Mathematical Biosciences 225 (2010) 53–58

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Mathematical Biosciences

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Immunodominance analysis through interactions of CD8+ T cells and DCsin lymph nodes

Yu Wu, Lijin Xia, Mingjun Zhang, Xiaopeng Zhao *

Mechanical, Aerospace, and Biomedical Engineering Department, University of Tennessee, Knoxville, TN 37996, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 November 2009Received in revised form 26 January 2010Accepted 29 January 2010Available online 4 February 2010

Keywords:ImmunodominanceCD8+ T cellsDendritic cellAvidityMathematical modeling

0025-5564/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.mbs.2010.01.009

* Corresponding author. Tel.: +1 865 974 7682; faxE-mail address: xzhao9@utk.edu (X. Zhao).

Immunodominance is a common phenomenon observed in multiple epitopes immune systems. Previousstudies hypothesize that the competition among CD8+ T cell responses against different epitopes can beused to explain immunodominance. This paper proposes a mathematical model that describes thedynamics of CD8+ T cells primed by antigen-presenting dendritic cells (DCs) in the lymph nodes, andshows that the overall avidity of the interactions between peptide-specific T cells and cognate antigen-bearing DCs may determine the immunodominance. The model suggests the probability that a pep-tide-specific T cell be immunodominant is proportional to (1) the cognate T cell receptor (TCR) affinity,(2) the number of complexes of cognate peptide and major histocompatibility complex (pMHC) perDC, and (3) the half-life of cognate peptide-specific pMHC. The model predicts a threshold density ofpMHC complexes for T cell activation. These observations from the mathematical model are consistentwith experimental studies in the open literature. For DC-based vaccine design, the model suggests a strat-egy of immunotherapy based on the injection of cognate antigen-pulsed DCs.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction lowed by a third phase of high motility and rapid proliferation [8].

Immunodominance is the phenomenon of hierarchical immuneresponse observed in multiple epitopes system. Only the CD8+ Tcells directed to a few epitopes will be immunodominant. Compe-tition and predation of the immune species interactions are be-lieved to be the reason of immunodominance [1–3]. In general,CD8+ T cell adaptive immune responses are initiated and shapedby dendritic cells (DCs) in paracortical region of the lymph node.Immature DCs keep on surveying for antigens in peripheral tissues.Once they come into contact with antigens, immature DCs phago-cytose them and degrade their proteins into small pieces of pep-tides. Then immature DCs become activated into mature DCs andbegin to migrate into the lymph node, where DCs present thosepeptides at their cell surface using major histocompatibility com-plex (MHC) molecules and act as antigen-presenting cells. Theantigen-specific CD8 T cells will be activated into proliferationand differentiation cycle, when they contact with the cognate pep-tide and major histocompatibility complex (pMHC) on DCs.

The cellular dynamics of CD8+ T cells priming by DCs in lymphnodes have been actively investigated over the past few years[4–13]. It is shown that T-cell priming by DCs in lymph nodes oc-curs in three successive phases. During phase one, T cells undergotransient serial contacts with DCs, whereas phase two is character-ized by stable T cell–DC interactions and cytokine production, fol-

ll rights reserved.

: +1 865 974 6372.

The duration of phase one and T cell activation kinetics correlatedinversely with the number of complexes of cognate pMHC per DCand with the density of antigen-bearing DCs per lymph node [13].Avidity is used to describe multivalent interactions between cells.The overall avidity of the interaction influences the probability thatT cells will be stably captured by DCs [7]. Although antigen-bearingDCs are highly efficient in recruiting peptide-specific T cells andcould engage more than ten T cells simultaneously [7], T cell acti-vation will not occur below an antigen dose threshold determinedin part by pMHC stability [13]. T cells encountering a subthresholdantigen dose could not form prolonged contacts and continued tomigrate until they egressed from the lymph node without achiev-ing activation [13].

This paper proposes a mathematical model to describe thedynamics of interactions of CD8+ T cells and DCs in lymph nodes.The proposed model supports the idea that high-avidity CD8+ Tcells have a higher probability of defeating the low-avidity onesin the competition for priming and therefore will be immunodom-inant. The model also predicts that the threshold of pMHC dosetriggers T cell activation, which is consistent with the experimentalobservation [7,13].

2. Mathematical model formulation

We study the evolutionary dynamics in well-mixed popula-tions. Let D denotes the number of DCs in a lymph node, T1 be

54 Y. Wu et al. / Mathematical Biosciences 225 (2010) 53–58

the abundance of A-peptide-specific CD8+ T cell, and T2 be theamount of B-peptide-specific CD8+ T cell. Here only two generalpeptides A and B are considered. This assumption simplifies thediscussion without impairing the generality of the conclusions.The dynamics of the interactions are represented by the followingequations:

_D ¼ s� k1n1T1D� k2n2T2D� qD; ð1aÞ_T1 ¼ rk1n1DT1 � dT1; ð1bÞ_T2 ¼ rk2n2DT2 � dT2: ð1cÞ

In model (1), the overall avidity of the interaction between T cellsand DCs is defined by the product of average peptide-specific T cellreceptor (TCR) affinity ki and the average number of the cognatepMHC complexes per DC ni. In Eq. (1a), the DCs migrate at a constantrate s from inflamed tissues into the draining lymph node. Wechoose the rate at which 300 DCs will reach one lymph node in15 h after infection [13]. Since DCs have a natural lifespan, we as-sume q is the average per capita death rate, which implies that theprobability of cell death in lymph node at time t is given by an expo-nential distribution with an average cell half-life time t1=2 ¼ lnð2Þ=q.T cells receiving antigenic information and priming signals may, inturn, regulate the half-life of antigen-bearing DCs [4]. The timecourse of acquisition of lytic effector function parallels the kineticsof elimination of antigen-bearing DCs [14,15]. By assuming the DCclearance rate to be proportional to CD8+ T cell priming rate, andthus proportional to the overall avidity of the interaction betweenT cells and DCs [7], we show that DCs are cleared out for the regula-tion action by mature CD8+ T cells at rate kiniDTi.

In Eqs. (1b) and (1c), the antigen-specific CD8+ T cells arecleared out at rates dTiði ¼ 1;2Þ, which comprise the death ratedue to natural lifespan and the egress rate for leaving the lymphnode. Here we ignore the recruitment of antigen-specific CD8+ Tcells, as the frequency of precursor of cognate naive CD8+ T cellsin the T cell repertoire is very low (typically 1 in 105–106 T cells)[16,7]. Since the T cell proliferation cannot happen without prim-ing, we assume that the proliferation rate is proportional to thepriming rate with the ratio r. Definitions and values of the param-eters used in this paper are given in Table 1.

3. Dynamical analysis

There are three steady states for the system defined by Eq. (1).The three steady states are summarized in Table 2.

To study the local stability of the system, the Jacobian matrix Jevaluated at the first steady state P1 can be calculated as

J1 ¼

�q � k1n1sq � k2n2s

q

0 �dþ k1n1rsq 0

0 0 �dþ k2n2rsq

2664

3775: ð2Þ

Then, the three eigenvalues of J1 are

k1 ¼ �q; k2 ¼ �dþ k1n1rsq

; k3 ¼ �dþ k2n2rsq

: ð3Þ

The steady state P1 is asymptotically stable if and only if all of theeigenvalues are negative. So we have

ni <qdkirs

; i ¼ 1;2; ð4Þ

which sets a threshold for T cell activation.The Jacobian matrix evaluated at the second steady state P2 is

given by

J2 ¼

� k1n1rsd � d

r � dk2n2k1n1r

r k1n1rsd � q

� �0 0

0 0 d k2n2k1n1� 1

� �

266664

377775: ð5Þ

The characteristic equation detðJ2 � kIÞ ¼ 0 becomes

ðkþ AÞðk2 þ Bkþ CÞ ¼ 0; ð6Þ

where

A ¼ d 1� k2n2

k1n1

� �;

B ¼ k1n1rsd

;

C ¼ k1n1rs� dq:

ð7Þ

The three eigenvalues of J2 are

k1¼�A¼dk2n2

k1n1�1

� �;

k2;3¼12�B�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2�4C

p� �¼�k1n1rs�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4d3q�4d2k1n1rsþk2

1n21r2s2

q2d

:

ð8Þ

Similarly, the steady state P2 is asymptotically stable if and only ifall of the eigenvalues have negative real parts, thus we have the fol-lowing conditions for stability by the Routh–Hurwitz criterion

A; B;C > 0: ð9Þ

The Jacobian matrix evaluated at the third steady state P3 is givenby

J3 ¼

� k2n2rsd � dk1n1

k2n2r � dr

0 d k1n1k2n2� 1

� �0

r k2n2rsd � q

� �0 0

266664

377775: ð10Þ

The characteristic equation detðJ3 � kIÞ ¼ 0 becomes

ðkþ DÞðk2 þ Ekþ FÞ ¼ 0; ð11Þ

where

D ¼ d 1� k1n1

k2n2

� �;

E ¼ k2n2rsd

;

F ¼ k2n2rs� dq:

ð12Þ

The conditions for stability are

D; E; F > 0: ð13Þ

The stable regions of the three steady states are mutually exclusiveand cover the whole parameter space as shown in Fig. 1.

4. Evolution of population

The evolutions of T cell priming are studied by integrating sys-tem (1) numerically. Here we consider two conditions based on thehalf-life time of pMHC.

4.1. Infinite pMHC lifespan assumption

We first assume that pMHC has infinite lifespan, so that theaverage number of the pMHC complexes per DC ni are constants.The time courses of numbers of DCs and CD8+ T cells in 10 daysafter infection are plotted in Figs. 2–6. The average number of

Table 1Variables and parameters.

Symbol Interpretation Initial or default values Comments

D Number of DCs in a lymph node 10 EstimatedT1 Number of A-peptide-specific CD8+ T cells

in a lymph node1 Antigen-specific CD8+ T cells are present at very low frequency

(typically 1 in 105–106 T cells) in the T cell repertoire [7]T2 Number of B-peptide-specific CD8+ T cells

in a lymph node1 [7]

s Rate at which antigen-bearing DCs migrateinto draining lymph node

20 h�1 300 DCs could migrate into lymph node in 15 h [13]

q Average per capita death rate of DC 0.01 h�1 Half-life time 69.3 hn1 Average number of the A-peptide cognate

pMHC complexes per DCn1 ¼ 1

N

PNj¼1n0

1e�ðj�1Þ=sm1,

N ¼max n 2 Zjn 6 stf g þ 1, n01 is an

integer with value range of 0–4000;m1 ¼ 8:67

The half-life depends on the MHC affinity of A-peptide m1.Half-life for M-peptide is 6.01 h, corresponding to m1 ¼ 8:67[13]

n2 Average number of the B-peptide cognatepMHC complexes per DC

n2 ¼ 1N

PNj¼1n0

2e�ðj�1Þ=sm2,

N ¼max n 2 Zjn 6 stf g þ 1, n02 ranges

from 0 to 4000; m2 ¼ 3:40

The half-life depends on the MHC affinity of B-peptide m2.Half-life for C-peptide is 2.36 h, corresponding to m2 ¼ 3:40[13]

k1 Coefficient proportional to average TCRaffinity for A-peptide

1 � 10�6 h�1 To ensure that most CD8+ T cells can be primed within 24 h

k2 Coefficient proportional to average TCRaffinity for B-peptide

1 � 10�6 h�1

r Ratio of the CD8+ T cell proliferation rateinduced by cognate antigen-presenting DCto the CD8+ T cell priming rate

5 The rate of T cell proliferation is 5 times larger than the rate ofT cell priming (estimated)

d The sum of average per capita death rateand egress rate of CD8+ T cell

0.15 h�1 T cells may encounter draining sinusoids through which theycan egress from the lymph node and, if not activated by 20 h,90% of T cells will have left a given lymph node [17]

Table 2Steady states of system (1).

Steady state D T1 T2 Comments

P1sq 0 0 T cell activation will not occur below a threshold antigen dose

P2 dk1n1r

rsd �

qk1n1

0 A-peptide specific CD8+ T cell is immunodominant

P3 dk2n2r

0 rsd �

qk2n2

B-peptide specific CD8+ T cell is immunodominant

Fig. 1. Parameter regions of the stable steady state of system (1). The boundaries ofparameter region P1 set a threshold for T cell activation. Thus, T cells cannot beprimed if the average number of the specific peptide cognate pMHC complexes perDC is below the threshold. The boundary between parameter regions P2 and P3

reveals the criterion for dominance. The A-peptide-specific CD8+ T cells will bedominant if n1k1 > n2k2, vice versa.

0 2 4 6 8 100

200

400

600

800

1000

1200

Time (day)

Den

driti

c ce

ll

n1=55n1=65n1=100n1=300n1=1000

Fig. 2. Evolution of the number of DCs per lymph node, n2 ¼ 60. There are clearlythree phases for the DC population evolution. Both the duration of the primaryincreasing phase, and the steady DC population correlated inversely with theoverall avidity of the interaction.

Y. Wu et al. / Mathematical Biosciences 225 (2010) 53–58 55

B-peptide cognate pMHC per DC n2 are assumed to be 60 for Figs.2–4. We compare the outcomes with different average abundanceof A-peptide cognate pMHC per DC n1.

Figs. 3 and 4 clearly indicate the shift in immunodominance bychanging n1. When n1 ¼ 65, the A-peptide-specific CD8+ T cell willbe immunodominant as k1n1 > k2n2, while B-peptide-specific CD8+

T cell dominates with n1 ¼ 55. Thus, the overall avidity of the inter-action kini influences the probability that T cells will be stablycaptured by DCs, providing a mechanism for T cell competitionand dominance.

0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

1800

Time (day)

A−pe

ptid

e−sp

ecifi

c C

D8+

T ce

ll

n1=55n1=65n1=100n1=300n1=1000

Fig. 3. Evolution of the number of A-peptide-specific CD8+ T cell per lymph node,n2 ¼ 60. The latent phase is followed by a substantial proliferation phase, and then adecay and convergence phase. The latent duration correlated inversely with theoverall avidity of the interaction, while the steady T cell population is proportionalto the overall avidity.

0 2 4 6 8 100

500

1000

1500

Time (day)

B−pe

ptid

e−sp

ecifi

c C

D8+

T ce

ll

n1=55n1=65n1=100n1=300n1=1000

0 5 1010−20

10−10

100

1010

Fig. 4. Evolution of the number of B-peptide-specific CD8+ T cell per lymph node,n2 ¼ 60. B-peptide-specific CD8+ T cell dominates with n1 ¼ 55, as k1n1 < k2n2.

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (day)

Den

driti

c ce

ll

0 2 4 6 8 10

10−10

100

0 1 2 3 4 5 6 7 8 9 1010−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Pept

ide−

spec

ific

CD

8+ T

cel

l

Dendritic cellA−Peptide−specific CD8+ T cellB−Peptide−specific CD8+ T cell

Fig. 5. Evolution of the number of DCs and T cells per lymph node, n1 ¼ 5; n2 ¼ 10.Since n1 and n2 are both below the threshold value 15, T cell priming is suppressedand the population goes extinct.

0 2 4 6 8 100

50

100

150

200

250

300

350

400

450

500

Time (day)

Aver

age

pMH

C n

umbe

r per

DC

A−peptide cognate pMHC number n1B−peptide cognate pMHC number n2

Fig. 6. Evolution of the average number of the pMHC complexes per DC,n0

1 ¼ n02 ¼ 500; tA

1=2 ¼ 6:01 h;tB1=2 ¼ 2:36 h; tf ¼ 1=q ¼ 100 h ¼ 4:17 days. The trajec-

tory shows an exponential decay by the time tf , and then evolves to the steady stateniðtf Þ.

56 Y. Wu et al. / Mathematical Biosciences 225 (2010) 53–58

There are clearly three phases for the population evolution ofthe immunodominant species. The first phase corresponds to the

stage during which T cells undergo brief serial contacts and succes-sive stable interactions with the DCs. The abundance of DC perlymph node shows a significant increasing in this phase, while A-peptide-specific CD8+ T cell shows a latently increasing. The dura-tion of the first phase, typically half to 4 days, correlated inverselywith the overall avidity of the interaction k1n1. The typical periodfor n1 > 1000 is less than half a day, while T cell proliferation ob-served nearly 3 days after infection with n1 < 100. Phase one, cul-minating in peak value of DC number, is followed by a secondphase of a decay in DC number and a substantial proliferation inA-peptide-specific T cell. It is notable that the dynamics of prolifer-ation are very fast. The typical time duration for T cell booming isabout one day. The number of T cells then decays and makes atransition to a third phase characterized by converging towardequilibrium. The steady T cell population is proportional to theoverall avidity of the interaction kini.

To illustrate the steady state in which T cell will never be acti-vated, we set the average number of peptide-specific pMHC per DCbelow the threshold. Fig. 5 clearly show that both T cells becomeextinct, and DCs keep on increasing and approaching theequilibrium.

4.2. Limit pMHC lifespan

The physiological importance of pMHC lifespan for T cell immu-nity is unclear. The lifespan of pMHC complexes on DCs in lymphnode depends on the MHC affinity of cognate peptide [13]. pMHCcomplexes with higher affinity will have longer lifetime. Thetypical half-life time for M-peptide (peptide terminates in a methi-onine residue) specific pMHC complexes is 6.01 h, while for C-pep-tide (peptide with a carboxy-terminal cysteine) specific pMHCcomplexes is 2.36 h [13]. Thus the average number of pMHC com-plexes per DC ni should be time-varying.

To estimate the average number of pMHC complexes per DC,the decay of DC due to its natural lifespan should be taken intoconsideration. The average lifespan of DCs in lymph node can becalculated to be 1/q [18,19]. We assume for the sake of simplicitythat all DCs have identical lifespan tf ¼ 1=q, and the averagenumber of pMHC complexes per DC could be calculated as a func-tion of time.

We mark each DC in sequence, the first one which migrated intolymph node at time t ¼ 0 is labeled No. 1. The length of residencein lymph node for the jth DC is therefore given by t � ðj� 1Þ=s. Thepopulation of pMHC complexes per DC will perform exponential

0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

1000

Time (day)

A−pe

ptid

e−sp

ecifi

c C

D8+

T ce

ll

n01=300

n01=500

n01=1000

n01=3000

Fig. 8. Evolution of the number of A-peptide-specific CD8+ T cell per lymph nodewith limit lifespan of pMHC, n0

2 ¼ 60; tA1=2 ¼ 6:01 h; tB

1=2 ¼ 2:36 h; tf ¼ 1=q ¼ 100h ¼ 4:17 days. Since the steady T cell population decay over time, the minimumentrance number of pMHC per DC required for priming should be much higher thanthe threshold. The A-peptide-specific CD8+ T cell activation will happen if thesteady value of average pMHC number per DC n1ðtf Þ is higher than the thresholdqd=k1rs ¼ 15. For entrance number n0

1 ¼ 300, n1ðtf Þ ¼ 26:1 satisfies the criterion.However, there is a significant time delay for T cell priming compared to Fig. 3.

0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

1800

Time (day)

Den

driti

c ce

ll

tA1/2=9h

tA1/2=6h

tA1/2=3h

Fig. 9. Evolution of the number of DCs per lymph node with limit lifespan of pMHC,

Y. Wu et al. / Mathematical Biosciences 225 (2010) 53–58 57

decay with rates 1=mi, where mi ¼ ti1=2= ln 2 are the coefficients

proportional to cognate peptide-specific MHC affinity.For t < tf , all DCs in the lymph node are alive. Thus, the total

number of DCs migrated into the lymph node until time t is

N ¼max n 2 Zjn 6 stf g þ 1: ð14Þ

The average number of pMHC complexes per DC can be calculatedas

niðtÞ ¼1N

XN

j¼1

n0i e�

t�ðj�1Þ=smi ði ¼ 1;2Þ; ð15Þ

where n0i are the initial number of pMHC per DC at the entrance of

lymph node. For a typical half-life of 6.01 h and 2.36 h [13], theaffinity coefficients are m1 ¼ 8:67 and m2 ¼ 3:4, respectively.

If t P tf , the first mover is going to die, and the total number ofdeaths is

N1 ¼ max n 2 Zjn 6 sðt � 1=qÞf g þ 1: ð16Þ

So the total number of DCs remain in the lymph node until time t is

N2 ¼ N � N1: ð17Þ

The average number of pMHC complexes per DC is

niðtÞ ¼1

N2

XN

j¼N1þ1

n0i e�

t�ðj�1Þ=smi ði ¼ 1;2Þ: ð18Þ

Fig. 6 shows the evolution of ni with time. The curve performs anexponential decay before the average lifespan of DCs tf , and thenevolves to the steady state niðtf Þ. Thus, T cells will be primed ifthe steady value is higher than the threshold, niðtfÞ > qd=kirs.

The time courses of number of DCs and CD8+ T cells in 10 daysafter infection are plotted in Figs. 7–10. There are three similarphases. The major differences exist in phase three, where the stea-dy states shift along with time. The population evolution showssimilar relationship with the number of pMHC per DC in Figs. 7and 8. Due to the decayed average pMHC number per DC, the prim-ing process shows significant time delay compared to the infinitepMHC lifespan cases described in Section 4.1. To study the roleof pMHC lifespan on T cell priming, we depict in Figs. 9 and 10the evolution of DCs and A-peptide-specific CD8+ T cells with fixedhalf-life of B-peptide-specific pMHC tB

1=2 and variable half-life ofA-peptide-specific pMHC tA

1=2. It is clear that the duration of phaseone correlated reversely with the half-life of pMHC. As depicted in

0 2 4 6 8 100

500

1000

1500

Time (day)

Den

driti

c ce

ll

n01=300

n01=500

n01=1000

n01=3000

Fig. 7. Evolution of the number of DCs per lymph node with limit lifespan of pMHC,n0

2 ¼ 60; tA1=2 ¼ 6:01 h; tB

1=2 ¼ 2:36 h; tf ¼ 1=q ¼ 100 h ¼ 4:17 days. The populationevolution shows three similar phases to Fig. 2, with a much higher steady valuein phase three.

n01 ¼ 500; n0

2 ¼ 60; tB1=2 ¼ 2:36 h; tf ¼ 1=q ¼ 100 h ¼ 4:17 days. Three cases with

different half-life of A-peptide-specific pMHC are compared. The second phase ofdecay does not show up till 10 days with a relatively short half-life tA

1=2 ¼ 3 h.

Fig. 10, although identically distributed on each migrant DC at theentrance of lymph nodes, the pMHC with relative short half-lifewill be weeded out over time, and the duration of phase onemay tend to infinity, which means T cells will never been activatedwith too short half-life. Thus, the longest-lived pMHC will be dom-inant in a broad repertoire of peptides.

5. Conclusion

We have developed a mathematical model to study the dynam-ics of T cell–DC interactions in lymph nodes. The components inthe model are DCs and two varieties of antigen-specific CD8+ Tcells. The basic mechanisms of T cell priming observed in experi-ments are integrated into the model. The parameters are estimatedbased on available experimental data. Analysis and simulations ofthe model indicate that the immunodominance of adaptive CD8+ Tcell response may be determined by the overall avidity of the inter-actions between peptide-specific T cells and cognate antigen-bear-

0 2 4 6 8 100

200

400

600

800

1000

1200

Time (day)

A−pe

ptid

e−sp

ecifi

c C

D8+

T ce

ll

tA1/2=9h

tA1/2=6h

tA1/2=3h

Fig. 10. Evolution of the number of A-peptide-specific CD8+ T cell per lymph nodewith limit lifespan of pMHC, n0

1 ¼ 500; n02 ¼ 60; tB

1=2 ¼ 2:36 h; tf ¼ 1=q ¼ 100 h ¼4:17 days. Three cases with different half-life of A-peptide-specific pMHC arecompared. The duration of latent phase correlated inversely with the half-life ofpMHC, which a relatively short half-life tA

1=2 ¼ 3 h, the latency phase lasts for nearly8 days, followed by a slow priming process. The latency phase may tend to infinitewith a even shorter half-life tA

1=2, which suggests that T cell priming will neverhappen.

58 Y. Wu et al. / Mathematical Biosciences 225 (2010) 53–58

ing DCs, which comprises three major elements, (1) cognate TCRaffinity, (2) the density of complexes of cognate pMHC per DC,and (3) the lifespan of cognate pMHC. The peptide-specific T cellwith strongest avidity will become dominant over time. The varia-tion in overall avidity will likely shift immunodominance. Themost straightforward way to improve the overall avidity may beincreasing density of complexes of cognate pMHC per DC, whichcan be achieved by injecting highly loaded cognate peptide-pulsedDCs into lymph nodes. Since their pivotal role in determining thetype of immune response, DCs as targets for vaccine design canbe exploited in the treatment of various diseases. Our model alsopredicts a threshold for T cell priming, and provides a referenceto the lower bound dose for vaccine design. All the above conclu-sions are consistent with experimental observations as reportedin [7,13].

Acknowledgments

Yu Wu is grateful to Dr. Scott Lenaghan and Ms. Jing Wu fortheir helpful discussions. We are grateful to anonymous referees

for their extremely insightful comments which improved ourpaper.

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