Regulation of T Cell Expansion by Antigen Presentation Dynamics

Regulation of T cell expansion by antigen presentation dynamics

Andreas Mayera, Yaojun Zhangb, Alan S. Perelsonc, and Ned S. Wingreena,d,1

aLewis–Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544; bPrinceton Center for Theoretical Science, Princeton University, Princeton, NJ 08544; cTheoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545; and dDepartment of Molecular Biology, Princeton University, Princeton, NJ 08544

Edited by Arup K. Chakraborty, Massachusetts Institute of Technology, Cambridge, MA, and approved February 8, 2019 (received for review July 25, 2018)

An essential feature of the adaptive immune system is the neglected the role of antigen decay, our model readily explains proliferation of antigen-specific lymphocytes during an immune the dependence of T cell expansion on antigen affinity (6) and reaction to form a large pool of effector cells. This prolifer- on the kinetics of antigen administration (7). We thus identify ation must be regulated to ensure an effective response to the dynamics of presented antigen as a key regulator of the size infection while avoiding immunopathology. Recent experiments of an immune response. in mice have demonstrated that the expansion of a specific clone of T cells in response to cognate antigen obeys a strik- Results ing inverse power law with respect to the initial number of T A Simple Model Explains Power Law Dependence of Fold Expansion cells. Here, we show that such a relationship arises naturally from on Precursor Number. We propose a minimal model of T cell a model in which T cell expansion is limited by decaying lev- expansion in which T cell proliferation is regulated by the level of els of presented antigen. The same model also accounts for the presented cognate antigen (Fig. 1D). Our model has two exper- observed dependence of T cell expansion on affinity for anti- imentally motivated features: T cells proliferate at a saturated gen and on the kinetics of antigen administration. Extending the rate at high pMHC concentrations (3), and pMHC turnover model to address expansion of multiple T cell clones compet- leads to decaying antigen levels over time (8, 9). We describe ing for antigen, we find that higher-affinity clones can suppress the dynamics of the number of specific T cells, T , and cognate the proliferation of lower-affinity clones, thereby promoting the pMHCs, C , using the equations specificity of the response. Using the model to derive optimal vaccination protocols, we find that exponentially increasing anti- dT TC = α − δT [1] gen doses can achieve a nearly optimized response. We thus dt K + T + C conclude that the dynamics of presented antigen is a key reg- dC ulator of both the size and specificity of the adaptive immune = −µC . [2] response. dt Eq. 1 describes a population of T cells that proliferate at clonal expansion | T cells | precursor frequency | vaccination | power law rate αC /(K + T + C ) and die at rate δ. The rate of proliferation depends on how much antigenic stimulus the average T cell receives, which depends on the amount of presented antigen, uring an immune reaction, antigen-specific lymphocytes the number of competing T cells, and a parameter K related to Dproliferate multiple times to form a large pool of effector the affinity of the T cells for the antigen. The functional form cells. T cell expansion must be carefully regulated to ensure effi- cient response to infections while avoiding immunopathology. One challenge to regulation is that the number of naive T cells Significance specific to any given antigen is biased by the VDJ recombina- tion machinery and thymic selection (1), and the total number of Antigen-specific T cells proliferate multiple times during an T cells specific to an antigen further depends on previous infec- immune reaction to fight against disease. This expansion of tions with the same or similar pathogens (2). As a result, there T cells must be carefully regulated to ensure an effective are large variations in the number of precursor cells before an defense while avoiding autoimmunity. One challenge to regu- immune response. How does the adaptive immune system prop- lation is that the initial size of antigen-specific T cell clones can erly regulate its lymphocyte expansion given varying precursor be quite variable. Intriguingly, a recent study in mice found numbers? that the fold expansion of a T cell clone depends as an inverse + In a study by Quiel et al. (3), transgenic CD4 T cells specific power law on its initial size. We propose a simple mathe- for a peptide from cytochrome C were adoptively transferred matical model that naturally yields the observed power law into mice. The number of transgenic T cells was then tracked relation. Our model accounts for multiple experiments on T in response to immunization with cytochrome C (Fig. 1A). It was cell proliferation, suggests optimal vaccination protocols, and found that fold expansion of the transgenic T cells depends as highlights dynamics of presented antigen as a key regulator an inverse power law on the number of transferred precursor of the size of an immune response. cells (Fig. 1B). Here, we propose a simple mathematical model to describe the dynamics of T cell expansion. We show that the Author contributions: A.M., Y.Z., A.S.P., and N.S.W. designed research, performed power law behavior arises naturally when T cell proliferation is research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.y limited by decaying antigen availability, and we predict how the The authors declare no conflicts of interest.y power law exponent is related to the rates of lymphocyte prolifer- This article is a PNAS Direct Submission.y ation/death and of antigen decay. Our proposal adds to previous This open access article is distributed under Creative Commons Attribution-NonCommercial- modeling efforts to explain the surprising power law relation in NoDerivatives License 4.0 (CC BY-NC-ND).y terms of negative feedback regulation (4) or the active deple- 1 To whom correspondence should be addressed. Email: [email protected] tion (also known as grazing) of peptide–MHCs (pMHCs) by This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. T cells (5). Furthermore, in contrast to these previous studies, 1073/pnas.1812800116/-/DCSupplemental.y which did not explore the effect of T cell affinity for antigen and Published online March 8, 2019.

5914–5919 | PNAS | March 26, 2019 | vol. 116 | no. 13 www.pnas.org/cgi/doi/10.1073/pnas.1812800116 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 ae tal. et Mayer when [i.e., fulfilled longer no is time saturation transition C a for at conditions occurs two This level. uration concentration pMHC T C pMHC a from (TCR) receptor cell the T for dissocia- (11). individual complex cells the an on T of rate linearly the tion approximately parameter of depends The avidity which maximum. functional antigen, its overall of the one-half represents is rate ation small of limit that the Considering concentrations. antigen proportional is rate (large gens proliferation the antigen, to of maximal level a the at with proliferate specific and all stimulus rate abundant, sufficient is receive antigen cells T When parameters. term various proliferation the the Eq. perspective, in phenomenological Appendix, a (SI (10) From pMHCs to number bound the cells to T proportional of under is mechanistically proliferation motivated that be presumption the can rate proliferation the of fpHsi auaigfrTcl binding, cell T amount for the times, saturating small is for that, pMHCs Assume of dynamics. its stand expansion. cell plays T decay regulating this in by role implied key antigens a the presented below, apoptosis of show the we level As and changing (13). prominently (12) cells most ubiquitylation antigen-presenting activated mechanisms, by of of MHCs number of turnover a (8, the by experiments mediated in processing). directly is and measured 9), uptake been has antigen which of a decay, through This process established fast number relatively pMHC initial prior the (with time over to needed is stimulation SEs: antigen asymptotic continual and and parameters their proliferate, model on to Fitted MHCs time. on cells these over T present peptides stimulates and presented pMHC peptides, of cognate short decay into of to them Recognition leads process TCRs. µ pMHCs antigens, via of up Turnover pMHCs take proliferation. (3, cells to stimulation maintain Dendritic bind antigen model. cognate cells the to T of response surfaces. Schematic in vivo (D) in mice). proliferation individual bars, cell error dots, T mean; of geometric monitoring (crosses, for cells allows T mice recipient into clones (B 6). cell T transgenic of Transfer (A) 1. Fig. = (0)e (t (t Eq. ie h ipiiyo h oe,w a edl under- readily can we model, the of simplicity the Given C 1.2 ? ) eas nie eoe iiig ial,lwafiiyanti- low-affinity Finally, limiting. becomes antigen because , K and ycnrs,we elnmesaelrecompared large are numbers cell T when contrast, By α. ) ( ≈ 2 ± iiaino elepninb nie ea a xli h oe a eedneo odepnino h nta ubro ont cells. T cognate of number initial the on expansion fold of dependence law power the explain can decay antigen by expansion cell T of Limitation α 1 estecnetaino nie twihteprolifer- the which at antigen of concentration the sets T ecie o h ubro rsne niesdecays antigens presented of number the how describes −δ C 0.5/d, T atrsteefciedpnec fpoieainon proliferation of dependence effective the captures atro xaso tdy7a ucino h ubro precursor of number the of function a as 7 day at expansion of Factor (B) predictions. model with 3 ref. from data experimental of Comparison ) (t (t ) K t ota el rlfrt exponentially, proliferate cells T that so ), xoeta rlfrto rcesutlthe until proceeds proliferation Exponential . ? A a rv rlfrto u nya ufiinl high sufficiently at only but proliferation drive can ) δ ) Cognate antigen = rwhen or 0.22 Transgenic T cells ± C 1d and 0.21/d, C (t (t ) ? ' ) C ≈ D Antigen (0)e K C (0) hcee oe rt.The first]. comes whichever , transgenic = expansion E.(C ±SE).

−µt p T cells 10 Count

after MHC 6.7±1 easblwtesat- the below decays eladpH ubrv.tm o 0 n 000iiiltasei el coss emti mean; geometric (crosses, cells T transgenic initial 30,000 and 300 for time vs. number pMHC and cell T ) eto 1). section Text, SI efixed We . t C ? B Dendritic cell (t hnoeof one when ) T K K K T = shows (t uprbudfo fit from bound (upper 0. thus and ) ' ilrt fices fTcl ubri ie by given sets The is peak number data. cell their the T after of from numbers increase of inferred cell rate T readily tial param- of be few decay can The of model 1C). rate num- courses (Fig. the time the sizes the of by for population eters as divided cell well 7 T 0—as day transgenic day at of at cells T cells transgenic T of ber transgenic of number the precursor of number the transgenic between relationship the on data shown mental as Moreover, 1 1). (Fig. Fig. precursor (3) in al. et higher Quiel of at data experimental size population the in peak numbers. the of depen- rence law power amplification inverse cell population an T yields of naturally dence model the Thus, (SI time this beyond by given decay is expansion addi- fold plus peak small as the proliferation the S2), down Fig. cell Neglecting Appendix, slows decline. rapidly T to cells tional T continue of levels proliferation pMHC time, this After lwondet h iie fnt fteTclsfrteanti- the in for shown cells as two T S1 a the the Fig. between implies of cross-over latter affinity a the (with limited competi- and gen the to binding, to antigen due due for expansion slowdown cells of T slowdown among a tion implies case former hrceitctime characteristic ihEqs. with oho hs outpeitoso h oe odtu nthe in true hold model the of predictions robust these of Both efis osdrtecmeiinlmtdrgm,i hc the which in regime, competition-limited the consider first We Proliferation ). h oe a uniaieyacutfrteexperi- the for account quantitatively can model the B, 1 CD4 and T diinly Eq. Additionally, (0). ' T cell T implies 2, + 700). T C (t (0) el n odepninhr endas defined expansion—here fold and cells T PNAS t ? t ? ) ? = ≈ sstby set is | α ac 6 2019 26, March C T −

TCR (0) (0) 1 δ T(0) =30000 T(0) =300 T + C (t µ ( α (t ? ln )/T −δ 3 ? ) C T ) rdcsa ale occur- earlier an predicts ≈ /( (0) | (0) (0) α T −δ o.116 vol. (t . nteiiilTcell T initial the on +µ) ? hc together which ), α . − | α IAppendix, SI δ o 13 no. = hc sets which , δ 1.5 h ini- The . ± | 0.3/d, 5915 [3] [4]

BIOPHYSICS AND COMPUTATIONAL BIOLOGY α. The experimentally observed power law exponent of ≈ −0.5 concentration of free pMHC at which the proliferation rate of implies that α − δ ≈ µ, from which we can infer µ. We obtain an the T cells is half-maximal in the noncompetitive regime. Clearly, upper limit on the saturation constant K from the observation affinity is important whenever pMHC concentrations are near that the power law holds down to the smallest experimental pre- or below K . Applying our model to affinity-limited expansion cursor numbers (Fig. 1B), as the scaling would have broken down in the context of a pMHC concentration that is decreasing over if the condition K C (t ?) was not fulfilled for the smallest time makes specific predictions about how T cell proliferation precursor numbers (SI Appendix, Fig. S3). Finally, the model nat- depends on affinity. Additionally, we demonstrate that affinity urally recapitulates the observed biphasic time course of T cell can play an important role in expansion for mixtures of T cells of expansion: exponential proliferation at early times followed by a different affinities even if pMHC concentrations are higher than cross-over into a phase of more slowly exponentially decreasing all K values. T cell numbers. When proliferation of T cells is limited by their antigen Importantly, our model makes concrete predictions regard- affinity, low-affinity T cells stop proliferating earlier, and con- ing how the fold expansion depends on system parameters. For sequently, their fold expansion is smaller. In contrast to the example, according to Eq. 4, it predicts the dependence of the competitive regime (Eq. 3), the characteristic time t ? at which power law exponent of fold expansion on the rates of T cell exponential proliferation stops is now set by C (t ?) ≈ K . This proliferation/decay and antigen decay. It also predicts a uni- yields a logarithmic dependence of t ? on the binding constant form increase in fold expansion if C (0) is increased. This has 1 been observed experimentally (figure 5A in ref. 3): an increase t ? = ln(C (0)/K ), [5] in either the antigen dose or the number of antigen-presenting µ cells leads to a uniform multiplicative increase of fold expansion across precursor numbers. and we find that the fold expansion scales as Our model further predicts that a transfer of the same num- ? T (t ) (α−δ)/µ ber of transgenic T cells after a time delay tdelay relative ≈ (C (0)/K ) [6] to antigen administration will lead to a smaller fold expan- T (0) sion. During the time delay, pMHC decays, which implies that (i.e., as an inverse power law with respect to K ). C (tdelay) is lower than at the time of antigen administration −tdelayµ Zehn et al. (6) have studied the effect of antigen affinity on (t = 0) by a factor of e . Fold expansion scales with the + CD8 T cell expansion using adoptive transfer experiments. antigen concentration at the time of transfer (Eq. 4). Thus, + Assuming that Eq. 1 also describes the dynamics of CD8 T the fold expansions in the delayed transfer and simultane- ? cells, Eqs. 5 and 6 make testable predictions for how the tim- ous transfer experiments are related by T (tdelay)/T (tdelay) ≈ −t µ(α−δ)/(α−δ+µ) ? ing and magnitude of the peak expansion should depend on e delay T (t )/T (0) (i.e., fold expansion is lower affinity in these experiments. In their study, the expansion of by a factor that is exponential in the time delay) (SI Appendix, transgenic OT-1 T cells was tracked in response to an infec- Fig. S4). In contrast, if antigen grazing dominates as a source tion by Listeria monocytogenes. Different strains of the bacterium of early pMHC loss (5), then a later transfer of T cells should were engineered to express altered ligands with different affini- not impact fold expansion. Time delay experiments thus pro- ties for the transgenic T cells. This experimental design allowed vide a means to ascertain how much antigen loss is due to T direct study of how T cell expansion depends on antigen affinity cell-independent mechanisms and how much is due to grazing. in vivo. A reanalysis of the published data confirms our model Our model formulation has been kept deliberately simple to predictions that the fold expansion depends as a power law on highlight our proposed explanation for the power law relation antigen affinity (Fig. 2A, data points) and that peak population observed in ref. 3: namely, competition among an exponentially increasing number of T cells for an exponentially decreasing number of pMHC complexes. The model can be extended in a number of ways without altering the basic behavior. First, one A B might ask how the results change for a more complicated dynamics of the pMHC number that includes the initial uptake and processing of antigen by the antigen-presenting cells as was done in ref. 5. A simple analysis shows that the power law scaling con- tinues to hold as long as there is no new processing of antigen into pMHC for times larger than the smallest t ? (SI Appendix, SI Text, section 2). Second, one can modify the dynamical equations to include the added effect of T cell grazing of antigen (5, 14, 15) while keeping the other aspects of the model intact (SI Appendix, SI Text, section 3). Simulations and mathematical analysis of this modified decay plus grazing model show that Ligand SIINFEKL SAINFEKL SIYNFEKL SIIQFEKL SIITFEKL SIIVFEKL it exhibits similar dynamics and can also fit the experimental rel. EC50 1.0 2.7 4.1 18.3 70.7 680 data. Both models have in common the same robust mecha- nism for generating a power law: T cells proliferate at a constant Fig. 2. Limitation of T cell expansion by antigen decay can account for the rate until the exponential decay of cognate pMHCs causes power law dependence of fold expansion on affinity. (A and B) Comparison proliferation to cease. of data from an experiment with L. monocytogenes strains expressing different antigens (see legend for their amino acid sequence) (6) with model Dependence of T Cell Expansion on Antigen Affinity. T cells only predictions. (A) Factor of expansion of the transgenic T cells at day 7 relative respond to ligands that bind sufficiently strongly to their TCR, to day 4 vs. the relative concentration of different pMHCs needed to elicit which is the basis of the specificity of the adaptive immune sys- half-maximal IFN-γ response from the T cells (EC50 relative to antigen SIIN- FEKL’s ∼ 8 pM). (B) T cell number vs. time for the different strains. Fitted tem. Above this binding threshold, there is a large range of model parameters and asymptotic SE: α = 2.47 ± 0.13/d, µ = 3.1 ± 0.3/d, possible affinities that can stimulate T cell expansion. How does δ = 0.23 ± 0.06/d, C(4) = 103.22±0.17, and T(4) = 0.92 ± 0.10. The number the affinity of a T cell clone for the pMHC, as typically measured of transgenic T cells is calculated from their fraction f of total T cells as by a dissociation constant, affect its expansion? In our model, f/(1 − f), with the number at day 4 set to 1. K was set equal to the relative affinity is captured by the parameter K , which is equal to the EC50 values of the different ligands.

5916 | www.pnas.org/cgi/doi/10.1073/pnas.1812800116 Mayer et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 ae tal. et Mayer faster increase levels pMHC and whether on strongly (γ depends outcome tion con- example, For onset regime. the competitive both the on of depends extent expansion the fold and overall affects much ties proliferate above cells well T concentrations higher-affinity 3). the pMHC (Fig. affinities, faster Eq. for cell preference Appendix, even T the (SI all that, of ratio implies magnitude affinity this the the with by preferen- antigen, set enjoy to cells binding T high-affinity tial antigens, of for proliferation competing change are Eq. other not , Appendix of (SI does presence clones affinities individual the Appendix, different abundant, with (SI multiple are cells affinities T pMHCs to varying When model 1). of binding ref. section populations Following competitive cell clone? our cell T generalize T we affinities particular different a 10, of cells of T proliferation other of affect presence participat- the affinities broad, does different typically How prein- of is ing. clones the cell antigen T an However, different to many pMHCs. specific with the repertoire cell for T affinity fection particular a with 2B). courses (Fig. time affinities different full for the numbers and cell 2A) the T (Fig. of of affinity the dependence on observed expansion the fold fitted both the reproduces Nevertheless, other. closely each model of independent adjust to the ters Since numbers. measured the pMHC from and cell T ative in linear equation the makes Eq. in denominator the concentration pMHC 4 day ber the as well as IFN-γ detectable iikn nie yaiseryi nifcin[C infection an in pMHCs of early number dynamics increasing exponentially antigen but (C mimicking small numbers. cell a T by (µ initial driven pMHCs (B) liferation unequal of and number (A) decreasing equal exponentially from starting but (A large curves). a (dashed own by of their driven types on cells two T of the of mixtures expansion of high, ( A–D) expansion with affinities. of cells cell courses T T time all the exceed of concentrations Comparisons pMHC when even pMHCs, 3. Fig. (EC data pMHC 2B, To of (Fig. quantitatively. tion data set lower the we is fit end, can this affinity model when our Moreover, earlier points). reached are sizes CD AB = o ifrnilpoieaino el fdfeetaffini- different of cells T of proliferation differential How cells T of clone a of proliferation considered have we far, So δ T 3; = (4) C ihafiiyTclscnotopt o-fnt el o cesto access for cells T low-affinity outcompete can cells T High-affinity 0. rslower or ) otetm oredt.I h fnt-iie regime, affinity-limited the In data. course time the to K K 1 γ = epne(i.2.W hnfi h rates the fit then We 2). (Fig. response (0.5; = oteeprmnal eemndconcentra- determined experimentally the to EC ,adlow, and 1, 50 eddfroehl fTclst hwa show to cells T of one-half for needed ) 50 1 hnTclspoieae Parameters: proliferate. cells T than D) aus h uvshv ofe parame- free no have curves the values, a eapoiae by approximated be can T K 2 n hs needn fterel- the of independent thus, and = 0 fnte sldcre)wt the with curves) (solid affinities 10, .Hwvr hnTcells T when However, S20). C (4) K .Interestingly, S21). (t and ausaetaken are values n elnum- cell T and ) ∝ K Proliferation B) e γ + t .Competi- ]. and C IText, SI which , α α, Pro- D) = = µ, 1.1) 1.2 δ hta piie al oepooo scoet h experi- the to close is 4 (Fig. protocol protocol dose find exponential we used daily Indeed, mentally optimized expansion. fold an higher proliferation a that cell to cell T leads T with therefore, lev- levels rising and antigen these the synchronizes pro- stimulate better stimulation, input increasing tocol exponentially to cell The times. insufficient T later at become full population to for decay levels needed are pMHC els times than initial early While higher budget: at antigen are pMHCs finite of the levels contrast, of wasteful By high times. with all minimize schedules to at dosing cells stimulation T maximize with Ide- tandem and expansion. increase, in their competition rise numbers limit should cell can levels T pMHC pMHC prolifer- for ally, as to cells on, T cells Later of rate. T competition low maximal all even their for low, at very sufficient ate are are numbers the model concentrations cell of T pMHC Our potencies when different amplitudes. on the Early response for pre- protocols. explanation of correctly simple exponential model a order provides the our observed protocol, of the constant advantage the dicts the While fold 4C). over larger Fig. protocol in a of 4C) curve Fig. factor shows (blue in a experiment protocol curve by T (green single-shot ahead dosing the largest antigen 4C), and find the constant Fig. We to to in relative 4C). leads curve two (Fig. (orange input 6 expansion day increasing cell at exponentially numbers the cell different that T these on of impact kinetics markedly the exponen- experi- analyze pMHC the to we (iii) mimic 7, lead To ref. and of 4B). inputs protocol (Fig. d, mental These time 4 over d. levels over 4 pMHC different over input input constant pulse increasing (ii) antigen tially single antigen length, a total (i) 1-d equal 4A): of with (Fig. schedules cases different three but for input equations of system the rate cell T of dependence dosing? observed antigen this on with expansion consistent model increase our can instead dose Is doses larger single smaller vac- a Kinetics. multiple using spreading of into that Antigen administration shown on has antigen (7) cine Expansion mice with Cell study experimental T of Dependence expansion. cell T interesting T influence in differentially affinities, combine to In antigen concentrations ways limited. of pMHC and effects affinity numbers, the vs. cell cases, limited limiting affinity competition these of model: between effect the the of explain cases quantitatively for to numbers able on cell also initial is cells on cells T expansion of low-affinity dependence the high- of pMHCs proliferation, for expansion cells cell 3D). suppress (Fig. T T low-affinity thereby, of outcompete and can rate more cells T rises concentra- maximum T all concentration affinity pMHC the pMHC for if expansion proliferation than the contrast, saturate cell slowly as By that the T 3C). long levels (Fig. in initial to cells as model, role quickly affinity our major rises In a tion of expansion. play independent of not is d 2B), did 4 4 (Fig. affinity day affinities first that at different trans- suggests numbers with the comparable antigens which (6), to by al. expanded stimulated et clones Zehn when T of cell low-affinity their experiments T the on the of genic In proliferating expansion 3B). fold stop still (Fig. lowered cells would can a these in low- cells resulting of before T own, population cells smaller high-affinity T a However, of affinity of proliferation 3A). off onset (Fig. shut the effectively expan- low before fold in is happens difference sion overall expansion the ini- the high If the therefore, saturating and 3). of relatively high competition, (Fig. most very with antigen level, a clones pMHC from tial same cell declines the concentration T for pMHC two affinities of low and dynamics the sider oadesti usin eadatm-ayn nie input antigen time-varying a add we question, this address To law power the explain to used model same the summary, In CD8 ν (t + ) oEq. to elepnin h w xeiet eellimiting reveal experiments two The expansion. cell T 2 PNAS Eq. Methods, and (Materials | ac 6 2019 26, March | CD8 o.116 vol. sn h derived the using D) + elresponses. cell T .W simulate We 8). | o 13 no. recent A CD4 | + 5917 T

BIOPHYSICS AND COMPUTATIONAL BIOLOGY A 18), newly generated pMHCs likely play a small role compared with the turnover of the already presented pMHCs at the late D stages of infection analyzed in Fig. 2. The second assumption of the decay of presented pMHCs is experimentally well supported (8, 9) and has a known mechanistic basis (12, 13). Furthermore, our inferred decay constants are within the range of decay constants reported for different antigens bound to dendritic cells (8) and are also roughly compatible with direct measurements of the B decrease of the stimulatory capacity of antigen-presenting cells in transgenic mice after switching off inducible antigen produc- tion (9). One limitation of our model, particularly for replicating antigens, is that we have neglected any influence of the epitope- E specific pMHC density on the surface of an antigen-presenting cell, which may also play a role in determining T cell stimulation and expansion (19). C The kinetics of antigen administration has been shown to influence the magnitude of T cell (7) and B cell (20) immune responses. This finding has implications for the rational design of vaccination strategies (21), but open questions remain regarding how to optimize dosing for high magnitude of response and/or high affinity of the responding cells (20). Our modeling suggests that exponentially increasing doses are close to optimal for maximizing the magnitude of T cell response. We further find that selection for higher affinity is strongest when Fig. 4. Impact of antigen kinetics on T cell proliferation. (A) Antigen input T cells compete for antigen stimulation. Selection for affinity is schedules: single pulse, constant input, and exponentially increasing input thus predicted to be more stringent for lower or more slowly (increase each day by fivefold as in ref. 7). (B) Dynamics of pMHCs. (C) Dynamics of T cells. (D) The optimal schedule is close to the experimen- increasing antigen levels and for larger or faster growing prior tally used exponential schedule. The antigen input schedule over 4 d that T cell populations. This could have important implications for optimizes fold expansion at day 6 was computed numerically using a pro- patterns of immunodominance in primary vs. secondary infec- jected gradient algorithm (16). (E) Fold expansion for exponential schedules tions: In secondary infections, competition is expected to be as a function of the fold increase per day, with the experimental sched- stronger, as preexisting memory cells specific to the pathogen ule indicated by the dot. Parameters: α, µ, and δ as in Fig. 2; K = 10; are usually present in higher numbers and can also proliferate 6 C(0) = 0, T(0) = 100; and total administered antigen is 2 · 10 . faster. Looking ahead, our model could be further extended to + make it more realistic. Including single-cell stochasticity could parameters for CD8 T cells from Fig. 2. It is worth noting that help clarify how reproducible population-level expansion arises the experimental choice of fivefold increases of dose per day despite stochasticity at the single-cell level (22) and would also (7) is surprisingly close to optimal among exponential protocols allow connections to recent experimental studies, which have (Fig. 4E). revealed substantial heterogeneity of the immune responses of single cells (23). Furthermore, the model could be extended to Discussion account for the diverse compartments (different lymph nodes, Prompted by the surprising observation of a power law depen- the spleen, different tissues) (2) and T cell subtypes (4) involved dence of T cell fold expansion on initial cell numbers (3), we in an immune response. Spatial or cellular heterogeneity can cre- developed a simple mathematical model in which T cell pro- ate separate niches in which T cells compete for proliferation and liferation is stimulated by a dynamically changing number of survival, which may provide another layer of regulation of T cell cognate pMHC molecules and showed that it naturally yields expansion. the observed power law. We then explored more generally Among the open questions raised in our study, we highlight how T cell numbers, TCR affinity for antigen, and the dynam- two. First, other than regulating T cell expansion during an acute ics of pMHC presentation combine to regulate T cell expan- infection, how else might antigen presentation levels influence sion in different regimes. Testing these results against other T cell populations, and do they play a role in thymic selection experimental datasets, we found that our model correctly pre- or the dynamics of naive T cells competing for self-antigens? dicts a power law dependence of T cell expansion on affinity Specifically, recent work in ecology (24) suggests that T cell for low-affinity antigens (6), a quantitative relationship that grazing, which consumes antigens, could allow for the coexis- had not previously been appreciated. With regard to vaccina- tence of a diverse naive repertoire despite competition for self- tion, our model furthermore predicts the observed enhanced antigens. Second, regulation of T cell population dynamics can efficacy of stretching a fixed total antigen dose over several be achieved by tuning of system parameters. For example, smart days (7) and provides testable predictions for how to optimize control of the lifetime of presented pMHC complexes could antigen dosing. induce the “right” amount of T cell amplification. Might system The core of our model is that T cell expansion is regulated by parameters have evolved to be close to optimal, and/or could dynamically changing levels of presented antigens. The dynamics some parameters also be regulated during an immune response of pMHCs is assumed to be characterized by a fast process- to provide a robust response to infections while avoiding ing of antigens by antigen-presenting cells followed by a slower autoimmunity? decay. The first assumption of rapid processing seems well jus- tified for the subcutaneous injection of antigens used in ref. 3 Materials and Methods but might be more questionable in ref. 6, where live replicating Experimental Studies. We used previously published data from refs. 3 and bacteria are used, as this might lead to continued processing of 6. In both studies, transgenic T cells were transferred into recipient mice, new antigens by dendritic cells. However, as the bacterial load and their expansion in response to antigenic stimulation was subsequently declines rapidly after reaching a peak at 3 d postinfection (17, studied. Quiel et al. (3) studied the response of 5C.C7 CD4+ T cells to a

5918 | www.pnas.org/cgi/doi/10.1073/pnas.1812800116 Mayer et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 2 oh A uuaK(05 h n n uso H ls Imdae antigen II-mediated class MHC of outs and ins The cell (2015) biomarker T K stable Furuta and describing robust PA, a function is Roche rate general 12. dissociation TCR-ligand a (2017) Towards al. et (1995) M, Allard AS 11. Perelson RJ, Boer De 10. ae tal. et Mayer h yaiso h elpopulation, cell T C the of dynamics the Equations. IFN-γ Model by assessed response half-maximal were for responses needed cell dose peptide T different the probability. h. by with differ- quantified 5 loaded and the cells for ing for RMA-S antigen affinity with of cell stimulated T doses were assess they To ligands, cells. expansion ent endogenous relative the vs. measure spleen to transferred cytometry the of flow within in by cells substitutions quantified T was transgenic acid mice of of amino percentage The specific region. with antigenic the ovalbumin altered numbers. or mice quantitative cell ovalbumin CD8 the OT-1 T by into used known injected determined (6) with They was al. reference cells a et T against Zehn calibrated transgenic PCR of real-time number The cytochrome ride. pigeon of administration subcutaneous .Os ,vnSne M ahsD eos 20)Atgnpritnei required is persistence Antigen (2005) C Benoist D, Mathis HM, Santen van R, Obst 9. .Zh ,ChnC,Rie ,Wle 20)Etne rsnaino pcfi MHC- specific of presentation Extended (2004) P Walden Y, Reiter reactivity. CJ, immune Cohen D, determines Zehn kinetics 8. Antigen (2008) low- al. very to et be response P, T-cell can curtailed Johansen but expansion Complete 7. (2009) cell MJ T Bevan SY, CD4 Lee D, Antigen-stimulated Zehn (2013) 6. AS Perelson RJ, Boer De 5. rates differentiation log- vs. proliferation and of regulation inversely Feedback Generation, is (2011) al. expansion et cells: G, T-cell Bocharov T CD4 and 4. Antigen-stimulated memory frequency (2011) Human al. precursor et (2014) cell J, NP Quiel T Restifo 3. naive NA, of Yudanin role DL, The Farber (2012) 2. JJ Moon MK, Jenkins 1. ftefloigform: following the of , hogotteepninpaeo D+Tcl response. cell T CD4+ a of phase expansion the throughout rcsigadpresentation. and processing potency. cell T CD8+ of proliferation. antigen- of types other to compared cells dendritic cells. presenting mature by complexes peptide USA Sci Acad antigen. affinity complexes. peptide-MHC of grazing their by limited number. precursor on expansion T-cell USA CD4 Sci Acad of dependence the explains number. precursor to related linearly homeostasis. and compartmentalization magnitude. response immune dictating in recruitment 105:5189–5194. 108:3318–3323. ho Biol Theor J u Immunol J Eur Nature ecnie eemnsi it–et-yemdl for models birth–death-type deterministic consider We C Insight JCI 458:211–214. 175:567–576. dC dT dt dt a e Immunol Rev Nat .monocytogenes L. = = 34:1551–1560. + α(T ν 2:92570. (t el pcfi oa vlui peptide. ovalbumin an to specific cells T ) rcNt cdSiUSA Sci Acad Natl Proc , − C a e Immunol Rev Nat , µ(T K T n h ocnrto fpMHCs, of concentration the and , )T , C − 15:203–216. , K δ T )C Immunol J xrsigete wild-type either expressing , . Immunol J 14:24–35. x Med Exp J C 108:3312–3317. n lipopolysaccha- and 190:5454–5458. 188:4135–4140. 201:1555–1565. rcNatl Proc rcNatl Proc stain- [8] [7] 4 ofiA aleuirT igenN 21)Mtblctaeof rmt diversity promote trade-offs Metabolic (2017) responses. NS immune Wingreen cell T, Taillefumier T A, of Posfai resolution 24. Single-cell (2018) of M model Flossdorf A VR, (2007) Buchholz PD 23. Hodgkin C, geometry, Gend van size, MR, of Dowling matter ML, Turner A ED, delivery: Hawkins Vaccine 22. (2010) GT Jennings MF, initiation center Bachmann germinal during 21. availability antigen Sustained (2016) al. et HH, Tam 20. virus-specific of induction The and (1999) load LC Eisenlohr bacterial A, Measuring Porgador KA, (2011) Puorro EJ, T Wherry Brodnicki 19. O, Wijburg R, Strugnell monocytogenes. N, Listeria Wang well-adapted to 18. a responses How Immune (2015) (2004) AMA EG Walczak Pamer T, 17. Mora V, Balasubramanian resolved proposals A, Hasty Mayer antigen: to 16. responses cell T (2013) G peptide- Altan-Bonnet down-modulate versus K, cells Tkach T tolerance (2002) 15. of P Marrack Regulation JW, Kappler apoptosis: BC, Schaefer cell RM, Kedl Dendritic (2010) 14. J Hu R, Kushwah 13. infne yNFGatPY1300(oNSW n ...Prin of Energy Portions of Y.Z.). Department US and of N.S.W. auspices the (to 89233218CNA00000. Func- under PHY-1734030 Contract Biological performed Grant were of work Physics NSF this the by and for A.S.P.) funded Center (to the Lewis– tion R01-OD011095 and Grants a A.S.P.), (Y.Z.), NIH (to by Science Y.Z.), R01-AI028433 Theoretical (to supported for PHY-1607612 was Center Grant Princeton work NSF the This (A.M.), fellowship 2919.01. Sigler Betty Grant and Gordon Foundation Theo- and part R25GM067110, in of Moore Grant supported Institute NIH and PHY-1748958, Kavli Barbara Grant the NSF Santa by at California, of discussions experimental University from Physics, providing started retical for was Grossman work Zvi This and data. Bocharov Gennady and sions ACKNOWLEDGMENTS. 4. section Text, SI .I h text, the 3 In functions 3). the section regarding Text, assumptions and simplifying SI of Appendix, number (SI a numbers make we cell T on depend may 2 rate section influx a time-varying Text, at a SI to die lead cells processing T its 1). and section T other Text, with SI competition parameter Appendix, antigen, a of and availability rate the cells, a on at proliferate depend to can stimulated are which cells T pMHCs, to binding By ueial,w iuaeteeeutosa ecie in described as equations these simulate we Numerically, . namdlecosystem. model a in Immunol Adv death and division lymphocyte randomized of times. consequence a as regulation immune patterns. molecular and kinetics until steadily vaccination. rise to responses antibody Responses enhances expression: epitope attained. are increasing epitope of of levels high function excessively a as CTL with infected mice in responses immune 4:812–823. organized. is system immune engagements. long through vivo. in APCs on complexes MHC immunity. aitoso hc r osdrdin considered are which of variations µ, rcNt cdSiUSA Sci Acad Natl Proc Immunol J 137:1–41. .TepH opee ea tarate a at decay complexes pMHC The ). 185:795–802. PNAS hsRvLett Rev Phys etakGr thank We K eae oteafiiyo h C o MC(SI pMHC for TCR the of affinity the to related urOi Immunol Opin Curr rcNt cdSiUSA Sci Acad Natl Proc 104:5032–5037. | a e Immunol Rev Nat a Immunol Nat ac 6 2019 26, March 118:028103. gieAtnBne o epu discus- helpful for Altan-Bonnet egoire ´ itramonocytogenes. Listeria rcNt cdSiUSA Sci Acad Natl Proc 3:27–32. Immunol J etos1– sections Text, SI Appendix, SI 25:120–125. 10:787–796. | ν 112:5950–5955. (t o.116 vol. δ IAppendix, (SI pMHCs of ) h paeo antigen of uptake The . 163:3735–3745. | µ(T i Exp Vis J a e Immunol Rev Nat 113:E6639–E6648. o 13 no. , IAppendix, SI C , α(T K 54:1–10. ,which ), | , C α, 5919 , K ν ), ,

BIOPHYSICS AND COMPUTATIONAL BIOLOGY