A deterministic model for the processing and presentation of bacteria-derived antigenic peptides

ARTICLE IN PRESS

Journal of Theoretical Biology 250 (2008) 532–546 www.elsevier.com/locate/yjtbi

A deterministic model for the processing and presentation of bacteria-derived antigenic peptides Jozef Janda1, Gernot Geginat Institut fu¨r Medizinische Mikrobiologie und Hygiene, Fakulta¨t fu¨r Medizin Mannheim der Universita¨t Heidelberg, Theodor-Kutzer-Ufer 1-3, 68167 Mannheim, Germany Received 27 July 2007; received in revised form 19 October 2007; accepted 23 October 2007 Available online 3 December 2007

Abstract The amount and the dynamics of antigen supply to the cellular antigen processing and presentation machinery differ largely among diverse microbial antigens and various types of antigen presenting cells. The precise influence, however, of antigen supply on the antigen presentation pattern of cells is not known. Here, we provide a basic deterministic mathematical model of antigen processing and presentation of microbial antigens. The model predicts that different types of antigen presenting cells e.g. cells presenting or crosspresenting exogenous antigens, cells infected with replicating microbes, or cells in which microbial antigen synthesis is blocked after a certain period of time have inherently different antigen presentation patterns which are defined by the kinetics of antigen supply. The reevaluation of existing experimental data [Sijts, A.J., Pamer, E.G., 1997. Enhanced intracellular dissociation of major histocompatibility complex class I-associated peptides: a mechanism for optimizing the spectrum of cell surface-presented cytotoxic T lymphocyte epitopes. J. Exp. Med. 185, 1403–1411] describing the processing and presentation of two antigenic peptides derived from the p60 proteins of the facultatively intracellular bacterium Listeria monocytogenes shows that p60 proteins accumulating intracellularly during bacterial infection of cells play no measurable role as substrate for the cytosolic antigen presentation pathway. r 2007 Elsevier Ltd. All rights reserved. Keywords: Mathematical model; Bacteria; Listeria monocytogenes; Antigen presentation; Antigen processing; MHC class I

1. Introduction The control of microbial infections requires T cells recognizing pathogen-derived antigenic peptides displayed by MHC class I and II molecules, respectively. Cells present endogenous and exogenous antigens (Ag) via different pathways. Endogenous Ag are presented in the context of MHC class I molecules via the cytoplasmic Ag presentation pathway (Yewdell and Bennink, 2001) while exogenous Ag are primarily presented in the context of MHC class II molecules via the endosomal Ag presentation pathway (Li et al., 2005). However, professional Ag presenting cells (APC), in particular dendritic cells (DC), Corresponding author. Tel.: +49 6213832679; fax: +49 6213833816.

E-mail address: [email protected] (G. Geginat). 1 Current address: Department of Immunology, Ludwig Institute for Cancer research, Lausanne, Switzerland. 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.10.025

also have the ability to cross-present exogenous Ag in the context of MHC class I molecules (Larsson et al., 2003). The primary prerequisite for Ag presentation and subsequent recognition by T cells is the availability of Ag. The amount as well as the dynamics of the supply of Ag to the cellular Ag processing and presentation machinery differ largely among diverse microbes and various types of APC. In general, the rate of Ag derived from bacterial or protozoan pathogens that replicate intracellularly increases during the course of infection. This aspect has been studied in detail for the intracellular bacterium Listeria monocytogenes which replicates in the cytoplasm of the host cell with a doubling time of approximately 60 min (Villanueva et al., 1994). In some other instances microbes persist in cells in a non-replicating state as e.g. the bradyzoites of the intracellular parasite Toxoplasma gondii (Dubey et al., 1998). Ag derived from those slowly replicating microbes resembles cellular Ag that is expressed at a rather constant rate over an extended period of time.

ARTICLE IN PRESS J. Janda, G. Geginat / Journal of Theoretical Biology 250 (2008) 532–546

In contrast to the situation of Ag synthesis within the APC the presentation of exogenous Ag depends on the singular uptake of Ag by external loading (Li et al., 2005), a mechanism which results in a strict limitation of the amount of Ag available. After uptake of Ag an important function of professional APC, particularly DC, is the transport of Ag from the site of Ag uptake to the site of T cell priming in regional lymph nodes or the spleen. Uptake of exogenous Ag, e.g. bacterially or virally infected apoptotic cells, triggers the maturation of DC enhancing the immunogenicity of DC but at the same time also preventing further Ag uptake (Guermonprez et al., 2002). Thus when DC prime naı¨ ve T cells in lymphoid organs the presentation of exogenous Ag will be dependent solely on the initial Ag load. The cascade of events resulting in the display of MHC class I-bound antigenic peptides on the cell surface is well understood. It includes the proteolytic cleavage of Ag by the cytoplasmic proteasome, translocation of short peptides into the endoplasmic reticulum, assembly of MHC class I heavy chain, b2-microglobulin, and peptide to the ternary peptide/MHC complex, and subsequent transport of complexes to and display on the cell surface (Yewdell and Bennink, 2001). Cross-presentation of exogenous Ag in the context of MHC class I molecules also requires the proteasomal degradation of Ag, binding to nascent MHC class I molecules, and the transport of peptide/MHC complexes to the cells surface (Shen and Rock, 2006). Basically, the presentation of exogenous antigenic peptides by MHC class II molecules requires similar processing steps as the processing of endogenous Ag, which however take place in different cellular compartments. After external uptake of Ag proteolytic cleavage by endosomal proteases occurs and after binding to MHC class II molecules peptide/MHC complexes are transported to the cell surface of the APC (Li et al., 2005). Once arrived on the cell surface both MHC class I and II molecules exhibit a half-life which depends on the specific peptide/MHC combination (Shen and Rock, 2006). So far numerous MHC class I ligands derived from endogenous self or tumour Ag or various intracellular pathogens have been identified (Rammensee et al., 1997). In many instances also the number of antigenic peptides per APC has been determined, which can vary between less than 10 and up to more than 10,000 peptides per cell. The study of virally and bacterially infected cells revealed that the number of naturally processed antigenic peptides per APC depends on the Ag synthesis rate, the half-life of Ag, and the peptide processing efficacy of the specific antigenic peptide studied (Pamer, 1994; Princiotta et al., 2003; Qian et al., 2006; Sijts et al., 1996a, b; Sijts and Pamer, 1997; Villanueva et al., 1995). The assessment of the correlation between Ag synthesis and the half-life of Ag with Ag presentation is a very demanding task because at least the presentation of viral Ag depends in part on so-called defective ribosomal translation products, which represent misfolded or aberrantly synthesized proteins. Due to their

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short half-life these instable translation products are perfect substrates for the Ag processing machinery (Princiotta et al., 2003; Reits et al., 2000; Schubert et al., 2000). Their inhomogeneity and instability, however, renders the quantification of the number and the half-life of instable translation products a challenging task (Yewdell, 2007). In order to better understand the stochiometry and dynamics of Ag processing and presentation of microbederived Ag versus endogenous Ag on one hand and of exogenously acquired Ag versus endogenously synthesized Ag on the other hand we developed a basic mathematical model of Ag presentation. The application of the model to the Ag processing of two antigenic peptides derived from the p60 of L. monocytogenes shows a good fit with experimental data and suggests that stable p60 proteins accumulating during bacterial infection of cells play no measurable role as a substrate for the cytosolic Ag presentation pathway. 2. Materials and methods 2.1. Experimental data The Ag presentation model was applied to experimental data previously published by Sijts and Pamer (1997). The L. monocytogenes p60-derived, Kd-presented antigenic peptides p60217–225 and p60449–457 were quantified after isolation from cells infected with the L. monocytogenes 43251 strain. Peptides were isolated from cells by acidic extraction, separated by HPLC, and subsequently quantified with peptide-specific CD8 T cells (Sijts et al., 1996b; Sijts and Pamer, 1997). 2.2. Computational simulation of Ag presentation of microbial Ag In order to understand the influence of the kinetics of Ag supply on the formation of peptide/MHC complexes in APC, simulations were performed using a basic deterministic model of Ag processing (Fig. 1A). In this model, we described the multitude of steps involved in Ag processing and presentation by a limited number of well-defined parameters, which can be experimentally determined for a specific microbially expressed T cell Ag, namely the microbial growth rate g (defined by the microbial doubling time tg), the microbial Ag production rate a, the Ag decay rate b (defined by the Ag half-life ta), the peptide processing efficacy w (defined as the ratio of generated peptide/MHC complexes per degraded Ag), and the peptide/MHC complex decay rate d (defined by the halflife of the complex tc). For the simulation of the cell surface presentation of peptide/MHC complexes in addition to the above parameters also the transport time D that finally assembled peptide/MHC complexes require to travel from the endoplasmic reticulum to the cell surface was taken into account. We assumed that microbial mRNA synthesis

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antigens

microbes

peptide/MHC complexes

Δ surface δ transport time

γ microbial growth rate

α antigen production rate

. g(t) = g(t)

β

χ

δ

antigen decay rate

peptide processing efficacy

peptide/MHC complex decay rate

. a(t) = g(t) − a(t)

. c(t) = a(t) − c(t)

Model III

>0, >0, g0>0, a0=0 g(t)=g0et

=0, >0, g0>0, a0=0 g(t)=g0

=0, =0, g0=0, a0>0 g(t)=0

log10 (antigen/cell)

log10 (antigen/cell)

Model II

log10 (antigen/cell)

Model I

time

time

time

Fig. 1. Mathematical model of Ag presentation: (A) The interdependency of bacterial replication, Ag synthesis, Ag decay, processing of antigenic peptides, and the decay of peptide/MHC complexes are represented as a system of ordinary differential equations. (B) The differential equations were solved for distinct physiological scenarios defined by the microbial replication rate g and the Ag synthesis rate a resulting in models I, II and III (see materials and methods section). Shown is the kinetics of intracellular Ag supply for the different models. Model I assumes that Ag are synthesized by microbes that replicate intracellularly (g40, a40). Model II assumes that Ag are synthesized intracellularly without replication (g ¼ 0, a40). Model III assumes that APC are initially loaded with a limited amount of exogenous Ag a0 which subsequently decays (g ¼ 0, a ¼ 0, a040).

is already in a steady state and therefore we did not include terms describing transcription and degradation of mRNA. All these influence factors are accounted for in the microbial Ag production rate a that can be determined straightforwardly. The model was primarily designed to describe Ag processing and presentation early after infection of cells. Therefore, and also with the purpose to keep the model as simple as possible we did not take into account cell death of infected cells and a possible limitation of Ag presentation by the number of available nascent MHC molecules. Table 1 summarizes the input and output parameters of the Ag presentation model. By choosing specific starting conditions different physiological situations can be represented (Fig. 1B). Model I assumes unrestricted microbial growth (microbial growth rate g40, Ag synthesis rate a40, initial number of microbes per cell g040, initial loading of Ag a0 ¼ 0). Model II assumes that no microbial replication occurs and the number of microbes expressing an Ag remains constant over time as it can be imagined for cells presenting Ag from nonreplicating microbes as well as for cellular e.g. tumour Ag (microbial growth rate g ¼ 0, Ag synthesis rate a40, initial number of microbes per cell g040, initial loading of Ag

a0 ¼ 0). Model III describes Ag presentation if Ag synthesis does not occur within the APC, but cells are exclusively loaded with preformed external Ag (microbial growth rate g ¼ 0, Ag synthesis rate a ¼ 0, initial number of microbes per cell g0 ¼ 0, initial loading with Ag a040). Symbolic solutions for the number of microbes per cell g(t), Ag per cell a(t), and peptide/MHC complexes per cell c(t) were derived for all the three scenarios (see Appendix A). All computations were done using MathematicaR 5.0 (Wolfram Research, IL, USA) on a personal computer. 3. Results 3.1. Modelling of the kinetics of Ag supply under different physiological situations The Ag supply of APC depends on the Ag presentation pathway (e.g. cytoplasmic versus endosomal pathway) and the source of the Ag (e.g. Ag from replicating versus nonreplicating microbes, preformed exogenous Ag). In order to estimate the impact of the kinetics of Ag supply on Ag presentation the total number of peptide/MHC complexes per cell was simulated for all three Ag presentation models,

ARTICLE IN PRESS J. Janda, G. Geginat / Journal of Theoretical Biology 250 (2008) 532–546 Table 1 Input and output parameters used in the Ag presentation model Parameter

Description

Dependency

g(t)

Number per cell of microbes expressing the Ag at time t Number per cell of microbes expressing the Ag at t ¼ 0 Microbial doubling time (min)

Output

g0 tg

g a(t) a0 a ta b c(t) cs(t) c0 w tc d D

Microbial growth rate Number of Ag per cell at time t Number of Ag per cell at time t ¼ 0 Ag synthesis rate per microbe Ag half-life (min) Ag decay rate Number per cell of peptide/MHC complexes at time t Number per cell of surface peptide/ MHC complexes at time t Number per cell of peptide/MHC complexes at time t ¼ 0 Peptide/MHC complexes per Ag processed Peptide/MHC complex half-life (min) Peptide/MHC complex decay rate Time required for peptide/MHC complexes to reach the cell surface (min)

Input, g(0) ¼ g0 Input Replication: tg 4 0 No replication: tg-N g ¼ ln 2/tg Output Input, a(0) ¼ a0 Input Input b ¼ ln 2/ta Output Output Input, c(0) ¼ c0 Input Input d ¼ ln 2/tc Input

using various numerical values for the parameters defining the cellular Ag supply. In model I the total Ag synthesis depends on the number of microbes g0 initially infecting the cell, the intracellular microbial doubling time tg, and the microbial Ag synthesis rate a (Fig. 2, model I). In addition, the number of Ag per cell is also influenced by the Ag half-life ta (Fig. 3A, model I). In this model the number of Ag per cell increases as long as microbial replication continues. Variation of the microbial doubling time affects the number of Ag and peptide/MHC complexes of cells much more than changes in the initial infectious dose g0 or the Ag synthesis rate a. In model II no replication occurs and the Ag supply in an infected cell depends on the initial number g0 of microbes per cell expressing the Ag and the microbial Ag synthesis rate a only (Fig. 2, model II). In this scenario the intracellular level of Ag and peptide/MHC complexes converges towards a steady state defined by the initial multiplicity of infection of cells, the Ag synthesis rate and also the Ag half-life ta (Fig. 3A, model II). In model III Ag supply is primarily defined by the initial uptake of Ag (a0) (Fig. 2, model III) which is subsequently degraded in a first-order kinetics depending on the Ag halflife ta (Fig. 3A, model III). Taken together, these simulations show how the Ag presentation pattern of cells is dependent on the kinetics of Ag supply that differs largely among various sources of Ag.

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3.2. Modelling of the generation of peptide/MHC complexes under different physiological situations The generation of peptide/MHC complexes depends upon the degradation of Ag. In addition to the Ag half-life ta, the generation of peptide/MHC complexes is determined by the peptide processing efficacy w. After assembly, peptide/MHC complexes again decay with a first-order kinetics defined by the peptide/MHC complex half-life tc. In order to estimate the relative impact of these factors on Ag presentation, simulations were performed for all three Ag processing scenarios with various numerical values for ta, w and tc (Fig. 3A). The Ag half-life ta reciprocally influences the abundance of Ag and peptide/ MHC complexes in cells. In general, a longer Ag half-life yields higher Ag levels and because less Ag are degraded lower levels of peptide/MHC complexes. The impact of the Ag half-life on the formation of peptide/MHC complexes, however, depends also on the scenario studied. In model I a shorter Ag half-life always results in a decrease of Ag and an increase of peptide/MHC complexes. If no replication occurs (model II), both, the number of Ag per cell and the number of peptide/MHC complexes per cell converge towards a steady state. Under steady state conditions the number of Ag per cell depends on the Ag half-life while the number of peptide/MHC molecules is independent of the Ag half-life (see also Appendix, Eqs. (19) and (20)). In this circumstance, the Ag half-life only determines how fast the steady state is reached. In model III, after external Ag loading of cells the Ag half-life determines how long Ag persists. If Ag decay rapidly peptide/MHC complexes are formed quickly, but after depletion of the Ag pool formation of new peptide/ MHC complexes rapidly subsides. On the other hand, formation of peptide/MHC complexes is delayed if Ag decay slowly, but complexes will persist as long as the decaying peptide/MHC complexes can be replaced until the intracellular Ag pool is depleted. Similar to the Ag half-life, also the impact of the peptide/MHC complex half-life is more pronounced if Ag supply is limited (model III4model II4model I). For the modelling of the cell surface presentation of peptide/MHC complexes also the decay of peptide/MHC complexes during transport from the endoplasmic reticulum to the cell surface has to be taken into account. This decay depends on the time D required to reach the cell surface and the half-life tc of peptide MHC molecules (Eq. (33)). In order to estimate the relative impact of these factors on the cell surface presentation of peptide/MHC complexes simulations were performed with various numerical values for tc and D (Fig. 3B). The simulation of surface peptide/MHC complexes (Fig. 3B) shows that the peptide/MHC complex half-life influences the number of surface peptide/MHC complexes much stronger than the total number of cellular peptide/MHC complexes. This is particularly evident if the half-life of peptide/MHC complexes is shorter than the transport time from the

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antigens

peptide/MHC complexes

Model I 7 6 5 4 3 2 1 0 1

microbes

Model III

n/a

n/a

τg

n/a

g0

n/a

α

120 min

0

3

7 6 5 4 3 2 1 0 1

6

9

12 7 6 5 4 3 2 1 0 1

1

0

log10 (copies per cell)

Model II

3

6

7 6 5 4 3 2 1 0 1

9

3

6

0

12 7 6 5 4 3 2 1 0 1

150

0

1

9

12

3

6

9

12

150

0

3

n/a

6

9

n/a

hours post infection

12 7 6 5 4 3 2 1 0 1

a0 10000

0

3

6

9

12

Fig. 2. Simulation of the influence of Ag supply on the formation of peptide/MHC complexes. In model I (intracellular replication of microbes) Ag supply is defined by the initial number of microbes per cells g0 expressing the Ag, the microbial doubling time tg, and the Ag synthesis rate a. In model II (intracellular Ag synthesis without replication) Ag supply is defined only by the number g0 of Ag-expressing microbes per cell and the Ag synthesis rate a. In model III (external loading only) Ag supply is defined exclusively by the initial Ag load a0. The number of microbes per cell (bold lines), proteins per cell (narrow dotted lines), and peptide/MHC complexes per cell (narrow solid lines) were calculated using default values (tg ¼ 60 min, g0 ¼ 1, 1 a ¼ 100 microbe1 min1; ta ¼ 100 min; a0 ¼ 10,000; w ¼ 10 ; tc ¼ 120 min). The parameters indicated on the right were varied in order to evaluate their impact on Ag presentation. For selected curves the numerical values of the variable parameter are indicated. Parameter ranges: microbial doubling time tg ¼ 15, 30, 60, 120; initial number of microbes per cell g0 ¼ 1, 2, 4, 8; Ag synthesis rate a ¼ 150, 300, 600, 1200; initial number of Ag per cell a0 ¼ 10,000, 20,000, 40,000, 80,000. No graph is shown if the simulation was independent of the variable parameter (n/a, not applicable).

endoplasmic reticulum to the cell surface. This effect of the peptide/MHC complex half-life again is particularly prominent under conditions of limited Ag supply (model III4model II4model I). In summary, the above analysis demonstrates that the kinetics of Ag supply modulate the relative impact that other parameters as the half-life of Ag or peptide/MHC complexes have on Ag presentation.

3.3. Modelling of peptide processing ratios Multiple antigenic peptides can be processed from a single protein. In this situation the ratio of the numbers of extractable antigenic peptides has been calculated with the intention to infer the relative peptide processing efficacy (Sijts et al., 1996b). In order to study the kinetics of the ratio of two peptides processed from the same Ag we

ARTICLE IN PRESS J. Janda, G. Geginat / Journal of Theoretical Biology 250 (2008) 532–546

peptide/MHC complexes antigens

log10 (copies per cell)

Model I 7 6 5 4 3 2 1

1024 min

1024 min

0

peptide/MHC complexes antigens

6

9

0

3

7 6 5 4 3 2 1

6

6

7 6 5 4 3 2 1

7 6 5 4 3 2 1 9

3

7 6 5 4 3 2 1

6

9

7 6 5 4 3 2 1

7 6 5 4 3 2 1

100 min

0

3

6

9

12

9

12

3

6

9

3 6 9 hours post infection

12

3

6

9

3 6 9 hours post infection

3

6

9

12

1 min

3

6

9

12

τc

8 min

0 7 6 5 4 3 2 1

12

12

τc

0

12

9

1/1024

7 6 5 4 3 2 1

3

6

χ

0 7 6 5 4 3 2 1

τa

1024 min

0

12

100 min

0

1024 min

7 6 5 4 3 2 1

8 min

0

12

6

1 min

0

12

4 min

0

3

1/1024

0

12

1 min

3

1024 min

7 6 5 4 3 2 1 9

7 6 5 4 3 2 1

1024 min

0

12

1/1024

0

log10 (copies per cell)

3

7 6 5 4 3 2 1

Model III

Model II 7 6 5 4 3 2 1

537

3

6

9

12

Δ 100 min

0

3

6

9

12

Fig. 3. Simulation of the influence of Ag half-life, peptide processing efficacy, peptide/MHC stability, and the surface transport time of peptide/MHC complexes on Ag presentation. (A) The numbers of proteins per cell (dotted lines) and peptide/MHC complexes per cell (solid lines) were calculated using 1 default values (tg ¼ 60 min, g0 ¼ 1, a ¼ 100 microbe1 min1; ta ¼ 100 min; a0 ¼ 10,000; w ¼ 10 ; tc ¼ 120 min). The parameters indicated on the right were varied in order to evaluate their impact on Ag presentation. For selected curves the numerical values of the variable parameter are indicated. Parameter 1 ranges: Ag half-life ta ¼ 1, 2, 4, 8y,1024 min; peptide processing efficacy w ¼ 11; 12; 14; . . . ; 1024 peptide/MHC complex half-life tc ¼ 1, 2, 4, 8,y,1024 min. No graph is shown if the simulation was independent of the variable parameter (n/a, not applicable). (B) The numbers of Ag per cell (dotted lines) and surface peptide/MHC complexes per cell (solid lines) were calculated using the default values indicated above. The peptide/MHC complex half-life tc and the endoplasmic reticulum to cell surface transport time D were varied in order to evaluate their impact on Ag presentation. Parameter ranges: tc ¼ 1, 2, 4, 8,y,1024 min; D ¼ 10, 20, 30,y,100 min.

simulated the processing of two peptides derived from a hypothetical Ag with an intracellular half-life of 120 min. The simulation was performed for the total number of peptide/MHC complexes per APC. For this simulation we assumed that peptide2 is processed with a 10-fold higher

processing efficacy than peptide1 but that the half-life of peptide/MHC complexes formed with peptide1 is 10-fold longer than with peptide2. In order to estimate the relative impact of the microbial doubling time tg, the Ag half-life ta, the peptide processing efficacy w, and the peptide/MHC

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peptide ratio (r21=peptide2/peptide1)

r21(t)

lim r21(t) t∞

Model I 10 9 8 7 6 5 4 3 2 1 0

Model II

Model III

n/a

n/a

15 min

0

3

10 9 8 7 6 5 4 3 2 1 0

6

9

12 10 9 8 7 6 5 4 3 2 1 0

20 min

0

3

10 9 8 7 6 5 4 3 2 1 0

6

9

12

0

3

6

1/2

0

3

9

0

10 9 8 7 6 5 4 3 2 1 0

1/1

6

9

12

3

6

9

12

3

6

9

20 min

0

12

1/2

3 6 9 hours post infection

3

12

6

9

12

10/1 χ2/χ1

0

3

10 9 8 7 6 5 4 3 2 1 0

1/1

0

τa

10 9 8 7 6 5 4 3 2 1 0

10/1

0

12

10 9 8 7 6 5 4 3 2 1 0

20 min

10 9 8 7 6 5 4 3 2 1 0

10/1

10 9 8 7 6 5 4 3 2 1 0

τg

6

9

12

1/1

1/2

0

3

6

τc2/τc1

9

12

Fig. 4. Simulation of the influence of microbial doubling time, Ag half-life, peptide processing efficacy, and peptide/MHC stability on the relative abundance of two antigenic peptides derived from the same Ag. The relative abundance of two hypothetical antigenic peptides is represented as peptide ratio r21. The time-dependent peptide ratio (r21(t), solid lines) and the limit of r21(t) for t-N (dotted lines) were calculated using default values 1 , w2 ¼ 1, tc1 ¼ 450 min, w2 ¼ 1, tc2 ¼ 45 min). In (tg ¼ 60 min, g0 ¼ 1, a1 ¼ a2 ¼ 100 microbe1 min1, ta1 ¼ ta2 ¼ 120 min, a01 ¼ a02 ¼ 10,000, w1 ¼ 10 order to evaluate the impact of the microbial doubling time tg, Ag half-life ta, the peptide processing efficacy w, and the half-life of peptide/MHC complexes tc these parameters were varied. For selected curves the numerical values of the variable parameter are indicated. Parameter ranges: tg ¼ 15, 30, 1 60,y,150; ta1 ¼ ta2 ¼ 20, 40, 60,y,200; w1 ¼ 11; 12; 13; . . . ; 10 (-w2/w1 ¼ 1:1, 2:1,y,10:1); tc1 ¼ 45, 90, 135,y,450 min (-tc2/tc1 ¼ 1:1, 1:2,y,1:10). No graph is shown if the simulation was independent of the variable parameter (n/a, not applicable).

complex half-life tc, these parameters were varied (Fig. 4). Interestingly, for t-0 the initial ratio of two peptides derived from the same Ag is independent of the model situation studied and is only defined by the ratio of the processing efficacies of the two peptides (see also Eqs. (15), (24), and (30)). However, in all three models over time the more stable peptide accumulates and eventually outnumbers the less stable but more efficiently processed peptide (Fig. 4). If the half-lives of both peptide/MHC complexes are identical the peptide ratio remains constant

over time and is given by the ratio of the processing efficacies w2/w1 of both peptides. Under conditions of continuous Ag synthesis without microbial replication (model II) the peptide ratio converges towards a value which only depends on the ratios of the peptide processing efficacies w2/w1 and the peptide/MHC complex half-lives tc2/tc1. In the model I with microbial replication the peptide ratio converges towards a value that in addition also depends on the microbial doubling time tg. In models I and II the Ag half-life ta itself does not influence the value

ARTICLE IN PRESS J. Janda, G. Geginat / Journal of Theoretical Biology 250 (2008) 532–546

towards which the peptide ratio converges, but determines how fast this value is reached. In model III in which Ag is provided by initial external loading the peptide ratio does not converge towards a defined value but over time is strongly biased towards the peptide that forms the more stable peptide/MHC complexes. Taken together, these simulations demonstrate that the ratio of two antigenic peptides processed from the same Ag depends not only on the ratio of the peptide processing efficacies but also on the half-lives of the peptide/MHC complexes as well as on the kinetics of Ag delivery. Vice versa these simulations implicate that the relative processing efficacy of two peptides with largely different half-lives of peptide/MHC complexes cannot be directly inferred from the relative abundance of antigenic peptides in infected cells. 3.4. In silicio modelling of the Ag processing and presentation of L. monocytogenes-derived antigenic peptides In order to validate the Ag presentation model, we decided to simulate the presentation of the processing and presentation of L. monocytogenes-derived antigenic peptides. The processing of the L. monocytogenes p60-derived Kd-presented antigenic peptides p60217–225 and p60449–457 has been studied in detail (Harty and Pamer, 1995; Sijts

log10 (copies per cell)

Model I 7 6 5 4 3 2 1 0 −1

p60449-457

3

6

9

12

ratio (p60449-457/p60217-225)

Model III 7 6 5 4 3 2 1 0 −1

p60449-457 p60217-225

0

3 6 9 hours post infection bacteria

20 18 16 14 12 10 8 6 4 2 0

et al., 1996b; Sijts and Pamer, 1997). L. monocytogenes 43251 replicates intracellularly with a doubling time (tg) of 60 min, and secretes p60 with a rate (a) of 58 molecules per min per bacterium. Intracellular p60 decays with a halflife (ta) of 90 min. From 35 decayed p60
molecules one 1 p60217–225/Kd complex is generated w ¼ 35 . After peptide/ MHC complexes are formed they again decay with a peptide specific half-life. While p60217–225/Kd complexes are rather stable with a half-life (tc) of 6 h, p60449–457/Kd complexes are less stable and decay with a half-life of 1 h. The simultaneous quantification of both antigenic peptides from infected cells revealed that p60449–457 is much more abundant in infected cells. In this situation application of the Ag processing model allows to calculate the relative peptide processing efficacy (w2/w1) of two peptides. We used a data set from Sijts and Pamer (1997) that shows that between 3 and 4 h post infection the p60449–457/p60217–225 ratios are 15:1 and 13:1, respectively. Applying these data to Eq. (14) and solving for the relative peptide processing efficacy (w2/w1) yields relative peptides processing efficacies of 22.4 and 20.7 for p60449–457/p60217–225. For the further calculation we used the mean (21.5) of both values, yielding a processing efficacy of one p60449–457/Kd complex per 1.63 p60 molecules degraded. Using these data the per cell number of bacteria, p60 molecules, p60217–225/Kd complexes, and p60449–457/Kd complexes over time was Model II

7 6 5 4 3 2 1 0 −1

p60217-225

0

539

protein

12

p60217-225

0

3

p60449-457

6

9

12

peptides

model I

model II model III

0

3 6 9 hours post infection

12

Fig. 5. Modelling of the processing of antigenic peptides derived from the p60 protein of L. monocytogenes. (A) Shown is the simulated time-dependent change of the numbers of intracellular L. monocytogenes per cell (bold line), p60 molecules per cell (dotted line), p60449–457/Kd complexes, and p60217–225/ Kd complexes per cell (narrow lines as indicated) for all three Ag presentation models. (B) Shown is the simulated change of the p60449–457/p60217–225 ratio over time. Calculations were performed using the following numerical values: tg ¼ 60 min, g0 ¼ 1.25, a1 ¼ a2 ¼ 58 microbe1 min1, ta1 ¼ ta2 ¼ 90 min, 1 1 a01 ¼ a02 ¼ 10,000, w ¼ 35 , tc1 ¼ 360 min, w2 ¼ 1:6 , tc2 ¼ 60 min.

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2000 1800 1600 1400 1200 1000 800 600 400 200 0

p60217-225per cell

p60217-225per cell

540

2 4 6 hours post infection

8

20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0

2 4 6 hours post infection

8

+Ab

0

p60449-457per cell

p60449-457per cell

0

2000 1800 1600 1400 1200 1000 800 600 400 200 0

20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0

2 4 6 hours post infection

8

+Ab

0

2 4 6 hours post infection

8

Fig. 6. Matching experimental and simulated Ag presentation data. Experimental data (filled circles) of the numbers of p60217–225 (A, B) and p60449–457 (C, D) peptides per cell at different time points after infection of cells with L. monocytogenes were taken from Sijts and Pamer (1997). Peptides were extracted either in the absence of antibiotics (A, C) or after inhibition of bacterial protein biosynthesis with antibiotic (+Ab) tetracycline 4 h post infection (B, D). Experimental data (filled circles) were overlaid with calculated curves (lines). The kinetics of antigenic peptides after inhibition of bacterial protein synthesis was either calculated using model III, assuming that previously accumulated p60 molecules are further processed and presented after addition of antibiotics (solid lines), or alternatively by a first-order decay kinetics assuming that after inhibition of bacterial protein synthesis peptide/MHC complexes decay without further Ag processing (broken lines). The number of p60449–457 peptides per cell in C and D was calculated assuming either a processing 1 (narrow lines). The values of all other parameters are identical to Fig. 5. efficacy (w2) of 1/1.6 (fat lines) or 3:5

simulated for all three Ag presentation scenarios (Fig. 5A) and the peptide ratio r21 ¼ p60449–457/p60217–225 was calculated (Fig. 5B). Fig. 5B demonstrates how the p60449–457/p60217–225 ratio will depend on the time post infection and the Ag presentation model. We next compared the prediction of the Ag presentation model to experimental data from Sijts and Pamer (1997). Fig. 6A shows that the calculated number of p60217–225/Kd complexes per cells reasonably well describes the experimental data between 2 and 5 h post infection. At time points later than 5 h post infection much less antigenic peptides were experimentally extracted than predicted by the model, probably due to the L. monocytogenes-mediated cytopathic effect on infected cells or due to limited availability of MHC class I molecules. Also the predicted number of p60449–457/Kd complexes per cell correlates with the experimental data set during the first 4 h after infection (Fig. 6C, bold line). The model predicts the experimentally determined number of antigenic peptides per cell much less accurately if instead of the calculated relative processing 1 1 efficacy of 1:6 for p60449–457 the previously reported value 3:5 (Sijts et al., 1996b) is used (Fig. 6C, narrow line). Sijts and Pamer (1997) also provided a data set for cells in which p60 synthesis is blocked by treatment with the antibiotic tetracyclin 4 h post infection. Assuming that

after antibiotic inhibition of protein synthesis antigenic peptides are generated from previously accumulated p60 we used model III (see Appendix, Eq. (32)) to simulate these particular experimental conditions. Remarkably, the model does not at all fit with the experimental data (Fig. 6B, D), which, as will be discussed later in this particular scenario in the presence of antibiotics is probably not due to cell death or limiting supply of MHC class I molecules. Alternatively we applied Eq. (31) that describes the decay of peptide/MHC complexes in the absence of further Ag processing. Remarkably, now the model fits fairly well with the data set (Fig. 6D, fat broken line) suggesting the interpretation that the large amount of rather stable p60 (ta90 min) accumulating in infected cells does not play an important role as a substrate for the cytosolic Ag processing pathway. Also in this situation the prediction of the model for p60449–457 is much more accurate if for the relative peptide processing efficacy the 1 calculated value of 1:6 instead of the previously published 1 value 3:5 is used. 4. Discussion To understand the relation between Ag supply and naturally processed antigenic peptides under different

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physiological situations we developed a descriptive mathematical model which describes the interdependency of microbial replication, Ag synthesis, Ag degradation, Ag processing, and the formation and decay of peptide/MHC complexes as a system of linear differential equations. Probably the most interesting prediction of the model is that the Ag presentation pattern of peptides is strongly influenced by the kinetics of Ag delivery that differs largely under various physiological situations. Depending on the initial conditions that define the kinetics of Ag supply the model yields different symbolic solutions, indicating that different types of APC e.g. cells presenting or cross-presenting exogenous Ag, cells infected with replicating microbes, or cells presenting Ag from nonreplicating microbes have inherently different Ag presentation patterns. For example, in the situation of exponential increase of Ag supply (model I), as it can be expected in cells infected with rapidly replicating microbes the maximum number of peptide/MHC molecules per cell will rapidly increase over time as long as microbial growth continues (see Appendix, Eq. (9)). In this scenario, a long Ag half-life or a short halflife of the peptide/MHC complex will influence Ag presentation, but will not totally prevent presentation of most MHC-binding peptides that are processed with a reasonable efficacy. The relatively weak influence of the half-life of peptide/MHC complexes on their number in cells infected with exponentially growing microbes is intuitively understandable if one considers that under these conditions the majority of peptide/MHC molecules are always newly built. Similarly, in models describing the epidemic spread of infections it has been noted that under conditions of exponential increase of the number of infected individuals the lifespan of the infected individuals has a minor influence on the spread of the epidemic as the majority of the infected individuals always are newly infected (Renshaw, 1991). If the Ag supply is very limited, however, as in the case of APC that are loaded once by uptake of an external Ag, the half-lives of Ag and peptide/MHC complexes become very important determinants for the maximum number of peptide/MHC complexes per cell. The half-lives of Ag and peptide/MHC complexes also define how long Ag presentation persists. As peptide/MHC molecules are assumed to decay with a similar kinetics on their way to the cell surface a long half-life is particularly important for cell surface presentation of peptide/MHC complexes. Our Ag presentation model (I) fairly well predicts the kinetics of the generation of p60449–457/Kd and p60217–225/Kd complexes in L. monocytogenes infected cells during the early exponential growth phase of intracellular bacteria. The model also predicts a time-dependent change of the ratio of both peptides in infected cells which also has been shown experimentally by Sijts et al. (1996b) and Sijts and Pamer (1997). This change of the ratio of p60449–457/Kd and p60217–225/Kd complexes which have largely different halflives (Sijts and Pamer, 1997) makes it impossible to directly

541

infer the relative Ag processing efficacies of the two antigenic peptides from the abundance of the antigenic peptides in infected cells. Application of Eq. (14) (see Appendix) to the data published by Sijts and Pamer (1997), however, allows to calculate that in L. monocytogenesinfected cells approximately 20-fold more p60449–457/Kd than p60217–225/Kd complexes are generated. As approximately one p60217–225/Kd complex is generated per 35 degraded p60 molecules this yields an extremely high processing efficacy of approximately one newly generated p60449–457/Kd complex per 1.6 degraded p60 molecules, which is significantly higher 1 than the ratio of 3:5 reported previously (Sijts et al., 1996b). This value is several orders of magnitude higher than the generation of cell surface peptide/MHC complexes from a virally expressed model Ag for which an average processing 1 efficiency of 2000 has been determined (Princiotta et al., 2003). Intracellular L. monocytogenes replicate with a doubling time of approximately 60 min. Thus after infection with an average of 1.25 bacteria per cell 5 h post infection an individual cell already harbours between 32 and 64 bacteria and will die rapidly. Thus at later time points the number of peptide/MHC complexes per cell will be strongly influenced by the L. monocytogenes-mediated cytopathic effect, which might explain why at time points later than 4 h post infection the number of extractable antigenic peptides is lower than predicted. In addition also the availability of peptide MHC molecules (Chang et al., 2005) might limit the number of extractable peptides. As our model describes the early phase of infection reasonably well we—as others before us (Bulik et al., 2005)—avoided the use of more sophisticated models which take into account cell death of infected cells and possible limitations of Ag presentation by the number of available nascent MHC molecules. It would probably be possible to precisely model the generation of antigenic peptides in L. monocytogenesinfected cells at time points later than 4 h post infection by taking into account the killing of infected cells and/or a limited availability of MHC class I molecules. As quantitative experimental data, however, are neither available for the kinetics of infection-dependent cell death (which probably would be dependent on the multiplicity of infection and the duration of infection) nor for the synthesis rate of MHC class I molecules in Listeria-infected cells the numerical values used for the simulation would be highly speculative. A more advanced model also would require to take into account the competition of different peptides for the same MHC molecule as well as the stability of peptide/MHC complexes displayed by killed cells. Also these values, however, are not readily available. Because of our limited knowledge of these parameters we deliberately designed our basic models I–III for settings in which cell death or availability of peptide/MHC molecules are not limiting. The reported (Sijts et al., 1996b; Sijts and Pamer, 1997; Villanueva et al., 1994) unlimited exponential increase of the numbers of intracellular L. monocytogenes,

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p60 molecules, and extractable p60-derived antigenic peptides between 0 and 4 h after low dose infection with roughly 1 bacterium per J774 cell indicates that in this early infection phase neither cell death nor the availability of MHC class I molecules limits the generation of L. monocytogenes-derived antigenic peptides. Death of infected cells due to growth of intracellular L. monocytogenes can be prevented by the antibiotic tetracyclin that rapidly inhibits bacterial protein synthesis and blocks further bacterial replication. The rate of dead cells among antibiotic-treated infected cells is not significantly higher than among non-infected cells (approximately 5% (Skoberne et al., 2002) and it therefore can be assumed that the influence of dying cells on the average number of antigenic peptides per cell would also be less than 5%. Thus, it seems justified to use the basic model III for the modelling of time points later than 4 h post infection if further bacterial replication is inhibited by the presence of antibiotics. After antibiotic treatment the number of extractable antigenic peptides, in the case of p60217–225 levels well below the maximum number of extractable p60217–225 peptides which without antibiotic inhibition increases until 7 h post infection or, in the case of p60449–457, rapidly declines below the 4 h value. Because after the blockade of Ag synthesis and bacterial replication by antibiotic treatment the numbers of extractable p60217–225 and p60449–457 peptides per cell do not exceed the amount of antigenic peptides extractable at the 4 h time point immediately before addition of the antibiotic, it seems hardly imaginable that in this setting Ag processing is limited by MHC class I synthesis which itself is not affected by antibiotic treatment. Thus, even if a more sophisticated model which also takes into account the killing of cells and/ or the limited availability of MHC molecules would be applied, it should yield similar results as the basic model if cell death can be neglected and the availability of MHC class I molecules does not limit Ag processing. The intracellular half-life of naturally occurring T cell Ag varies between days (e.g. influenza virus nucleoprotein (Princiotta et al., 2003)) and a few minutes (e.g. listeriolysin O from L. monocytogenes (Schnupf et al., 2006)). Remarkably, antigenic peptides are very efficiently processed also from the stable influenza nucleoprotein within a period of time that theoretically only would allow for the decay of a few proteins. This paradoxon was solved by the demonstration of the presence of unstable incomplete and/or misfolded translation products, which account for a significant portion of newly synthesized viral proteins (Princiotta et al., 2003; Schubert et al., 2000). Interestingly, our Ag presentation model predicts that the impact of the half-life of Ag on the Ag presentation pattern of APC depends significantly on the kinetics of Ag supply. While in the early phase of presentation an unstable Ag is always expected to yield more peptide/MHC complexes per cell, over time the outcome depends on the model situation studied. In cells infected with replicating microbes (model I), a peptide derived from an unstable Ag will always be more

abundant than a peptide derived from a more stable Ag (see Appendix, Eq. (9)). In the presence of constant continuous Ag synthesis (no replication, model II), as it can be assumed for Ag from non-replicating microbes and also for tumour Ag, the abundance of a peptide under steady state conditions will be independent of Ag stability (see Appendix, Eq. (20)). In the model with external loading of Ag only (model III) a peptide derived from an unstable Ag will be presented more rapidly but will persist shorter than an antigenic peptide derived from a more stable Ag. Thus from model III it can be predicted that after Ag uptake by external loading or by cross-presentation antigenic peptides will be presented most efficiently if they are rapidly degraded and yield stable peptide/MHC complexes. Interestingly, this prediction is in line with the observation that the cross-presentation of L. monocytogenes-derived antigenic peptides requires the uptake by DC of a instable but not finally processed Ag from infected cells (Janda et al., 2004). As can be inferred from Eq. (26), under these conditions (tcbta-bbd) the persistence of peptide/MHC complexes will be mostly determined by the decay rate d of the peptide/MHC complex which correlates inversely with the complex half-life tc. In this regard, it is interesting to know that peptide/MHC class II complexes of DC are very stable (Lanzavecchia et al., 1992; Zehn et al., 2004), thus fulfilling the prerequisite for optimal presentation of exogenous Ag as discussed above. The experimental data from Sijts and Pamer (1997), who also monitored the abundance of p60449–457/Kd and p60217–225/Kd in L. monocytogenes-infected cells after blockade of bacterial protein synthesis by antibiotic treatment, show that the processing of Listeria-derived antigenic peptides is coupled to bacterial protein biosynthesis. In this situation the simulation (model III, with c040, Eq. (32)) does not match the experimental data, but the decrease of the number of peptides per cell after antibiotic inhibition of bacterial protein biosynthesis can be modelled reasonably well by a first-order decay kinetics (see Appendix, Eq. (31)). This strongly suggests that the large number of p60 molecules which accumulate in infected cells and which have an intracellular half-life of 90 min (Villanueva et al., 1994) are not a main substrate for the generation of p60-derived antigenic peptides. Antibiotic inhibition of bacterial protein synthesis and replication prevents cell death and in this special situation allows the application of the basic model III to time points later than 4 h when unprotected cells begin to die due to the cytopathic effect exerted by the rapidly growing number of intracellular L. monocytogenes. In contrast to viral Ag the existence of previously postulated ‘‘immunoribosomes’’ (Yewdell, 2007; Yewdell and Nicchitta, 2006) could not explain the dependency of the processing of bacteriaderived Ag from bacterial protein biosynthesis because in the infected cell bacterial protein biosynthesis is separated from cellular protein synthesis. However, a possible explanation would be the existence of a still hypothetical rapidly degraded fraction of instable p60 molecules which

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due to the short half-life up to know might have escaped from observation. Activation of naı¨ ve T cells requires prolonged antigenic stimulation (Iezzi et al., 1998) and also requires a certain threshold level of Ag presentation (Viola and Lanzavecchia, 1996). Findings from various infectious disease models suggest that the stability of a certain peptide/MHC complex correlates with the immunogenicity of the antigenic peptide in vivo. Measurements of the affinities of various peptides to the HLA-A molecules and their dissociation rates have revealed that immunodominance is more closely correlated to the stability of a peptide/MHC complex than to its binding affinity (van der Burg et al., 1996). Lipford et al. (1995) have prepared a series of amino acid substitutions to the ovalbumin 257–264 octapeptide, which differentially stabilized H2-Kb molecules. When these substituted peptides were used to prime CD8 T cells in vivo, the stability of peptide-Kb complexes directly correlated with the magnitude of the elicited CD8 T cell response. Similarly, the immunogenicity of two EBV-specific CD8 T cell epitopes derived from nuclear Ag 4 correlates with the stability of the peptide/ MHC complex (Levitsky et al., 1996). Also in the murine L. monocytogenes infection model, the immunodominance of a T cell specificity correlates well with the stability of peptide/ Kd complexes (Busch and Pamer, 1998; Pamer et al., 1997; Sijts and Pamer, 1997). We think that the reported requirement of stable peptide/MHC complexes for the induction of CD8 T cell responses might be due the involvement of cross-presenting cells e.g. DC in the induction of the CD8 T cell response. Model III predicts that stable peptide/MHC complexes would remain significantly longer on the cell surface of cross-presenting cells suggesting an advantage for stable over unstable peptide/MHC complexes for the induction of a primary T cell response by DC. Numerous experimental studies addressed quantitative aspects of Ag processing and presentation but so far only a few mathematical models have been developed to describe the kinetics and stochiometry of Ag processing and presentation. Bulik et al. (2005) used a mathematical model to re-evaluate data for the surface presentation of OVA257–264/Kb peptide/MHC complexes that were previously reported from Yewdell’s lab (Princiotta et al., 2003). The model used by Bulik et al. (2005) describes virally infected cells in which after infection of cells Ag are translated from viral mRNA which accumulate and subsequently degrade with a specific half-life. The model incorporates the possibility of synchronous processing of stable and unstable form of an Ag and it also accounts for the time that is required to transport peptide/MHC complexes to the cell surface. Similar to our model it does not take into account cell death of infected cells as well as the potential limitation of Ag presentation by limited availability of nascent MHC molecules. In contrast, to the model reported by Bulik et al. (2005) our model focuses on the influence of the kinetics of Ag supply and the stability of peptide/MHC complexes on the Ag presentation pattern of cells. In our model, we did not take into account the

543

synthesis and decay of mRNA. In contrast to mRNAs in eucaryotic cells the half-life of procaryotic RNAs is very short (Selinger et al., 2003) and thus mRNA levels will quickly reach a steady state. Further, in contrast to viruses that rely on the protein synthesis machinery of the host cell, protozoa and bacteria provide their own metabolism and if bacterial mRNA synthesis is not tightly regulated mRNA levels may even be already at steady state levels when entering the host cell. Thus, the kinetics of constitutively expressed microbial Ag can be adequately described by the microbial doubling time and/or the microbial Ag synthesis rate only, which avoids the necessity to make assumptions regarding the transcription rate and the half-life of mRNA. In respect to the different scenarios described here, the kinetics of Ag delivery in virally infected cells is most similar to model II in which Ag accumulates until a steady state is reached. Also the Ag level of virally infected cells will reach a steady state, which, however, in addition to the dependency upon the protein translation rate and the protein half-life will also be influenced by the mRNA transcription rate and the mRNA half-life. Particularly, if mRNA exhibits an extended half-life this will also result in a prolonged accumulation phase for proteins with a short half-life, resulting in a different kinetics of Ag supply and antigen presentation until the steady state is reached. In summary, here we provide a basic mathematical model for the simulation of the processing and presentation of microbe-derived Ag. On the one hand, the model can be applied to diverse experimental situations and be used for the calculation of basic parameters of Ag processing as e.g. the half-lives and processing efficacies of peptide/MHC complexes. On the other hand, basic calculus also allows deriving general predictions for the Ag presentation patterns of APC under different physiological situations. These predictions can be tested experimentally and by the application of more accurate experimental techniques might yield additional insights into the physiology of Ag processing and presentation which eventually will result in more refined computational models with even better predictive potential. Acknowledgments This work was supported by Deutsche Forschungsgemeinschaft (DFG) grants GE 1081/2-1 and GE 1081/2-2. We thank V. Lindenstruth, University Heidelberg and R. Holtappels, University Mainz for discussing the Ag presentation model.

Appendix A A.1. Model I with microbial replication The basic steps involved in Ag presentation were described by a set of linear differential equations that are

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544

based on the following basic assumptions. Model I assumes that a microorganism, e.g. a bacterium expressing a specific Ag replicates intracellularly with a doubling time tg which defines the intracellular replication rate g,

The relative strength of Ag presentation of two peptides derived from two different Ag can be described as the timedependent ratio r21(t) of the abundance of the peptide/ MHC complexes formed by both peptides

g ¼ ln 2=tg .

r21 ðtÞ ¼ c2 ðtÞ=c1 ðtÞ.

(1)

Under the assumption that in the first few h post infection cells are not killed by replicating intracellular microbes and that after infection the mRNA level of microbes already is in a steady state or at least rapidly reaches a steady state the microbial Ag synthesis rate depends only on the number of intracellular microbes. The microbial growth rate is given by _ ¼ ggðtÞ. gðtÞ

As peptide ratios are independent of absolute peptide numbers they are expected to be less influenced by systematic errors occurring during peptide quantification experiments and thus should be a more robust experimental parameter than absolute peptide numbers. The peptide ratios can be further simplified for t-N and t-0:

(2) lim r21 ðtÞ ¼

a2 b2 w2 ðg þ b1 Þðg þ d1 Þ , a1 b1 w1 ðg þ b2 Þðg þ d2 Þ

Intracellular microbes synthesize the Ag with a defined synthesis rate a, which subsequently is degraded in a firstorder kinetics with a specific rate b which depends on the half-life ta of the Ag

t!1

b ¼ ln 2=ta .

Using Eq. (3), Eq. (12) can also be written as a2 ta1 w2 lim r21 ðtÞ ¼ . t!0 a1 ta2 w1

(3)

The rate of Ag synthesis is then given by _ ¼ agðtÞ  baðtÞ. aðtÞ

(4)

From the degraded Ag, only a peptide-specific fraction w is correctly processed, and after transport into the endoplasmic reticulum forms peptide/MHC complexes. In the model the possible limitation of the formation of peptide/MHC complexes by limited supply of nascent MHC class I molecules is not accounted for in order to keep the model as simple as possible (Bulik et al., 2005). All other peptides which are not protected by complex formation with MHC molecules are rapidly degraded (Reits et al., 2003). The peptide/MHC complexes subsequently decay again with first-order kinetics with a specific decay rate d defined by the half-life tc of the peptide/MHC complex d ¼ ln 2=tc .

(5)

Thus the rate of formation of peptide/MHC complexes is given by c_ðtÞ ¼ bwaðtÞ  dcðtÞ.

(6)

Solving the system of linear differential equations defined by Eqs. (2), (4) and (6) yields the following terms that describe the time-dependent evolution of the numbers of microbes g(t), Ag a(t), and peptide/MHC complexes c(t) per APC under conditions of unrestricted microbial replication. The constant g0 defines the initial number of microorganisms per cell gðtÞ ¼ g0 egt , aðtÞ ¼

g0 a gt ðe  ebt Þ, bþg

cðtÞ ¼

g0 abw ðbðegt  edt Þ ðb þ gÞðb  dÞðg þ dÞ þ gðebt  edt Þ þ dðebt  egt ÞÞ.

(10)

lim r21 ðtÞ ¼

t!0

a2 b2 w2 . a1 b1 w1

(11)

(12)

(13)

If both peptides are derived from the same Ag (a1 ¼ a2 and b1 ¼ b2) Eqs. (11) and (12) obtain the more simple forms: lim r21 ðtÞ ¼

w2 ðg þ d1 Þ , w1 ðg þ d2 Þ

(14)

lim r21 ðtÞ ¼

w2 . w1

(15)

t!1

t!1

A.2. Model II without microbial replication If no intracellular replication occurs (g ¼ 0), the number of microbes expressing the Ag remains constant and is defined by the initial number of microbes g0 only. In this case, the system of differential equations yields more simple terms for the numbers of microbes, Ag, and peptide/MHC complexes per cell gðtÞ ¼ g0 ,

(16)

aðtÞ ¼

g0 a ð1  ebt Þ, b

(17)

cðtÞ ¼

g0 aw ðdðebt  1Þ þ bð1  edt ÞÞ. dðb  dÞ

(18)

(8)

In the model without microbial replication the load of cells with Ag and peptide/MHC complexes converges for t-N towards a steady state: ga (19) lim aðtÞ ¼ 0 , t!1 b

ð9Þ

t!1

(7)

lim cðtÞ ¼

g0 aw . d

(20)

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The ratio of peptide/MHC complexes for t-N and t-0 is given by lim r21 ðtÞ ¼

t!1

a2 w2 d1 a2 w2 tc2 ¼ , a1 w1 d2 a1 w1 tc1

a2 b2 w2 a2 ta1 w2 lim r21 ðtÞ ¼ ¼ . t!0 a1 b1 w1 a1 ta2 w1

(21) (22)

If both peptides are derived from the same Ag Eqs. (21) and (22) can be further simplified: w d1 w tc2 lim r21 ðtÞ ¼ 2 ¼ 2 , t!1 w1 d2 w1 tc1

(23)

w2 . w1

(24)

lim r21 ðtÞ ¼ t!0

A.3. Model III with Ag loading only Cells can be loaded with Ag in the absence of intracellular replication (g ¼ 0) and intracellular Ag synthesis (a ¼ 0). The initial Ag load of cells is defined by a0. In this case the system of differential equations yields the following terms for the numbers of Ag and peptide/MHC complexes per cell aðtÞ ¼ a0 ebt , cðtÞ ¼

a0 bw dt ðe  ebt Þ. bd

c2 ðtÞ a02 b2 w2 ðb1  d1 Þðed2 t  eb2 t Þ ¼ . c1 ðtÞ a01 b1 w1 ðb2  d2 Þðed1 t  eb1 t Þ

t!0

a2 b2 w2 a2 ta1 w2 ¼ . a1 b1 w1 a1 ta2 w1

c2 ðtÞ w2 ðb  d1 Þðed2 t  ebt Þ ¼ , c1 ðtÞ w1 ðb  d2 Þðed1 t  ebt Þ

lim r21 ðtÞ ¼ t!0

w2 . w1

(32)

A.4. Cell surface expression of peptide/MHC complexes The above equations describe the evolution of the total number of peptide/MHC complexes per cell but do not take into account the time that peptide/MHC complexes require to reach the cell surface. Physiologically, however, T cells are able to interact with peptide/MHC displayed on the cell surface of cells only. If peptide/MHC complexes decay exponentially with the decay rate d during the time D required to reach the cell surface the number of surface peptide/MHC complexes cs ðtÞ can be calculated from the above equations for c(t) by cs ðtÞ ¼ cðt  DÞedD .

(33)

Bulik, S., Peters, B., Holzhu¨tter, H.G., 2005. Quantifying the contribution of defective ribosomal products to antigen production: a model-based computational analysis. J. Immunol. 175, 7957–7964. Busch, D.H., Pamer, E.G., 1998. MHC class I/peptide stability: implications for immunodominance, in vitro proliferation, and diversity of responding CTL. J. Immunol. 160, 4441–4448. Chang, S.T., Linderman, J.J., Kirschner, D.E., 2005. Multiple mechanisms allow Mycobacterium tuberculosis to continuously inhibit MHC class II-mediated antigen presentation by macrophages. Proc. Natl. Acad. Sci. USA 102, 4530–4535. Dubey, J.P., Lindsay, D.S., Speer, C.A., 1998. Structures of Toxoplasma gondii tachyzoites, bradyzoites, and sporozoites and biology and development of tissue cysts. Clin. Microbiol. Rev. 11, 267–299. Guermonprez, P., Valladeau, J., Zitvogel, L., Thery, C., Amigorena, S., 2002. Antigen presentation and T cell stimulation by dendritic cells. Annu. Rev. Immunol. 20, 621–667. Harty, J.T., Pamer, E.G., 1995. CD8 T lymphocytes specific for the secreted p60 antigen protect against Listeria monocytogenes infection. J. Immunol. 154, 4642–4650. Iezzi, G., Karjalainen, K., Lanzavecchia, A., 1998. The duration of antigenic stimulation determines the fate of naive and effector T cells. Immunity 8, 89–95. Janda, J., Scho¨neberger, P., Skoberne, M., Messerle, M., Ru¨ssmann, H., Geginat, G., 2004. Cross-presentation of Listeria-derived CD8 T cell epitopes requires unstable bacterial translation products. J. Immunol. 173, 5644–5651. Lanzavecchia, A., Reid, P.A., Watts, C., 1992. Irreversible association of peptides with class II MHC molecules in living cells. Nature 357, 249–252. Larsson, M., Fonteneau, J.F., Bhardwaj, N., 2003. Cross-presentation of cell-associated antigens by dendritic cells. Curr. Top. Microbiol. Immunol. 276, 261–275. Levitsky, V., Zhang, Q.J., Levitskaya, J., Masucci, M.G., 1996. The life span of major histocompatibility complex-peptide complexes influences the efficiency of presentation and immunogenicity of two class

(27)

(28)

(29) (30)

If APC are initially loaded with peptide only which forms a defined number of initial peptide/complexes c0 or further Ag processing by APC is blocked the decay of preformed peptide/MHC complexes is defined by cðtÞ ¼ c0 edt .

a0 bw dt ðe  ebt Þ þ c0 edt . bd

(26)

If both peptides are derived from the same Ag (a01 ¼ a02, a1 ¼ a2, b1 ¼ b2) Eqs. (27) and (28) can be further simplified to Eqs. (29) and (30), respectively: r21 ðtÞ ¼

cðtÞ ¼

References

The initial ratio (t-0) of the abundance of the two peptide/MHC complexes is defined by lim r21 ðtÞ ¼

The sum of Eqs. (26) and (31) thus describes the Ag presentation by APC that are simultaneously loaded with peptide and Ag, a scenario which also occurs if after a period of time further Ag synthesis in APC is blocked

(25)

Because in model III degraded Ag are not replaced the intracellular Ag level does not converge towards a steady state. The time-dependent ratio of the peptide/MHC complexes formed by two different antigenic peptides processed from two different Ag is defined by r21 ðtÞ ¼

545

(31)

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