The relative contributions of infectious and mitotic spread to HTLV-1 persistence

Human T-lymphotropic virus type-1 (HTLV-1) persists within hosts via infectious spread (de novo infection) and mitotic spread (infected cell proliferation), creating a population structure of multiple clones (infected cell populations with identical genomic proviral integration sites). The relative contributions of infectious and mitotic spread to HTLV-1 persistence are unknown, and will determine the efficacy of different approaches to treatment. The prevailing view is that infectious spread is negligible in HTLV-1 persistence beyond early infection. However, in light of recent high-throughput data on the abundance of HTLV-1 clones, and recent estimates of HTLV-1 clonal diversity that are substantially higher than previously thought (typically between 104 and 105 HTLV-1+ T cell clones in the body of an asymptomatic carrier or patient with HTLV-1-associated myelopathy/tropical spastic paraparesis), ongoing infectious spread during chronic infection remains possible. We estimate the ratio of infectious to mitotic spread using a hybrid model of deterministic and stochastic processes, fitted to previously published HTLV-1 clonal diversity estimates. We investigate the robustness of our estimates using three alternative estimators. We find that, contrary to previous belief, infectious spread persists during chronic infection, even after HTLV-1 proviral load has reached its set point, and we estimate that between 100 and 200 new HTLV-1 clones are created and killed every day. We find broad agreement between all estimators. The risk of HTLV-1-associated malignancy and inflammatory disease is strongly correlated with proviral load, which in turn is correlated with the number of HTLV-1-infected clones, which are created by de novo infection. Our results therefore imply that suppression of de novo infection may reduce the risk of malignant transformation.

HTLV-1 persistence are unknown, and will determine the efficacy of different 23 approaches to treatment. 24 The prevailing view is that infectious spread is negligible in HTLV-1 proviral load 25 maintenance beyond early infection. However, in light of recent high-throughput data 26 on the abundance of HTLV-1 clones, and recent estimates of HTLV-1 clonal diversity 27 that are substantially higher than previously thought (typically between 10 4 and 10 5 28 HTLV-1 + T cell clones in the body of an asymptomatic carrier or patient with 29 HAM/TSP), ongoing infectious spread during chronic infection remains possible. 30 We estimate the ratio of infectious to mitotic spread using a hybrid model of 31 deterministic and stochastic processes, fitted to previously published HTLV-1 clonal 32 diversity estimates. We investigate the robustness of our estimates using two 33 alternative methods. We find that, contrary to previous belief, infectious spread 34 persists during chronic infection, even after HTLV-1 proviral load has reached its set 35 point, and we estimate that between 100 and 200 new HTLV-1 clones are created and 36 killed every day. We find broad agreement between all three methods. 37 The risk of HTLV-1-associated malignancy and inflammatory disease is strongly 38 correlated with proviral load, which in turn is correlated with the number of HTLV-  infected clones, which are created by de novo infection. Our results therefore imply 40 that suppression of de novo infection may reduce the risk of malignant transformation. 41 mechanism are unknown, and have major implications for drug development and 48 clinical management of infection. We estimate the ratio of infectious to mitotic spread 49 during the infection's chronic phase using three methods. Each method indicates 50 infectious spread at low but persistent levels after proviral load has reached set point, 51 contrary to the prevailing view that infectious spread features in early infection only. 52 Risk of disease in HTLV-1 infection is known to increase with proviral load, via 53 mutations accrued from repeated infected cell division. Our analyses suggest that 54 ongoing infectious spread may provide an additional mechanism whereby chronic 55 infection becomes malignant. Further, because antiretroviral drugs against Human 56 than the viral reverse transcriptase used in infectious spread, a lack of sequence 107 variation implies that infectious spread is rare. Third, many HTLV-1 + clones have been 108 observed at multiple time points separated by several years [9,17], and a long-lived 109 clone is very unlikely to be maintained by repeated proviral integration through 110 infectious spread at the same integration site, especially since there are no hotspots 111 of HTLV-1 integration [9]. 112 113 However, these three observations do not necessarily imply that infectious spread is 114 negligible [14], particularly when we consider the total number of clones in the host 115 and the very small proportion of clones that can be sampled. First, estimates of the 116 number of clones have increased over time [9,11,13,15,17,19], and current 117 estimates give approximately 10 4 -10 5 clones in the circulation of ACs and patients 118 with HAM/TSP [10, 21,22]. Second, apparent sequence uniformity may result from 119 repeated detection of sister cells from a small number of expanded clones. That is, 120 because of the limitations of sampling, there is a strong bias to detection of the large 121 clones which expanded through mitosis. Finally, the repeated observation of specific 122 clones over many years does not rule out persistent infectious spread. The 123 observation of a temporary but dramatic PVL reduction in a patient with HAM/TSP 124 following treatment with the reverse transcriptase inhibitor lamivudine [23] implies that 125 infectious spread remains important in HTLV-1 persistence, at least in some cases. 126 127 Even when taking recent estimates of clonal diversity into account, there is still good 128 reason to believe that mitotic spread is predominant, because the 10 4 to 10 5 clones 7 (created by infectious spread) present in one host consist of approximately 10 11 130 infected cells (maintained by mitotic spread). However, this consideration ignores the 131 possibility that clones may be continuously created by infectious spread and killed by 132 the immune response and natural death. 133

134
The aim of this study was to quantify the rate of infectious spread, and thus the ratio 135 of infectious spread to mitotic spread during chronic infection. We first estimated 136 HTLV-1 clonal diversity in 11 subjects using our previously developed method [10]. 137 We next developed a deterministic and stochastic hybrid model of within-host HTLV-138 1 persistence that we fitted to clonal diversity estimates. We further used two 139 alternative approaches to quantify the rate and to ensure robustness of our estimates. 140 First, we developed a simplified model to approximate the upper bound of the rate. 141 Second, we adapted a method originally developed to model naïve T cell dynamics. 142 We find broad agreement between estimates from all methods. We conclude that, 143 during chronic infection, a given HTLV-1-infected cell in the peripheral blood is 144 substantially more likely to be derived by mitosis of an existing clone than by de novo 145 infection, although infectious spread continues throughout chronic infection with an 146 average of 175 new clones created every day. 147 8

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Data sets 149 We apply all three methods described below to previously obtained high-throughput 150 data on HTLV-1 clonality [9]. Each HTLV-1 dataset quantifies the abundance of HTLV-151 1-infected T cell clones in ex vivo peripheral blood mononuclear cells, without selection 152 or culture. We studied 11 subjects, where each subject had three blood samples taken  Table S1 gives the notation used in the three modelling approaches that follow.

179
We consider a system with S(t) clones, where a given clone i has frequency xi(t) at 180 time t. We have the following ordinary differential equations (ODEs) for each clone: 181 is the total number of infected cells summed over all clones at 183 is the proliferation rate of infected cells (i.e. the rate of mitotic spread) 184 which is half maximal when N(t) = K (see supplementary information) and δ is the 185 death rate of infected cells [ Figure 2B]. 186

187
The dynamics of small clones, where random effects are important, will not be 188 adequately described by a deterministic model. Since small clones contain most 189 information about infectious spread, it is important to model these clones accurately, 190 and so we use a discrete stochastic model, in which we consider multiple potential 191 states of each clone and their corresponding probabilities over time.   , will be used below to calculate the expected number of 247 clones at time t [ Figure 2C], which in turn will enable our model to be fitted to HTLV-1 248 clonal diversity estimates [ Figure 2D]. [ Figure 3B], we can summarise Equation (3) using multiple, simpler differential 261 equations below 262  We model the expected number of clones S(t) at time t using by adding the total 275 number of clone "births" b(t) over time (that is, the number of infectious spread events), 276 and subtracting the total number of clone extinctions E(t) over time. b(t) is given by 277 where rI is the per-capita rate of infectious spread, xj(t) is the expected frequency of 279 the j th clone to be born since t = 0 (i.e. Note that b(t) and E(t) are increasing functions since rI, xj(t)  0, and because a clone 283 frequency of zero is an absorption state for the random variable Xj(t). Taking (11) and 284 (12) together we calculate the number of clones S(t) as 285 where S0 is the number of clones at time zero [ Figure Figure 2B-D]. We thus partition our system of HTLV-1 294 master equations [ Figure 2B]. We propagate these systems alternatively and 296 concurrently using "Strang splitting" [supplementary information] [34]. The 297 deterministic system described in Equation (1) has S(t) ordinary differential equations. 298 Since the S(t) can exceed 10 5 , we group clones into categories based on the order of 299 magnitude of their abundance. 300

301
We model the dynamics of clones in the body, and not only the blood, because this 302 allows us to model clone extinction. If zero cells of a particular clone are observed or 303 estimated in the blood, this does not necessarily imply that the clone is extinct, 304 because cells in that clone could remain in the solid lymphoid tissue, which contains 305 98% of lymphocytes. We model clones in the body as a whole to avoid this difficulty, 306 which necessitates the assumption that the clonal population structure in the blood is 307 representative of the HTLV-1 clonal structure in the whole body.   336 We considered a simplified model of HTLV-1 persistence that does not describe 337 individual clone dynamics. If S(t) and N(t) are the number of clones and number of 338 infected cells respectively at time t, and rI, is the per-capita rate of infectious spread, 339 we have the following differential equation 340 where δS(t) is the clone death rate at time t. The first term of Equation (14)  ,ˆf Provided fmax is sufficiently small, then ˆs mall  (which is less than or equal to δ) can be 375 approximated by δ. The error incurred by this approximation decreases as fmax is 376 reduced, and so the infectious spread rate will be best approximated by Further, using a two-tailed binomial test, we found little evidence that this change was 454 significantly different from zero (p = 1 for observed and p = 0.07 for estimated). We 455 therefore make the approximation that HTLV-1 clonal diversity remains unchanged in 456 the chronic phase of infection, after the proviral load has reached steady state. individually. Large clones are modelled deterministically using a system of ordinary 461 differential equations, whereas smaller clones are modelled stochastically by solving 462 each clone as a random variable governed by a birth-death process [ Figure 2B]. The 464 per-capita rate of infectious spread and the expected number of infected cells are then 465 combined to model the birth of new clones (11), whereas the extinction probability of 466 each clone is used to calculate expected clone death (12). The birth and death (or 467 extinction) of clones provide an estimate of the number of clones at equilibrium (13) 468 [ Figure 2C], and it is this value that is fitted to our estimates of HTLV-1 clonal diversity, 469 to infer the per-capita rate of infectious spread [ Figure 2D].  Table 1]. These nine estimates per patient were averaged to calculate the 474 mean rate for each individual. Between individuals, the mean estimated rate of 475 infectious spread was 7.7 × 10 -10 per day, ranging from 2.1 × 10 -10 to 1.7 × 10 -9 per 476 day [ Figure 6A], i.e. varying by almost an order of magnitude. While this per-capita 477 rate is very low, it translates to an average of 175 (range 39 -456) new clones created 478 per day [ Figure 6B]. Therefore the hybrid model predicts that infectious spread is not 479 limited to initial infection, but persists at a low level throughout the chronic phase. 480 Given an estimate of the rate of mitotic spread of 3.2 × 10 -2 per day, our infectious 481 spread estimates imply an average ratio of infectious to mitotic spread of 2.4 × 10 -8 482 (6.6 × 10 -9 -5.3 × 10 -8 ) [ Figure 7]. 483 484 Within individuals the standard deviation between samples in the infectious spread 485 rate was relatively small, with an average of 2 × 10 -10 (5.4 × 10 -11 -4.1 × 10 -10 ) [ Table  486 either proviral load or with the estimated diversity during the chronic phase (this may 488 be due to our 11 patients providing insufficient power). However, unsurprisingly, the 489 estimated number of new clones per day was correlated with both proviral load (R 2 = 490 0.62) and strongly correlated with the estimated diversity (R 2 = 0.99) [ Figure S1]. of the rate ranged between individuals from 2.8 × 10 -9 to 1.7 × 10 -8 per infected cell 514 per day, and thus (given a rate of per-capita mitotic spread of 0.0316 cells per day) 515 estimates of the ratio RSupremum ranged between 8.7 × 10 -8 and 5.5 × 10 -7 [ Figure 6A]. 516 The estimated number of new clones per day using the supremum estimates are 517 unsurprisingly much larger than those of the hybrid, ranging from 516 to 4804, i.e. 518 approximately an order of magnitude higher [ Figure 6B]. 519 520 We further estimated the more restrictive upper bounds of the ratio max f R from Equation 521 (21) for multiple fmax values between 1 and 1000 [ Figure 6A]. These estimates assume 522 that the cell death rate applies to clones with frequencies less than or equal to fmax, 523 and that larger clones do not contribute to the rate. Inaccurate estimates of the clonal diversity may play a significant role but calculations 573 using an alternative, widely used estimator provided even smaller estimates of clonal 574 diversity, and therefore yield an even lower ratio. 575 28 Discussion 576 577 The relative contribution of infectious and mitotic spread to HTLV-1 viral persistence 578 has not previously been estimated, and this has been a long-standing problem in the 579 field. For many years, it was believed that the virus persisted solely by oligoclonal 580 proliferation of latently infected cells, and that infectious spread contributed little if 581 anything to persistence. However, three observations have brought this belief into 582 question. First, the strong, persistently activated host T-cell response to HTLV-1 583 implied that the virus is not latent but is frequently expressed in vivo. Second, high-584 throughput analysis revealed that a typical host carries between 10 4 and 10 5 clones, 585 not ~100 clones as was previously believed. Third, treatment with the antiretroviral 586 In summary, we develop three methods, which have the potential to be applied to a 685 range of areas, and use them to quantify the role of de novo infection in maintaining 686 HTLV-1 viral burden at equilibrium. We find that on average 5 x 10 9 new infected cells 687 are produced every day; of these the vast majority (>99.9%) will arise from division of 688 an existing infected cell and will thus have the same proviral integration site as their 689 mother cell, but a small minority (about 175 cells per day) will arise from infectious 690 transmission and will contain a novel proviral integration site. These estimates suggest 691 that ongoing infectious spread may be a mechanism for malignant transformation that 692

R
Ratio of infectious to mitotic spread derived from value of π and fitted values of rI See Table 1 724