circle) and is uncoated. The rcDNA is transported to the nucleus (purple circle) and repaired to generate cccDNA. This cccDNA is the transcriptional template for all viral RNAs, including pre--genomic (pgRNA), which is transported to the cytoplasm, encapsidated, and converted into rcDNA by error--prone reverse transcription. The encapsidated rcDNA can be transported back into the nucleus to form more cccDNA (intra-cellular amplification), or enveloped and released as virions that can infect hepatocytes (extra--cellular amplification). B: Structure of the mathematical model. This is a single compartment model representing the burden of cccDNA in the liver, Y, over the course of infection. The cccDNA burden can increase due to amplification (intra--and extra--cellular), where b is a measure of the within--host replicative capacity of cccDNA. cccDNA can be cleared from the liver due to natural cell death, at rate d, cytolytic immune responses at rate δ, and non--cytolytic immune responses at rate c. Proliferation can also result in loss of cccDNA at rate (1--q)(d+δY), where q is probability that an individual cccDNA survives mitosis. C: Representation of the model dynamics and key results, where the numbers give the most likely values inferred by fitting the mutation and evolutionary rates to the model. The darker the colours on the figure the higher the cccDNA burden (reds) and the stronger the immune response (blues).

RESULTS
We developed a mathematical model describing the number of cccDNA molecules in the liver that is independent of infected cell frequency, and accounts for intra-- and extra--cellular cccDNA amplification and loss of cccDNA during hepatocyte mitosis (Fig  1B  and Methods, Eqs 1 and 2). Using this model we derived expressions for the viral generation time, defined as the typical time for one cccDNA molecule to generate another cccDNA molecule at time t since infection, g(t) (Eq 5), and the neutral rate of evolution at time S(t) (Eq 6). At equilibrium, we show that the lifespan of cccDNA, ! , is equal to the virus generation time, ! , which is given by the neutral mutation rate divided by the neutral rate of evolution, ! (Eq 8). The notation used throughout is given in Table 1. Per capita replicative capacity, defined as the per capita growth rate of cccDNA when few cells are infected and in the absence of infected cell death or loss of cccDNA due to non--cytolytic immune responses.

R0
The basic reproductive rate of cccDNA (the number of cccDNA molecules a single cccDNA produces in its lifetime in an otherwise uninfected population of hepatocytes) b= βΚ Replicative capacity of cccDNA (a rescaled measure of the per capita replicative capacity) d Natural death rate of hepatocytes δ Additional death rate of infected hepatocytes due to cytolytic immune responses c Loss rate of cccDNA due to non--cytolytic immune responses q Probability that a cccDNA molecule survives mitosis µ Mutation rate of cccDNA (substitutions per site per reproduction) Lifespan of cccDNA The lifespan of cccDNA molecules most likely changes over the course of HBV infection, and will be influenced by host and viral factors 18 , including the rate of hepatocyte proliferation 19,20 . Early in infection cccDNA is transcriptionally active and translation of pre--core/pgRNA results in detectable levels of hepatitis B e antigen (HBeAg) in the periphery that associates with high HBV DNA levels (viral load --VL) 21 . In later stages of infection after seroconversion and genesis of anti--HBe antibodies there is a loss of HBeAg and more efficient immune targeting of infected cells 22 , leading to a reduction in VL and a shortening of cccDNA lifespan. This HBeAg NEG phase of infection is often associated with the emergence of precore mutations that limit HBeAg expression 23 .The higher hepatocyte death rates during HBeAg NEG CHB infection will induce hepatocyte proliferation 21 . Although the extent to which cccDNA is lost during hepatocyte mitosis is uncertain 8 , unless all cccDNA episomes survive mitosis, the increased proliferation rate of infected cells will shorten the average lifespan of cccDNA 20,24,25 . From the published estimates for the mutation 10   The distributions for cccDNA in stable HBeAg POS and HBeAg NEG chronic infection are based on the neutral mutation rate and rate of neutral evolution (orange and blue lines, respectively). If the cccDNA burden during HBeAg NEG infection is not stable, but gradually falling (i.e. the basic reproduction number, R 0 , is less than one) the lifespan will be slightly less than inferred here. The upper estimate reflects the maximum likely cccDNA lifespan when few cells are infected, based on the neutral rate of evolution during HBeAg--postive infection and assuming no cccDNA survives mitosis (q=0; green line). The shorter lifespan of cccDNA during HBeAg NEG compared to HBeAg POS infection can be explained by higher rates of cccDNA clearance (Eq 9). This may reflect changes in the immune environment due to HBe--antigen seroconversion that is associated with increased cytolytic and non--cytolytic immune responses ( and c respectively). Mutational changes in the virus that limit HBeAg expression may also affect HBV replication and stability of cccDNA 23 . Increased host immune responses during HBeAg NEG infection could push the basic reproduction number, R 0 , of cccDNA below one (Eq 3) due to the higher clearance rates of cccDNA molecules, and also due to . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
(which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint reduced replicative capacity, b, of cccDNA. If R 0 <1, the number of cccDNA will not reach a stable level but will continually decline. In this non--equilibrium situation the lifespan of cccDNA may be less than our inferred 26 days since the viral generation time will be greater than the lifespan of cccDNA (Fig 3 and methods). Our model suggests cccDNA lifespan can be up to two times longer when few cells are infected compared to when most cells are infected cells (see methods; compare Eqs 9 and 10). When few cells are infected there is less cell death due to cytolytic immune responses, a lower rate of hepatocyte proliferation to maintain the number of hepatocytes, and consequently reduced loss of cccDNA via mitosis of infected cells. This is of more than theoretical interest, because when estimating how long it will take to deplete the cccDNA reservoir on treatment, it is the lifespan of cccDNA when relatively few cells are infected that is important since treatment is known to reduce the cccDNA load. The maximum expected cccDNA lifespan, corresponding to HBeAg POS infection, few infected cells, and no cccDNA surviving mitosis, is 123 days (71--472 days; Fig 2, green line). Reports for duck hepatitis B virus (DHBV) show a high proportion of cccDNA survives mitosis 25 . In contrast, for HBV recent experimental 8,24 and modelling 20 results suggest that relatively few cccDNA molecules survive mitosis, making this longer lifespan a reasonable expectation.

Dynamics of the mathematical model
To demonstrate the behaviour of our model we present examples of the dynamics when no cccDNA survives mitosis (q=0, Fig 3; see S1 Fig for model dynamics when q=1). We used parameters that are compatible with our estimated cccDNA generation times (61 days during HBeAg POS infection and 26 days during HBeAg NEG infection). Since hepatocytes are long--lived we defined the natural death rate as d=0.002 per day throughout and, for simplicity, we set c=0 under the assumption that cytolytic responses have greater antiviral activity than non--cytolytic responses. We assume a neutral mutation rate =2x10 --5 s/s/c 10 . The model dynamics when q=1 are similar to the case where q=0, apart from the lifespan of cccDNA in the early stages of infection is predicted to be higher if q=1 (see below). A graphical representation of the results is given in Fig  1C, and a summary of the parameters in Table 2.

HBeAg POS infection
The replicative capacity of cccDNA, b, was chosen to be 0.3/day so that the peak number of cccDNA molecules in the liver is reached at approximately 3 months since infection, in line with reported observations 26 . The death rate of infected cells due to cytolytic immune responses, , was determined assuming a cccDNA generation time at equilibrium of 61 days, and solving Eq 9 for (giving =0.006 per day if q=0; the associated R 0 is 30). Under these assumptions, during the first few months of infection the cccDNA burden (number of cccDNA divided by the maximum number of cccDNA) increases rapidly, leading to a short viral generation time predicted by the model of 3.3 days (Eq 11, Fig 3, S1 Fig). A recent study estimated an eclipse period of approximately 3 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint All rates are given per day, and generation times are listed as days 2 Where three values are given these refer to the alternative parameters used for the different trajectories presented in The neutral rate of evolution is also predicted to be high during this early stage of infection due to the short generation time.
As infection progresses, the viral generation time increases due to fewer susceptible target cells (Eq 5), in line with results in epidemiology 27 , and this in turn reduces the evolutionary rate (Eq 6). This dependency of evolutionary rate on epidemiological dynamics has been noted in a previous simulation study on within--host viral infection 28 , but is generally an underappreciated factor influencing evolutionary rates. At equilibrium, the estimated viral generation time and cccDNA lifespan are the same, and it is this equivalency that enables us to determine these parameters from the neutral rate of evolution, independent of the parameters of the model (see Methods). Due to the long lifespan of infected hepatocytes, a high cccDNA burden is reached in the model. This is in line with observations that most hepatocytes are infected at peak infection 29 .
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HBeAg NEG infection
We assumed the transition from HBeAg POS to HBeAg NEG occurs after an arbitrary amount of time after HBeAg POS equilibrium is reached and associates with a reduced cccDNA generation time from 61 to 26 days. If this reduced generation time is not accompanied by a decrease in replicative capacity, only a modest fall in the cccDNA burden is predicted (Fig 3,  . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint estimated 10--fold reduction in the ability of cccDNA to reproduce during HBeAg NEG compared to HBeAg POS infection. In Fig 3, the orange line shows the model dynamics given this decline in b, and when R 0 =1 during HBeAg NEG infection (i.e. = 0.034 per day and b=0.038 per day). In this case, the cccDNA burden falls at a relatively modest rate. Perhaps more likely is that R 0 < 1 and the number of cccDNA molecules continues to decline. The green line shows the dynamics if R 0 =0.7 ( =0.050). However, even with this modest increase in , the number of cccDNA is predicted to fall rapidly. The difficulty in explaining low but steady VL using standard within--host virus models, and the sensitivity of VL to model parameters when R 0 is close to one, have been acknowledged previously, particularly in relation to HIV--1 infections [32][33][34] . Possible explanations for the low numbers of cccDNA during HBeAg NEG infection and low rates of spontaneous cure include the existence of a small number of hepatocytes that are susceptible to infection, resulting in low numbers of cccDNA molecules even if R 0 is high 32 , or the existence of a metapopulation--type partitioned structure in the liver, which enables the cccDNA to persist when R 0 is low 34 .

Estimated time to eradicate cccDNA on treatment
When few cells are infected, the inferred cccDNA lifespan is 123 days during HBeAg POS infection if q=0. Even with this longer estimate for cccDNA lifespan, if there are 10 12 cccDNA molecules at the start of treatment (see methods), we would expect the reservoir to be depleted after less than ten years of treatment (Eq 13, Fig  4A). Moreover, if treatment is initiated during HBeAg NEG CHB the time to eradicate cccDNA is predicted to be even faster (only 1.5 years) with a lifespan of 26 days, and a lower number of cccDNA molecules (2x10 9 ) in the liver at the start of treatment. However, these predictions are in stark contrast to what is observed in the clinic, where a high proportion of individuals remain infected after many years of continuous treatment 35 and there is no appreciable difference in treatment mediated cure in HBeAg NEG or HBeAg POS patients 36,37 . The discrepancy may arise due to our estimated cccDNA lifespan being too short. An estimated lifespan of 236 days during HBeAg--postive CHB still lies within our 95% confidence interval, and would give a time to eradication, and hence sterilizing cure between 18 and 36 years (Eq 13). However, this does not explain the long time to eradicate cccDNA during HBeAg NEG infection. Alternative explanations include ongoing (albeit reduced) cccDNA amplification during NA treatment (b>0) 37,38 , or the presence of a long--lived subset of infected hepatocytes 24,39 . To evaluate these two scenarios, we modelled cccDNA dynamics in CHB patients on treatment assuming different levels of viral replication (Fig  4B) or a subset of long-lived cells (Fig  4C,  S2  Fig). The dynamics of cccDNA are sensitive to the amount of replication, making it unlikely that ongoing amplification alone explains the failure of treatments to eliminate cccDNA. Apart from a narrow range of replicative capacities, either a high and steady cccDNA burden, or relatively rapid cccDNA elimination, is predicted on treatment. The existence of a long--lived population of infected hepatocytes is more robust to differences in model parameters, with a gradual increase in the time to eradicate cccDNA as the death rate of long--lived cells is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint increased, making a long--lived population a more parsimonious explanation for the slow decline in the HBV reservoir. However, since the decay dynamics of the reservoir on treatment can be complex, and differ between individuals 38 , a combination of factors most likely explains the clinical observations.

DISCUSSION
We provide a new model to estimate the HBV cccDNA lifespan based on reported mutation and within--host evolutionary rates 10,14,16 . The lifespan of cccDNA is an important component of the half--life of the cccDNA reservoir, which describes how the population of cccDNA molecules in an individual declines over time. We predict an average cccDNA lifespan of 61 days during HBeAg POS CHB compared to only 26 days in the HBeAg NEG phase of infection. Although estimates for the mutation and evolutionary rates for HBV are associated with high levels of uncertainty, our predicted lifespan is in agreement with in vitro studies showing a 40 day half--life of HBV cccDNA 2 and an estimated half--life of 33--57 days in woodchucks and ducks in vivo 40,41 . As far as we are aware, this is the first time cccDNA lifespan has been estimated during untreated infection. The lower lifespan during HBeAg NEG infection is consistent with a study in which VL data during therapy was fitted to a mathematical model, concluding that the turnover of infected cells is higher if therapy is initiated during HBeAg NEG infection 22 , although our predictions for cccDNA persistence are longer 22 . The shorter cccDNA lifespan during HBeAg NEG CHB may reflect host immune responses, with our model suggesting a doubling of the clearance rate compared to HBeAg POS infection. However, this increased clearance rate is predicted to have a modest effect on the total number of cccDNA molecules. As well as inferring the lifespan of cccDNA, we inferred cccDNA replicative capacity (a combined measure of intra and extra--cellular amplification). Our results predict an approximate ten--fold reduction in replicative capacity between HBeAg POS and HBeAg NEG phases of infection. This can explain the lower cccDNA levels reported in HBeAg NEG CHB 30,31 , and is consistent with observations that the replicative capacity of cccDNA in the HBeAg NEG � � � . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint phase of infection is reduced compared to HBeAg POS infection 30 . This may reflect immune control at the level of the viral epigenome, but without cell death 42 . Our estimates for cccDNA lifespan have implications for curative treatment strategies. If NA therapy inhibits all cccDNA amplification, we would predict HBV to be cured after 1 to 10 years of continuous treatment. However, this is not observed in the clinic, with only 1% of individuals clearing HBsAg each year 35 . Possible explanations for this discrepancy are that NAs do not inhibit all intra--and extra-cellular amplification 37,38 , or the existence of long--lived infected cells 24,39 . Our model is consistent with the presence of long--lived infected cells providing the most parsimonious explanation for sustained infection on treatment. There is growing evidence that there is negligible intra--cellular cccDNA amplification in human HBV infection 8 , and since NA treatment will inhibit the genesis of viral particles this will prevent extra--cellular amplification. Furthermore, the dynamics of cccDNA clearance is sensitive to the assumed amplification rates, and therefore if amplification alone explains the dynamics we would expect to see a proportion of individuals clearing infection within 1--2 years of starting treatment. The presence of long--lived HBV infected cells has parallels with the HIV reservoir, where long--lived latent--infected CD4 + T cells prevent cure 43 . Distinguishing between residual amplification and long-lived infected cells will help define the expected impact of treatment strategies that prevent cccDNA replication, compared to those directly targetting cccDNA. As HBV evolution will only occur if there is cccDNA amplification, it may be possible to distinguish between these two mechanisms by measuring the rate of cccDNA evolution whilst on treatment. Our estimates of cccDNA persistence and amplification provide insights into mechanisms underlying CHB and will inform our understanding of how spontaneous or therapeutic clearance may be achieved. Given different infection profiles among individuals, and limited datasets available for our model, the confidence intervals of our estimations are wide. Our analysis exemplifies the power of modelling as a tool to inform therapeutic interventions and highlights the need for genomic studies to determine HBV evolutionary rates in CHB.

METHODS
To derive estimates of HBV cccDNA lifespan using the neutral mutation rate and the rate of evolution we developed a deterministic mathematical model describing the dynamics of cccDNA during the course of treatment naïve CHB. We used this model to derive expressions for viral generation time and neutral rate of evolution, both of which are predicted to change during the course of infection. Finally, we derived expressions for the lifespan of cccDNA during (i) stable CHB and (ii) when the proportion of infected cells is low, as would be expected in early stages of infection or in the first few months of NA treatment. A within--host model of HBV dynamics HBV cccDNA can replicate via intra--cellular and extra--cellular routes (Fig  1A), with a reported copy number between 1--50 molecules within a single hepatocyte nucleus 2,24,44-47 (the higher estimates tend to be for DHBV and lower estimates for human . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint HBV 8 ). Since cccDNA can be lost during mitosis, we modelled the number of cccDNA copies in the liver, rather than the number of infected cells. To do this, we implicitly assume that viral production is proportional to the number of cccDNA molecules. This is a reasonable assumption since VL has been reported to associate with increasing cccDNA copy numbers 30,48 . We describe the number of copies of cccDNA in the liver at time t since infection, N(t) as: where the first term describes the increase in cccDNA due to intra--and extra--cellular amplification. We assume that the rate of increase is density dependent, with a maximum per capita growth rate per day and a maximum possible number of cccDNA, K. We assume K is constant since proliferation ensures the number of hepatocytes in the liver remains stable during infection 21 , and since the maximum number of copies of cccDNA that can persist within each hepatocyte is virally controlled [48][49][50] . The second term describes the rate at which cccDNA is lost due to the natural death of hepatocytes and the host immune response, under the assumption that cccDNA is randomly distributed among infected hepatocytes. We assume that hepatocytes, and therefore cccDNA, have a natural death rate d per day. Infected hepatocytes (and hence cccDNA) have an additional death rate per day due to cytolytic immune responses, and cccDNA is lost at rate c per day due to non--cytolytic immune responses. The final term describes the loss of cccDNA due to cell proliferation. Uninfected and infected hepatocytes are assumed to proliferate at the rate ( ) per day, and hence cccDNA will be exposed to proliferation at rate, ( ), with a probability q that a cccDNA molecule will survive mitosis. Since the maximum possible number of cccDNA, K, is constant, proliferation and cell death are balanced, hence: = + ( ) (Eq1b) A complete expression for the dynamics of N(t) can be found by solving Eq 1b for and substituting into Eq 1a. To simplify further, we consider the cccDNA burden in the liver, ( ) = ( ) , rather than the total number of cccDNA molecules, giving us: is a rescaled measure of cccDNA replicative capacity. From this equation we can calculate the basic reproductive rate of cccDNA, R 0 , which is defined as the number of new cccDNA molecules a single cccDNA molecule will produce in a susceptible population of hepatocytes: . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint (Eq3) If R 0 <1, then the infection cannot be sustained in the long term. At equilibrium, the cccDNA burden is given by: which is equivalent to the cccDNA burden during stable chronic infection. Our model considers the number of cccDNA molecules independent of their distribution within cells. This is similar to the "single copy" modelling assumption used in 25 , in which only one cccDNA molecule can persist in a cell, and which was shown to produce almost identical dynamics to one in which multiple copies of cccDNA are explicitly modelled within infected cells 25 .

An expression for the neutral rate of HBV evolution
In a large well--mixed viral population, and in the absence of selection, the rate of evolution at time t is given by ( ) = ( ), where is the (neutral) mutation rate, measured per site per viral generation, and ( ) is the generation time 51 . For our within--host model of HBV infection, g is equivalent to the typical amount of time it takes for one cccDNA molecule to replicate another molecule. This is similar to the meaning of generation time in demography and epidemiology 27,52,53 , and which from Eq 2 is given by: ( ) = 1 1 − ( ) (Eq5) At time t since initial infection, the neutral substitution rate is therefore given by: Since intra--and extra--cellular amplification involve an error--prone reverse transcription step, we have assumed they have similar mutation rates. Substituting ! into Eq 6, we can find an expression for the neutral rate of evolution rate at equilibrium:

Lifespan of cccDNA during steady state infection
In our model, at equilibrium the generation time of HBV will be equal to the typical cccDNA lifespan, ! . At equilibrium the number of cccDNA molecules remains constant, and therefore the rate at which cccDNA is produced is equal to the rate at which cccDNA is lost due to infected cell death, non--cytolytic clearance of cccDNA, and proliferation of infected cells. Since the reciprocal of the production rate is equal to the generation time, and the reciprocal of the rate cccDNA is lost is the typical lifespan of cccDNA, at equilibrium, viral generation time and cccDNA lifespan are . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint identical ( ! = ! ). This relationship holds because of our assumption of constant death rate and hence exponentially distributed lifetimes of cccDNA 27 ; see 27,52 for how this changes for different distributions. Using the equivalence of ! and ! , the lifespan of cccDNA at equilibrium can be determined from the mutation and neutral evolution rates by rearranging the first part of Eq 6: ! = ! ! ! (Eq8) Substituting the expression for S E from Eq 7 into Eq 8, we can write an expression for the lifespan of cccDNA at equilibrium based on the model parameters:

The lifespan of cccDNA when few cells are infected
If infection increases the death rate of hepatocytes, then the level of proliferation (to replace eliminated cells) will be larger the more cells are infected. Consequently, the lifespan of cccDNA when few cells are infected (e.g. during early phases of infection or during spontaneous clearance of infection, or after prolonged successful suppressive treatment) may differ from the lifespan during HBeAg POS or HBeAg NEG steady state infection. By setting Y<<1 in equation 2, we can derive an expression for cccDNA lifespan when the copy number or burden is low: (Eq10) Comparing the expressions for ! and !≪! , we can see that if all cccDNA survives mitosis (q=1) or infection has a minimal effect on the death rate of infected cells ( =0), then cccDNA lifespan remains unchanged during infection (as long as d and c don't change). However, if these conditions are not met, then the lifespan of cccDNA when few cells are infected, !≪! , can be up to double the lifespan during chronic stable infection, ! , for identical model parameters (e.g. when q=c=d=0, and b>> ). As we noted above, the cccDNA lifespan is only equivalent to the generation time at equilibrium. Using equation 5, when few cells are infected, the generation time is given by: (Eq11) This has also been observed in the epidemiological literature 27 . Combining equations 3, 10 and 11 we see that: . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint . https://doi.org/10.1101/2020.02.04.20020362 doi: medRxiv preprint If ! > 1 and few cells are infected (i.e. the number of cccDNA is increasing) the life expectancy of cccDNA will be greater than the viral generation time, whereas if ! < 1 the life expectancy will be less than the viral generation time. This might be the case if, for example, increased immune responses associated with HBeAg NEG infection push ! below one. Estimating the generation time and lifespan of cccDNA from within--host evolutionary rates. During stable chronic infection, the lifespan of cccDNA, L E , equals the viral generation time, g E , with ! = ! (Eq 6). Although the mutation rate of HBV has not been determined, for avian hepadnavirus it has been estimated at 2x10 --5 s/s/c (in the range 0.8x10 --5 to 4.5x10 --5 ; 10 ). Since we are interested in the neutral rate of evolution, we assume that a third of all mutations in non--overlapping reading frames are synonymous, and that synonymous mutations are neutral or nearly neutral 54 , giving a neutral mutation rate of around 0.67x10 --5 s/s/c (0.3x10 --5 to 1.5x10 --5 ) in non-overlapping reading frames. To incorporate the uncertainty associated with this estimate, we assumed the probability of the true mutation rate is log--normally distributed with mean 10 --5.2 and standard deviation 10 0.2 . Using longitudinal HBV sequence data, rates of evolution for non--overlapping regions of the genome were generated using a relaxed clock method, inferring 16.1x10 --8 (8.1 x10 --8 , 25.5 x10 --8 ) substitutions per site per day (s/s/day) for HBeAg POS and 38.9 x10 --8 (27.2 x10 --8 , 51.5 x10 --8 ) for HBeAg NEG chronic infection (the numbers in brackets give the 5% and 95% highest posterior density (HPD) intervals; see Table  5 in 14 ). In a separate study, using data from 55 , the synonymous rate of evolution in non--overlapping genomic regions was estimated as half of the overall rate of evolution 16 . Assuming synonymous mutations are neutral, and that the ratio of synonymous to nonsynonymous evolutionary rates is constant during infection, we therefore take the neutral within--host rates of evolution during the HBeAg POS and HBeAg NEG phases of infection to be half the rates of evolution reported in 14 for non-overlapping reading frames. This gives a neutral rate of evolution of 8.0x10 --8 (4.0x10 --8 , 12.7x10 --8 ) s/s/day during the HBeAg POS phase, and 19.5 x10 --8 (13.6 x10 --8 , 25.8 x10 --8 ) s/n/day during the HBeAg NEG phase. We assumed the probability distributions of these rates are normally distributed, with the standard deviation calculated using the difference between the estimated rate and the 5% HPD. We randomly sampled from each of the probability distribution functions (PDFs) for the mutation rate and substitution rates, and used these values to calculate the generation time of cccDNA during HBeAg POS and HBeAg NEG CHB. This was repeated 100,000 times, from which the probability distributions for cccDNA generation time during HBeAg POS and HBeAg NEG chronic infection were estimated using the built in SmoothKernalDistribution function in Mathematica 56 . Assuming the number of cccDNA rapidly reaches equilibrium during HBeAg POS and HBeAg NEG infection, the virus generation will provide an approximation of the cccDNA lifespan during stable chronic infection (Fig 3).
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Time to cccDNA eradication on treatment
Apart from when treatment is first initiated, the number of infected cells on treatment will be relatively low. Assuming eradication in our model is achieved when fewer than one cccDNA molecule persists, and there is no cccDNA replication whilst on treatment, the time to eradication can be approximated by: (Eq13) where !"!# is the number of cccDNA when therapy is initiated and Ln is the natural logarithm. To determine reasonable values for !"!# , we multiplied the number of hepatocytes in a human liver by the number of cccDNA per hepatocyte during untreated infection. There are about 1.4x10 8 hepatocytes per gram of human liver 57 , and an adult human liver is around 1.5kg, giving approximately 2x10 11 hepatocytes in total. In a recent study, an average of 6.3 copies of cccDNA per hepatocyte were found during chronic HBeAg POS infection, and 0.01 per hepatocyte during HBeAg NEG infection 30 , which gives a total of approximately 1x10 12 copies of cccDNA during HBeAg POS infection and 2x10 9 copies of cccDNA during HBeAg NEG infection.

Model assuming a subset of long--lived hepatocytes
If a proportion, , of hepatocytes are long--lived, the dynamics of cccDNA in 'normal' infected cells, Y[t], and in long--lived infected cells Z[t], are given by: where ! , ! and ! represent the natural death rates, cytoloytic death rate and clearance rates of the very long--lived cccDNA.