Mechanistic Model of Natural Killer Cell Proliferative Response to IL-15 Receptor Stimulation

Natural killer (NK) cells are innate lymphocytes that provide early host defense against intracellular pathogens, such as viruses. Although NK cell development, homeostasis, and proliferation are regulated by IL-15, the influence of IL-15 receptor (IL-15R)-mediated signaling at the cellular level has not been quantitatively characterized. We developed a mathematical model to analyze the kinetic interactions that control the formation and localization of IL-15/IL-15R complexes. Our computational results demonstrated that IL-15/IL-15R complexes on the cell surface were a key determinant of the magnitude of the IL-15 proliferative signal and that IL-15R occupancy functioned as an effective surrogate measure of receptor signaling. Ligand binding and receptor internalization modulated IL-15R occupancy. Our work supports the hypothesis that the total number and duration of IL-15/IL-15R complexes on the cell surface crosses a quantitative threshold prior to the initiation of NK cell division. Furthermore, our model predicted that the upregulation of IL-15Rα on NK cells substantially increased IL-15R complex formation and accelerated the expansion of dividing NK cells with the greatest impact at low IL-15 concentrations. Model predictions of the threshold requirement for NK cell recruitment to the cell cycle and the subsequent exponential proliferation correlated well with experimental data. In summary, our modeling analysis provides quantitative insight into the regulation of NK cell proliferation at the receptor level and provides a framework for the development of IL-15 based immunotherapies to modulate NK cell proliferation.


Introduction
Effective immunity against pathogens requires the rapid expansion of lymphocytes capable of an appropriate response. This is illustrated by the substantial numeric increase of murine NK cells early during viral infections [1][2][3][4][5][6]. Vigorous expansion of human NK cells has also been observed during viral infections, including human cytomegalovirus and Hanta virus [7][8]. This viral-induced NK cell proliferation is driven by IL-15 and augmented by signaling through NK cell activation receptors that recognize infected cells [2][3][4][5]. IL-15 not only mediates NK cell proliferation during viral infections but also plays a critical role in NK cell development and homeostasis. Indeed, mice deficient in IL-15 lack NK cells [9] while mice given exogenous IL-15 [5] or that constitutively overexpress IL-15 (e.g., transgenic IL-15 mice [10]) have elevated NK cell numbers. Despite its key role in NK cell physiology, the influence of IL-15 receptor (IL-15R) signaling on NK cell responses (such as proliferation) has not been mechanistically studied and characterized at the cellular level.
The IL-15 receptor is composed of three distinct subunits, a, b, and c. The b and c subunits, shared with the IL-2 receptor, bind IL-15 with intermediate affinity (K d = 10 29 M) and mediate IL-15 signaling [11]. The a subunit is specific to the IL-15 receptor, binds IL-15 with high affinity (K d = 10 211 M), and associates with IL-15Rbc to form the high affinity trimeric IL-15 receptor [12][13]. Constitutive expression of IL-15Rbc is essential for NK cell development and homeostasis. In contrast, IL-15Ra expression is very low on resting NK cells and is upregulated following NK cell activation (e.g., IL-15 stimulation) [3,14]. IL-15 binding to its receptor activates a number of downstream molecules including Janus kinases and STAT transcription factors as well as Akt, PI(3)K, MAPK kinases, and Ras GTPase, ultimately promoting NK cell development, homeostasis, and proliferation [15]. The contributions of IL-15Ra have been best characterized in the trans-presentation of IL-15 (from cells that make both IL-15 and IL-15Ra to NK cells or CD8 T cells) [16][17][18]; however, a number of studies have implicated a role for IL-15Ra in cis-presentation of IL-15 [19][20][21] or participation in a trimeric complex with IL-15Rbc on NK cells or CD8 T cells [12,22].
We have previously examined the NK cell proliferative response to IL-15 at the population level with a two-compartment mathematical model representing quiescent and actively dividing NK cells [23]. Using experimentally derived rate constants, this model was able to accurately predict IL-15-mediated NK cell expansion over time, including changes in NK cell accumulation when IL-15 stimulation was reduced. This approach demonstrated that quiescent and dividing NK cells have distinct division and death rates, which could account for the experimentally observed time delay to first division. Although this modeling approach provided a powerful tool to characterize and understand cytokinedriven proliferation of a population of NK cells, it did not provide mechanistic insight into the regulation of IL-15-stimulated NK cell proliferation at a cellular level. The potential of more mechanistic models to enhance our understanding of the regulation of cellular proliferation is illustrated by several quantitative cellularlevel studies focused on IL-2 stimulation of T cells [24][25][26][27][28]. These studies demonstrated that the most important parameters regulating IL-2-stimulated T cell proliferation were IL-2 concentration, receptor density, and the surface retention of receptor complexes.
We propose that a mechanistic mathematical model incorporating receptor level kinetics will provide a potent approach to understand the influence of IL-15R signaling on NK cell proliferation and will supplement fundamental understanding garnered from qualitative models [1][2][3][4][5][6] and prior quantitative population-level models [23]. Therefore, we present a mathematical model incorporating IL-15R binding and trafficking parameters that modulate IL-15/IL-15R complex levels. Based on model predictions, we were able to draw inferences about NK cell population dynamics and to compare these conclusions with independent experimental results. Observations from this model provide novel mechanistic insights into the factors regulating IL-15-driven NK cell proliferation including the potential contributions of IL-15Ra upregulation on NK cells in mediating the more rapid proliferation of dividing NK cells.

Intermediate affinity binding model
Quiescent NK cells express IL-15Rbc, which binds IL-15 with intermediate affinity. To model IL-15 stimulation of quiescent NK cells, we evaluated the parameters that alter IL-15Rbc and IL-15/ IL-15R complex numbers, including ligand binding and dissociation at the cell surface and in endosomes (Fig. 1A), the internalization, recycling, and degradation of ligand, receptors, and complexes (Fig. 1B), and the constitutive and induced synthesis of receptors (Fig. 1C). These parameters and interactions were incorporated into differential equations describing the IL-15 ligand, receptors, and complexes at the cell surface and in endosomes. We incorporated a number of simplifying assumptions into these differential equations. First, quiescent NK cells express sufficiently low levels of IL-15Ra [3,14] that we assumed that IL-15Ra expression on these cells was effectively zero. Second, we assumed that the expression of the common gamma chain was non-limiting on NK cells which allowed us to represent the intermediate affinity IL-15R as IL-15Rbc heterodimers while ignoring any potential minor contributions of IL-15Rb homodimers. Third, IL-15Rb subunits have a cytoplasmic motif targeting internalized receptors to lysosomes [29][30][31][32], so we assumed that internalized intermediate affinity receptors (R e ) and complexes (C e ) were completely degraded and not recycled back to the cell surface. In contrast, we assumed that free IL-15 in endosomes was recycled back to cell surface with a first order rate constant, k rec , in a similar manner to IL-2 [29] and other soluble ligands such as transferrin [32][33]. (This simplifying assumption minimized ligand depletion in our model which is consistent with observations in our prior in vitro studies [23]. However, ligand depletion might be relevant in some situations, including very low concentrations of IL-15 in small volumes of media in vitro or when increasing the receptor number in silico by substantially modulating the constitutive synthesis rate). Synthesis of IL-15Rbc was assumed to be both constitutive (k s ) and induced by cell-surface associated IL-15Rbc signaling (k syn ). Finally, we elected to ignore the spatial impact of trans-presented IL-15/IL15-Ra by stromal or dendritic cells on increasing the local effective concentration of IL-15 since we were evaluating the influence of IL-15 across a wide spectrum of ligand concentrations. We solved the system of differential equations for a range of IL-15 concentrations (3.9 ng/ml to 2000 ng/ml) using initial variable values and parameter estimates delineated in Tables 1 and 2

Author Summary
Natural killer (NK) cells are innate immune cells that are important in our bodies' initial defenses against pathogens, like viruses. NK cells rapidly proliferate early during viral infections to provide an expanded pool of effector cells to suppress the infection. This proliferative response is driven by a cytokine called interleukin-15 (IL-15); however, the influence of IL-15 and its receptor (IL-15R) in stimulating NK cell proliferation has not been quantitatively characterized at the cellular level. To better understand the factors controlling the vigorous expansion of NK cells during infections, we developed a mathematical model incorporating IL-15R binding and trafficking parameters that regulate the number of cell-surface IL-15/ IL-15R signaling complexes. The analysis of this model provided us with insight on how IL-15-driven NK cell expansion can be modulated through changes in receptor kinetics and expression. Based on model predictions, we were able to draw inferences about NK cell population dynamics and to compare these conclusions with quantitative experimental results. Our results and model have applicability to studies designed to manipulate cell responses in the context of immunotherapies.  Binding: IL-15 binds to receptors on the cell surface with rate k f and dissociate from IL-15R with rate k r . Inside the endosome, the on and off rates k fe and k re reflect modified binding affinity at lower pH. B. Trafficking: Free IL-15 receptors are constitutively internalized with rate k endo , and the ligand bound receptor complexes are internalized with rate k int . The IL-15 ligand recycles from the endosome back to the surface with rate k rec , and the receptors and complexes are sorted for degradation with rate k deg . C. Synthesis: The IL-15R synthesis is constitutive with rate V s and can be induced (k syn ) by signaling initiated by the surface complexes. doi:10.1371/journal.pcbi.1003222.g001

Estimate of NK cell fractional recruitment from model predictions of the cell cycle threshold
We hypothesized that the number and duration of IL-15/IL-15R complexes on the surface of a quiescent NK cell must cross a threshold to trigger sufficient downstream signaling to initiate cell division. In our model, we designated the cell cycle threshold as the minimum number and duration of cumulative complexes necessary for NK cell recruitment into the cell cycle. Because IL-15/IL-15R complex numbers rapidly reached steady state (,2 hrs; Fig. 2C), we made the simplifying assumption that the IL-15 regulated cell cycle threshold was equal to the product of steady state surface complex number and the experimentally determined time delay to first division (t) at low IL-15 concentrations (i.e., area-under-the-curve (AUC) of C s vs t at t = t).
The time delay to first division has been estimated for NK cells stimulated with various concentrations of IL-15 as the time at which the normalized mean division number of dividing NK cells is equal to one [23]. At the lowest concentration at which IL-15 stimulated NK cell proliferation was reliably measured (9 ng/ml), the experimentally determined time delay was 37.8 hours [23]. Our model predicted that stimulation with 9 ng/ml of IL-15 would result in 58 surface complexes/cell at steady state (Fig. 3A). Therefore, we estimated the cell cycle threshold (C s,threshold ) to be 2198 C s N h/cell (58.2 surface complexes/cell N 37.8 hours). Assuming this estimate of cell cycle threshold was a reasonable approximation of the minimal cumulative IL-15R stimulation required to initiate cell division, we predicted that increasing IL-15 concentrations would alter the likelihood that an individual NK cell would enter the cell cycle (at times greater than the time delay to first division) by increasing the cumulative number of steady state surface complexes.
This estimation of the cell cycle threshold for an individual NK cell does not directly enable the determination of the probability of NK cell recruitment into cell division. However, we propose that the likelihood that an individual cell will divide can be estimated by comparing the ''excess'' cumulative IL-15/IL-15R complex stimulation (C s N t) over the C s,threshold at time t (designated as the cell cycle momentum) to the IL-15/IL-15R complex stimulation necessary to achieve complete recruitment of a NK cell population. The number of steady state surface IL-15/IL-15R complexes on quiescent NK cells plateaus at 257/cell as IL-15 reaches saturating concentrations (e.g., 2000 ng/ml; Fig. 2C). Using previously described methods [23], we calculated that 64 hours was necessary for complete recruitment of NK cells into cell division following stimulation with 2000 ng/ml of IL-15 (Fig.  S2). Therefore, the IL-15R stimulus necessary for all NK cells in a population to be recruited into the cell cycle could be estimated as 16448 C s N h/cell (257 C s /cell N 64 h). Using this value, we were able to predict the fractional recruitment of quiescent NK cells into cellular division on a population level following IL-15 stimulation for time t by calculating the net positive cell cycle Influence of receptor binding, trafficking, and synthesis on steady state IL-15/IL-15R complex numbers Based on our initial observations, the number of surface IL-15/ IL-15R complexes appears to be a key parameter in regulating the initiation of IL-15-mediated NK cell division. The equation for C s demonstrates that the interactions that modulate the surface IL-15/IL-15R complex numbers are the binding (k f ), dissociation (k r ), and ligand-induced internalization rate constants (k int ). However, since the number of available free surface receptors also appears in the equation, parameters that affect receptor numbers (including k endo , k s , and k syn ) may indirectly influence steady-state complex numbers. Therefore, we varied the value of each of the candidate parameters while keeping the others constant in the simulations to identify the critical parameters that modulate the numbers of surface receptors and complexes.
First, we varied the value of k f (over a range from 0.01 to 100 of the parameter value determined from prior studies) and evaluated the perturbations to the surface receptor and complex numbers. An increase in the binding affinity between IL-15 and IL-15Rbc significantly reduced the number of free receptors and correspondingly increased the number of surface complexes (Fig. 4A). In contrast, increases in k r had the opposite effect on R s and C s (Fig. 4B). We subsequently investigated varying k f and k r at the same time while keeping the equilibrium dissociation constant  Next we evaluated the influence of the trafficking parameters k endo and k int . A decrease in the constitutive internalization rate (k endo ) was associated with an increase in surface receptors as fewer receptors were endocytosed (Fig. 4D). The accumulation of surface receptors resulted in greater formation of surface complexes (Fig. 4D). Conversely, an increase in k endo resulted in the reduction of steady-state surface receptors and complexes as the receptors were more rapidly internalized to endosomes (Fig. 4D). Decreases in the IL-15-induced internalization rate of IL-15/IL-15R complex (k int ) resulted in greater perturbations of steady state numbers of receptors and complexes than similar changes in k endo as fewer complexes were internalized (Figs. 4D and E). Indeed, large decreases in k int (.33 fold) reduced the internalization of complexes to such an extent that the complexes accumulated at the cell surface without coming to steady-state in our model. In contrast, increasing k int led to decreased surface complexes as complexes were internalized more rapidly (Fig. 4E). The lower steady state levels of surface receptors at high values of k int occurred as the reduction in surface complex numbers resulted in fewer dissociated complexes and decreased induced synthesis of new surface receptors. However, in contrast to increases in k endo , increases in k int led to the reduction but not the depletion of surface receptors.
Finally, we examined the impact of varying either the constitutive or induced receptor synthesis rates (k s and k syn ) on the steady state number of surface receptors and complexes. Changing the constitutive synthesis rate of receptors, k s , resulted in corresponding shifts in the steady state receptor and complex levels. In contrast to k s (a zero-order rate constant), k syn is a first order rate constant, and increases in k syn of more than 10-fold caused large perturbations in receptor and complexes numbers.
Interestingly, our analysis suggests that alterations in binding affinity can mimic the impact of changes in IL-15 concentration. For example, an NK cell maximally stimulated by 2000 ng/ml of IL-15 and an NK cell stimulated by 25 ng/ml of IL-15 in the context of a k f that is increased by 64 fold both maintain 257 surface complexes at steady state. This demonstrates that modifications that result in changes in k f or k r could theoretically reduce the cell's dependence on the concentration of IL-15.

High affinity binding model
We have previously observed that the population-based proliferation rates of dividing NK cells were substantially higher than proliferation rates for quiescent NK cells at various IL-15 concentrations [23]. Although our intermediate affinity binding model demonstrated that sufficient signaling from surface IL-15/ IL-15R complexes resulted in quiescent NK cells initiating cell division and enabled the quantitative estimation of NK cell fractional recruitment into the actively dividing subset, it was insufficient to account for the more rapid NK cell proliferation after an NK cell has started to divide. IL-15Ra is expressed on quiescent NK cells at very low levels, but it is upregulated following NK cell activation [3,14]. We hypothesized that the upregulation of IL-15Ra on NK cells might substantially alter the proliferative response of NK cells by increasing the binding affinity of IL-15 for its receptor. Since surface IL-15/IL-15R complexes served as a surrogate measure of IL-15-mediated signaling, increased surface complexes might be sufficient to account for the more rapid proliferation of dividing NK cells. Therefore, we modified our model to incorporate the upregulation of IL-15Ra on NK cells to evaluate whether this alteration was sufficient to account for the differences in proliferation dynamics between quiescent and dividing subsets of the NK cells.
Our initial modeling simulations demonstrated that IL-15 was in excess even at low ligand concentrations and that IL-15 depletion was minimal (Fig. 2E-F). These observations coupled with the extremely high binding affinity of IL-15Ra for IL-15 [13,34] were sufficient to justify the simplifying assumption that all IL-15Ra molecules were bound to IL-15 [18] and that the IL-15/ IL-15Ra complexes on the cell surface or in the endosome did not dissociate [16][17][18]. This assumption allowed us to mathematically represent IL-15/IL-15Ra complexes as ''ligands'' (L s ) capable of binding to IL-15Rbc with high affinity (Fig. 5A). Thus, our high affinity binding model incorporated the binding of both intermediate and high affinity ligands with distinct binding and dissociation kinetics at the cell surface and in endosomes ( Table 2). Because both ligands bind IL-15Rbc and IL-15Ra has a very short cytoplasmic tail [13], we assumed that the receptor complexes share identical trafficking parameters (Fig. 5B). Following internalization with the rate constant k int , we assumed that both complexes (C s and C s 9) were sorted for degradation with the rate constant k deg . Based on the similarity between IL-15Ra and IL-2Ra, we assumed that the degradation of IL-15Ra not associated with IL-15Rbc was negligible [29]. Since IL-15Ra has been shown to recycle back to the cell surface bound to IL-15 [16][17], we made the simplifying assumption that the high affinity ligand (IL-15/IL-15Ra) recycles to the cell surface with the rate constant k rec . In addition, we assumed that common signaling pathway between C s and C s 9 induces the synthesis of IL-15Rbc and IL-15Ra with rates k syn and k syn 9, respectively (Fig. 5C).
In light of these assumptions, we modified the previous model by incorporating variables and parameters (Table 2) associated with the high affinity ligand, IL-15/IL-15Ra, into the following system of differential equations.   Moreover, substantial numbers of high affinity complexes accumulated in the endosome following internalization (C e 9), due to both ligand-induced internalization of C s 9 and less dissociation of the high affinity ligand in the low pH endosomal environment (Fig. 6F). However, the most striking difference was that the inclusion of IL-15Ra on the NK cell substantially raised the total number of steady-state surface complexes (Fig. 6G). Assuming that the cumulative number of surface complexes (both C s and C s 9) was proportional to the magnitude of IL-15 mediated signaling, IL-15Ra on NK cells amplified the receptor signal much more at lower IL-15 concentrations (Fig. 6H), while higher IL-15 concentrations appeared to mask the impact of IL-15Ra on NK cells due to more effective intermediate affinity receptor binding at higher IL-15 concentrations.
The inclusion of IL-15Ra in the high affinity binding model modified the sensitivity of the model to changes in trafficking and synthesis parameters (Fig. S4). Large increases in the IL-15induced internalization rate (k int ) in both the intermediate and high affinity models were predicted to drive the total complexes on the cell surface to negligible levels (Figs. 4E and S3A). However, in contrast to the elevated steady state receptor numbers observed in the intermediate affinity binding model sensitivity analysis, decreased k int values led to depletion of free surface receptors as the increased numbers of surface complexes (Fig. S4B) stimulated the upregulation of the high affinity ligand, L s . Similarly, increasing the induced synthesis rate of the high affinity ligand (k syn 9) markedly decreased the numbers of surface receptors while increasing the total number of surface complexes (Fig. S4D). In contrast, large decreases in k syn 9 mimicked the intermediate affinity model predictions of receptor and complex numbers as the generation of the high affinity ligand was substantially reduced. Even in the context of the high affinity binding of IL-15 and IL-15Rabc, this parameter sensitivity analysis demonstrates that the NK cell proliferation response could potentially be further modulated by molecular or biochemical manipulations of the internalization and synthesis of the intermediate and high affinity IL-15Rs.

Formation of high affinity complexes facilitates exponential expansion of dividing NK cells
To quantify the magnitude of the NK cell proliferative response to IL-15, we measured 3 H-thymidine incorporation in NK cells incubated in various concentrations of IL-15 for 72 hours (Fig.  S5A). We plotted the thymidine values at each IL-15 concentration as a fraction of the maximum thymidine incorporation (observed at 2000 ng/ml IL-15). Above IL-15 concentrations of 7.8 ng/ml, the response increased exponentially until plateauing at values greater than 125 ng/ml, suggesting that NK cell surface receptors were nearing saturation at this IL-15 concentration.
To determine the functional dependence of NK cell division on IL-15-mediated signaling, we used cell surface IL-15/IL-15R complexes as a surrogate for IL-15 receptor signaling and transformed the abscissa of Figure S5A from IL-15 concentration to the corresponding steady state total surface complex numbers calculated in our high affinity binding model (Fig. 7A). This approach builds on previous studies modeling EGF-mediated fibroblast proliferation and IL-2-stimulated T cell division [24,35]. The increase in surface complex numbers (as the IL-15 concentration was increased) was associated with an exponential increase in the proliferation (i.e., thymidine incorporation) of activated NK cells until surface complex numbers reach saturation at 298 complexes/cell (Fig. 7A). These findings suggest that NK cells expand exponentially after starting to divide and provide additional support for our hypothesis that the magnitude of the NK cell proliferative response is dependent on the total number of steady state surface complexes In the intermediate affinity binding model, we determined that a threshold of 2198 C s N h/cell must be reached to initiate a sufficient number of downstream molecular interactions to initiate NK cell division. Assuming that this cell cycle threshold is also applicable to actively dividing NK cells, the cell cycle threshold and the steady state surface complex numbers from the high affinity binding model allow the calculation of interdivision time (minimal time to complete one cell division) after a cell begins to divide and the estimation of division rates at different IL-15 concentrations. We hypothesized that the increased number of surface complexes on actively dividing NK cells would allow them to traverse the cell cycle threshold in less time. We divided the cell cycle threshold by the total steady state surface complex numbers predicted in the high affinity binding model to calculate the interdivision time ( cell cycle threshold C s zC 0 s ) at four representative IL-15 concentrations (Table 3). Knowing the interdivision time enabled us to estimate the maximum number of times that an NK cell could have potentially divided during a fixed time period, ( t{t interdivision time ). We made the simplifying assumption that the time delay (t) of 32 hours is independent of IL-15 concentration based on previous experimental NK cell studies [23]. We determined the maximum number of cell divisions following stimulation with four representative IL-15 concentrations for 90 hours and compared these predictions with experimental results (Fig. 7B). (The caveat in the experimental studies is that maximal cell divisions at high IL-15 concentrations at later times may be underestimated given that tracking cell division with CFSE is limited to 7 or 8 divisions as the CFSE is diluted with each cell division). Our model's predictions accounted for the experimentally observed maximal number of NK cell divisions. This strong correlation supports our hypothesis that the cell cycle threshold is an inherent cellular parameter that regulates NK cell proliferation. The mean division rate of a population of NK cells provides a more comprehensive assessment of population dynamics than the estimate of the maximum number of cell divisions. We determined the division rate of the fastest dividing cohort of NK cells from the inverse of the interdivision time. However, cells proceed through multiple cell divisions asynchronously. Based on previously published experimental data [23], we made the simplifying assumption that the distribution of dividing NK cell cohorts was approximately Gaussian with regard to division progression (Fig.  S6). Therefore, we reasoned that the division rates within a population of NK cells varied in roughly a Gaussian distribution from a maximum value (represented by the inverse of the interdivision time) to very low values for cells that have divided only once (near zero at times .2t). The mean division rate represents the mean value of this Gaussian distribution estimated as ( 1 interdivision time {0)=2. Mean division rates calculated from the interdivision times at several different IL-15 concentrations (Table 3) were compared with experimentally derived division rates (Fig. 7B). The reasonably good correlation of the predicted   mean division rates and the experimentally derived division rates provides support for our hypothesis that the upregulation of IL-15Ra on NK cells amplifies the stimulatory signal of IL-15, facilitating the exponential expansion of the NK cell population.

Discussion
To quantitatively investigate IL-15-mediated NK cell proliferation at a cellular level, we developed a mathematical model incorporating IL-15R binding and trafficking parameters that modulate IL-15/IL-15R complex levels. Based on model predictions, we were able to draw inferences about NK cell population dynamics and compare these conclusions with experimental results. This approach, leveraging predictions from a mechanistic IL-15R model to make inferences about NK cell behavior on a population level, provided a number of unique insights into the regulation of IL-15-stimulated NK cell proliferation: 1) IL-15/IL-15R complexes on the cell surface are a key determinant of the magnitude of the IL-15 proliferative signal and function as an effective surrogate measure of IL-15R signaling, 2) the cumulative number and duration of IL-15/IL-15R complexes appear to cross a quantitative threshold prior to initiation of NK cell division, 3) upregulation of IL-15Ra on dividing NK cells substantially increases the number of total cell surface IL-15/IL-15R complexes, resulting in an increased division rate, and 4) high affinity binding mediated by IL-15Ra on NK cells particularly enhances cellular responses at low IL-15 concentrations.
In the development of our model, we hypothesized that surface IL-15/IL-15R complexes mediated IL-15 signaling. This hypothesis was based on prior computational and experimental studies in other systems that implicated surface complexes as the critical parameter in determining cellular responses with internalization attenuating receptor-mediated signaling [24,[35][36][37][38][39][40][41]. For example, fibroblasts transfected with an internalization defective EGF receptor mutant required 10-fold less EGF to stimulate a half maximal mitogenic response [35][36]. Similarly, in the type 1 angiotension (AT1) system, overexpression of an accessory protein (ATRAP) increased the internalization of AT1 receptor-angiotension II complexes resulting in decreased phosphorylation of STAT 3 and Akt and inhibition of DNA synthesis [39]. Furthermore, previous studies of the analogous IL-2 system in T cells have also focused on signaling and regulation of IL-2 receptor interactions at the cell surface [24,27,40]. In our work, the utility of steady state surface IL-15/IL-15R complexes in determining the cell cycle threshold and the interdivision time demonstrated that steady state cell surface IL-15/IL-15R complexes were an effective surrogate measure of IL-15R signaling. Indeed, when we modified the calculations to compare cell surface complexes, endosomal complexes, or a combination of both as surrogate measures of IL-15R signaling, the addition of endosomal complexes did not improve the correlation with experimental fractional recruitment data and resulted in worse correlation with experimental data on maximum division number (data not shown).
Our work suggests that NK cells accumulate a sufficient number of IL-15/IL-15R complexes over time before initiating cell division (i.e., cell cycle threshold). The concept of a threshold necessary to begin cell division has previously been proposed for IL-2 stimulation of T cell proliferation [28,42]. Indeed, Fallon and colleagues [24] reported that the growth rate of T cells was minimal below a threshold number of IL-2/IL-2R complexes per cell and plateaued when surface receptors were saturated. Using predictions of steady state cell surface IL-15/IL-15R complex numbers from the intermediate affinity binding model with an experimentally observed time delay to first division at a low IL-15 concentration, we were able quantitatively define the cell cycle threshold for NK cells as 2198 C s : h=cell ( i.e., area-under-thecurve (AUC) of C s vs t when t = t). We utilized this cell cycle threshold to make predictions of fractional recruitment of quiescent NK cells into the dividing population, maximum division numbers of NK cells stimulated with different IL-15 concentrations, and mean NK cell division rates. The correlation of these predictions with experimental results corroborated our approach to quantify the cell cycle threshold necessary for IL-15stimulated NK cell proliferation as well as our hypothesis that the cell cycle threshold is an intrinsic parameter in both quiescent and dividing subpopulations of NK cells and is independent of division number.
After crossing the threshold necessary to initiate cellular division, NK cells proliferate more rapidly in subsequent divisions. Indeed, population-based proliferation rates calculated for dividing NK cells at various IL-15 concentrations were approximately three times higher than those for quiescent NK cells [23]. Our intermediate affinity binding model facilitated the quantitative estimation of the cell cycle threshold necessary for quiescent NK cells to initiate cell division and of NK cell fractional recruitment into the actively dividing subset; however, it was insufficient to account for more rapid proliferation after an NK cell has started to divide. An evaluation of the impact of various binding, trafficking, and synthesis parameters on steady state numbers of surface IL-15/IL-15R complexes illustrated that increasing the binding affinity of IL-15 for its receptor resulted in higher numbers of surface complexes. One way that the effective binding affinity of IL-15 to IL-15Rbc can be physiologically modified is through the upregulation of IL-15Ra on NK cells following IL-15 stimulation. The incorporation of IL-15Ra upregulation on NK cells into our model resulted in significantly higher numbers of steady state cell surface IL-15/IL-15R complexes and was sufficient to account for the more rapid proliferation of dividing NK cells. The maximum division number and mean division rate, calculated with the total cell surface IL-15/IL-15R complexes (C s +C s 9) from the high affinity binding model, correlated well with experimental results. The impact of the upregulation of IL-15Ra on NK cells was greatest at low IL-15 concentrations, with higher IL-15 concentrations masking the influence of IL-15Ra due to increased formation of intermediate affinity receptor complexes.
Given the substantial structural similarities between IL-15 and IL-2 quaternary complexes, differences in IL-2 and IL-15 receptor signaling have been attributed to the much higher affinity binding of IL-15 to IL-15Ra (K D = 38 pM) in comparison to the binding affinity of IL-2 to IL-2Ra (K D = 4.8 nM) [22,34]. IL-2Ra captures IL-2 at the cell surface, enriching the surface-associated cytokine and facilitating IL-2 binding to the intermediate receptor (IL-2Rbc). In contrast, the higher affinity of IL-15Ra for IL-15 (with a substantially lower off rate of 0.05 h 21 compared to the k r = 72 h 21 for IL-2 from IL-2a) results in greater formation and persistence of IL-15/IL-15Ra complexes. The best characterized role of IL-15Ra is in trans presentation (from cells that make both IL-15 and IL-15Ra) resulting in tight control of IL-15 localization and effectively increasing the local concentration of IL-15 that an NK cell or a CD8 T cell experiences [16][17][18]. Although IL-15Ra is strongly upregulated on NK cells following activation [3,14], few studies have focused on the contributions of cis presentation of IL-15 by IL-15Ra [19][20][21]. Given its very high affinity for its ligand, we predict that IL-15Ra on NK cells would be more effective at sequestering IL-15 on the cell surface than IL-2Ra is at enriching IL-2, particularly at low ligand concentrations. Indeed, our model illustrates that IL-15Ra on NK cells amplifies IL-15R signaling to a greater extent at low IL-15 concentrations. Based on the conclusions from our model, we predict that IL-15 stimulated proliferation of NK cells that lack the ability to upregulate IL-15Ra will be less vigorous than the proliferation observed in wild type NK cells.
Our work with IL-15 and its receptor on NK cells builds on the pioneering experimental studies of Cantrell and colleagues [27] and prior computational modeling of the analogous IL-2 system in stimulating T cells [24]. These studies established that IL-2 concentration, receptor density, and the duration of receptorligand interactions were critical factors in IL-2 stimulated T cell proliferation. More recent computational studies [40], which focused on the binding kinetics of IL-2 to IL-2Ra and subsequent interaction with IL-2Rbc over short time periods (e.g., 10 min), demonstrated that the density of IL-2R subunits modulated the sensitivity of the cell's response to IL-2. This work provided novel insight into differences in IL-2 responses between effector T cells and regulatory T cells (which constitutively express IL-2Ra). Although IL-2 and IL-15 share the same intermediate affinity receptor (IL-2/IL-15Rbc), there are a number of differences between these cytokines including distinct individual high affinity receptors, affinity of IL-2 and IL-15 for their receptor subunits (discussed in the previous paragraph), differences in constitutive expression of the receptor subunits (e.g., NK are dependent on IL-15 and constitutively express high levels of IL-2/IL-15Rb), and T cell autocrine production of IL-2. Despite these differences, we were able to utilize insights gained from previous experimental and computational IL-2 studies in our evaluation of the impact of IL-15 receptor binding, trafficking, and synthesis parameters on NK cell proliferation.
The quantitative insights provided by mechanistic modeling of IL-15 and IL-15R interactions on a cellular level combined with computational analysis of NK cell population studies suggests that it may be possible to therapeutically manipulate the interaction of IL-15 and IL-15R to modulate NK cell responses in clinically relevant situations, such as intractable viral infections, cancers, or NK cell lymphoproliferative disorders. Recent studies have demonstrated the potential to modulate cellular proliferation via alterations in cytokine binding or receptor trafficking. For example, engineered IL-2 molecules optimized for binding to either IL-2Ra or IL-2Rb exhibited enhanced IL-2R signaling [43][44] while deficiencies in EGF receptor internalization resulted in increased cellular proliferation [35]. Predictions from our model will not only guide the development of novel therapeutic strategies to modulate IL-15R signaling but also provide testable hypotheses for future experiments including studies of NK cells with mutations or deficiencies in IL-15Ra.

Mice
Female C57BL/6 (B6) mice were obtained from the National Cancer Institute (Charles River, MA). They were maintained under specific pathogen-free conditions and used between 8 and 16 weeks of age. All experiments were conducted in accordance with institutional guidelines for animal care and use based on the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. The protocol was approved by the Animal Studies Committee at Washington University (#20110104).

Proliferation assays
Murine splenocytes were enriched for NK cells via negative selection (Miltenyi, CA) and cultured with murine IL-15 (Peprotech, NJ) as previously described [23]. CFSE-labeled splenocytes were plated in 96-well plates (2.5610 4 NK cells/well) and cultured for 4 days in 200 ml of growth media (RPMI 1640 medium supplemented with 10% fetal calf serum) containing various concentrations of murine IL-15 (PeproTech, Rocky Hill, NJ). Cells were harvested at different time points and analyzed by flow cytometry. The number of dividing NK cells and their proliferation rate were determined using precursor cohort analysis and the two compartment model analysis [23].
Thymidine incorporation assays were performed with B6 splenocytes as previously described [23,45]. Briefly, NK cells from B6 splenocytes were enriched by passage over nylon wool columns and cultured with murine IL-15 (Peprotech, United Kingdom) at concentrations ranging from 1 ng/ml to 2000 ng/ml (0.067 nM to 133 nM) for 72 hours. After 48 hours, 3

Estimate of the maximum number of NK cell divisions
Flow cytometry analysis was performed with a FACScalibur flow cytometer (BD Pharmingen) to detect CFSE dilution in NK cells stimulated by 9, 25, 50, or 75 ng/ml (0.60, 1.67, 3.33 or 5.00 nM) of IL-15 at various times. These concentrations of IL-15 were chosen based on experimental thymidine studies of NK cell proliferation (Fig. S5A) and CFSE-based studies of NK division rates (Fig. S5C) which demonstrated that the linear range around the EC 50 value on a semi-log response curve (representing the range of concentrations over which changes in the stimulus resulted in meaningful changes in response) was from ,8 ng/ml to 125 ng/ml. Above 125 ng/ml, thymidine incorporation plateaued out at maximal values, while the division rate plateaued at concentrations above 50 ng/ml. These values correlated well with the plateauing of the steady state surface complexes predicted by the high affinity model at IL-15 concentrations greater 75 ng/ml (Fig. S5B). OriginPro 7.5 was used to distinguish the CFSE peaks representing cell cohorts that divided the same number of times. The fraction of the population in each cohort was determined from the ratio of area under each CFSE peak to the total area under the curve. The maximum number of divisions that NK cell have completed was determined from the total number of CFSE peaks.

Modeling parameter estimation
To account for IL-15R and IL-15/IL-15R complex trafficking, we utilized previously published estimates of k endo , k int , and k deg of IL-2/IL-15bc in our model. We assumed that k rec of IL-15 and the IL-15/IL-15Ra complex from endosomes were similar to the recycling of transferrin, given that endosomal sorting of fluid phase components (or membrane associated components that are not retained in the endosome) is relatively ligand independent (with the exception of steric issues with large ligands). In addition, we calculated the ligand concentration inside endosomes using published estimates of the total endosomal volume (V e ) of fibroblasts based on the simplifying assumption that endosomes of mammalian cells have similar volumes.
The intermediate and high affinity binding models share the same trafficking parameters but differ in the binding and synthesis parameters. For the intermediate affinity ligand-receptor binding, we utilized the on and off rate constants of human IL-15/IL-15Rbc binding for the estimates of k f and k r [46]. Estimates of k re were made based on the studies of IL-2 binding that found that the off rate of IL-2 from its receptor was 8 times higher in the low pH environment of the endosome than on the cell surface [25]. We also assumed that the lower pH in the endosome increased the equilibrium dissociation constant, K d , of IL-15 in a similar manner to the IL-2 receptor system and calculated k fe based on our estimate of k re . Furthermore, varying the estimates of k fe and k re by one order of magnitude did not significantly change the solutions of the model parameters (data not shown). At steady state, the number of intermediate affinity surface receptors on unstimulated NK cells was determined by endocytosis and constitutive receptor synthesis. Thus, we estimated k s from k endo of IL-2Rbc based on the observation that approximately 800 intermediate affinity receptors were found on unstimulated NK cells [14]. The induced synthesis rate constant of the intermediate receptors, k syn , was derived from studies of human NK cells stimulated with IL-15 in vitro [14].
The high affinity ligand-receptor binding describes the association between IL-15 and IL-15Rabc. IL-15Rabc's high affinity for IL-15 matches that of IL-15Ra, whereas IL-15Rbc has only intermediate affinity for IL-15 [12]. Therefore, we made the simplifying assumption that the high affinity IL-15 ligand-receptor binding and dissociation rate could be approximated by the interaction between IL-15 and IL-15Ra. To simplify the computational analysis, we assumed that all available IL-15Ra were bound to IL-15, based on the excess of IL-15 and the very high affinity of IL-15 binding to IL-15Ra [18]. Therefore in our model, uniform binding of IL-15Ra and IL-15 results in the formation of a high affinity ligand, L. We modeled the formation of the quaternary complex of IL-15/IL-15Rabc as the association between the high affinity ligand (L) and the intermediate affinity IL-15R (IL-15Rbc). Therefore, we estimated k f 9 and k r 9 of L for IL-15Rbc from the binding between IL-15 and recombinant IL-15Ra [34], and the synthesis of IL-15Ra essentially represented the synthesis of the high affinity ligand. We estimated the synthesis rate constant of IL-15Ra from the fold increase under IL-15 stimulation in comparison to basal levels [14]. Variations in the value of k syn 9 over one order of magnitude did not significantly affect receptor occupancy. To quantitatively analyze ligand-receptor binding in the high affinity binding model, we made the simplifying assumption that the IL-15/ IL-15Ra complexes were evenly distributed on the cell surface and the interior of endosomes. The cellular and the endosomal membrane were approximated as spherical shells with thickness equal to that of a lipid bilayer. We utilized the volumes of the cellular and endosomal membrane to compute conversion factors j surf and j endo , which enabled us to quantify the spatial distribution of IL-15/IL-15Ra complexes. Based on the assumption that the affinity between the high affinity ligand (L) and the intermediate receptor decreases in lower pH environments, the association and dissociation rate constants of the high affinity ligand in the endosome, k fe 9 and k re 9 , were estimated in a similar manner as were k fe and k re of the intermediate affinity receptor.

Model simulation and determination of the accuracy of model predictions
The solutions to the ODEs were obtained from numerical simulations performed using Mathematica 7.0 (Wolfram Research, Champaign, IL). The quality of the model predictions were assessed by computing the normalized root mean square deviation (NRMSD) between model computations and observed experimental values. The sum of squared deviations (SSD) at different time points were computed, and NRMSD was calculated as a percentage value by the following formula: Undivided NK cells were excluded, and the CFSE profiles (black curves) represented the sum of different dividing cohorts. Each CFSE profile was fit with a Gaussian curve (red curves) using OriginPro 7.5 software, and the R 2 value was shown in each panel. These results were representative of three to four independent experiments. (EPS)