Mathematical Modelling of the Interaction of Chlamydia Trachomatis with the Immune System

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1 , Ocober 20-22, 2010, San Francisco, USA Mahemaical Modelling of he Ineracion of Chlamydia Trachomais wih he Immune Sysem Masoumeh Bagher Oskouei, Dann G. Malle, Ashkan Amirshahi, Graeme J. Pee, Absrac In his paper, we presen a wo dimensional mahemaical model capable of describing he process of he ascension of infecion o he upper genial rac in females and invesigaing he number of free chlamydial paricles and infeced cells. This model includes diffusion and chemoaxis erms corresponding o he moion of free chlamydial paricles and he migraion of immune cells oward he infecion sie in response o diffusible chemical cues such as IFN γ, respecively. The qualiaive resuls of he model reflec experimenally observed phenomena. Keywords: mahemaical model, Chlamydia rachomais, parial differenial equaion, immune RB EB Lysis Infecious EB hrs RB hrs hrs RB Replicaion Enry EB expands in size EB RB RB Division 1 Inroducion Chlamydia rachomais is an obligae inracellular bacerial pahogen ha infecs he genial and ocular mucosa of humans causing sexually ransmied disease and rachoma. I is esimaed ha 70-75% of endocervical infecions in women caused by C. rachomais are asympomaic and may persis for monhs o years [4]. The sequelae of C. rachomais genial rac infecions in women, namely chronic pain, pelvic inflammaory disease (PID), inferiliy and ecopic pregnancy are he mos cosly oucomes of any sexually ransmied infecion excep human immunodeficiency virus (HIV/AIDS), resuling in an esimaed $US4 billion in healh care coss per annum only Unied Saes alone [5]. C. rachomais is an inracellular, complex and mulifuncional process pahogen, unique among prokaryoes because of a biphasic developmenal cycle of replicaion in which he organism exiss in wo disincive forms; he Elemenary Body (EB), which is he infecious form, and he Reiculae Body (RB), which is he replicaing srucure. In his developmenal cycle (see Fig. 1), he infecious bu meabolically inacive elemenary body, nm in diameer, is endocyosed by eukaryoic cells and resides wihin a cyoplasmic inclusion. Wihin he Mahemaical Sciences Discipline, Queensland Universiy of Technology, Brisbane, Ausralia m.bagheroskouei@qu.edu.au Insiue of Healh and Biomedical Innovaion, Queensland Universiy of Technology, Brisbane, Ausralia Figure 1: Schemaic of he Chlamydia rachomais Serovar D Developmenal Cycle. inclusion, he EBs ransform ino he non-infecious bu meabolically acive reiculae body which is larger a nm in diameer. The RBs divide wihin he cell by binary fission and ransform back o he infecious form before being released o he cell exerior [1, 3]. Wilson [16] indicaes ha new EBs are released from each infeced cell following he replicaion process. This is an iniial indicaor of how he C. rachomais infecion can progress and subsequenly reach he upper genial rac. I is also imporan hough o noe ha he hos immune sysem has a significan role in he clearance of Chlamydia hroughou he infecion period. C. rachomais is subjeced o boh innae and adapive immune responses of he hos. Wilson for example, has shown ha CD4+ T cells play a significan role in adapive immuniy o Chlamydia rachomais infecion of he genial rac [14]. Moreover he recen sudies on human clinical samples and also animal models clearly show ha i normally akes 4 o 7 days for he immune response o be induced. The nex secion will focus more closely on how chlamydial paricles inerac wih immune cells. In addiion o he effec of immune sysem, he female mensrual cycle influences Chlamydia growh. The cycle has hree phases: follicular, ovulaory, and lueal (see Fig.2).The mensrual cycle is regulaed by luerinizing

2 , Ocober 20-22, 2010, San Francisco, USA Follicular Ovulaory Lueal Figure 2: Schemaic of hormone levels during he normal mensrual cycle (progeserone doed line, esrogen dashed line). hormone (LH), follicle-simulaing hormone (FSH), esrogen, and progeserone. A sharp drop in he levels of esrogen and progeserone resuls in a shedding of he endomerial lining of he uerus (see Fig.3). Esrogen and progeserone are produced in he ovaries and are regulaed by oher hormones produced in he brain. Of imporance o his sudy is he fac ha a simple vaginal infecion, such as a chlamydial infecion, will no affec he producion or regulaion of hese reproducive hormones. However, he hormones have a significan effec on Chlamydia growh. Recen sudies [7, 8] show ha suscepibiliy and inflammaion are induced in high progeserone environmens and also esrogen decreases inflammaion and proecion from infecion. The rae of infecion increases when he level of progeserone is higher [17]. Figure 3: Endomerial changes during he normal mensrual cycle. In viro, providing all he facors menioned above is no sraighforward. Moreover, here are he several differences in he Chlamydia infecion of laboraory animals such as mice, compared wih he infecion in humans such as lengh of infecion and he immune-evasion mechanism. Therefore a mahemaical model describing he ineracion beween C. rachomais and he immune sysem is an exremely useful way o obain furher insighs ino he dynamics of he infecion process, and he subsequen effecs and possible effecive conrol sraegies. In his paper we develop a mahemaical model o invesigae he ineracion beween C. rachomais and he hos immune sysem. We presen preliminary resuls regarding infecion dynamics and progression of infecion in he genial rac and o rack he moion of free chlamydial paricles oward he upper genial level. The remainder of his paper is organised as follows. A brief descripion of C. rachomais and immune sysem ineracions is provided in Secion 2. Secion 3 is a deailed descripion of our parial differenial equaion based mahemaical model of he immune response relaed o Chlamydia rachomais. Then we presen preliminary numerical resuls from simulaions of he mahemaical model. Finally, a discussion and conclusions regarding he model are made in Secion 5. 2 The hos immune sysem and C. rachomais The innae and adapive immuniy are wo essenial barriers agains Chlamydia. The innae immune sysem is a non-specific sysem ha provides immediae defence agains infecion. Macrophages, neurophils and dendriic cells are he key elemens of he innae immune sysem. Paern recogniion recepors (PRRs) are proeins on he surface of boh epihelial cells and circulaing cells of he innae immune sysem. The oll-like recepors (TLRs) and NOD-like recepors (NLRs) are he wo mos imporan families of PRRs and are able o recognise and bind pahogen-associaed molecular paerns (PAMPs). The adapive immune sysem, on he oher hand, is a specific sysem which develops afer firs conac wih a pahogen. Humoral immuniy (B lymphocyes) and cell mediaed immuniy (T lymphocyes) are wo arms of he adapive immune response (see Fig. 4).The B lymphocyes are riggered by anigen-presening cells (APCs), cells of he innae response or T cells. APCs simulae B cells o produce anibodies which bind o anigens on he surface of he invading pahogen o flag hem for neuralisaion. Similarly, T cells are acivaed by APCs, cells of he innae response or B lymphocyes. T lymphocyes consis of T helper cells (Th cells) and cyooxic T cells (CD8+ cells or killer cells). Th cells produce proinflammaory cyokines such as IL-4, IL-5, IL-6, and IFNγ which boh suppor he he humoral sysem and inhibi pahogen growh. Cyooxic T cells direcly aack and eliminae a pahogen. Alhough humoral immuniy is he principal immunological response effecive agains exracellular baceria, he major defensive and proecive response agains inracellular baceria is performed by T cells [11, 10]. Because Chlamydia is presen boh exraand inracellularly, boh of hese elemens of he immune sysem are imporan when modelling chlamydial infecions. In erms of C. rachomais infecion, when a chlamydial

3 , Ocober 20-22, 2010, San Francisco, USA A Humoral immune response Cellular immune response B lymphocye + Anigen Dendriic cells ASPC Eliminaing anigen Ac d T helper T helper eff Ac d cyooxic Cyooxic T eff Cyokines Killing infeced cell Figure 4: Schemaic showing componens of he immune sysem. paricle eners he body, epihelial cells are he firs line of defence agains he pahogen. PRRs on he surface of he epihelial cells bind o he anigens expressed on he surface of C. rachomais. This connecion simulaes he epihelial cells o secree chemokines and oher proinflammaory cyokines which iniiae circulaing cells of he innae immune response. Once he innae immune cells such as macrophages, dendriic cells and neurophils reach he infecion sie, hey are acivaed and heir PRRs bind o chlamydial anigens leading o pahogen desrucion. Acivaed macrophages and dendriic cells are able o express pahogen anigens bound o major hisocompaibiliy complex (MHC) proeins on he surface and o serve as APCs which can acivae he adapive immune cells. Under simulaion of acivaed dendriic cells, T cells move oward he sie of infecion and differeniae ino eiher cyooxic T cells (CD8+ T cells or T c cells) or helper T cells (CD4+ T cells). Cyooxic T cells (CD8+) are acivaed when heir recepors srongly inerac wih he MHC class I molecule presened a he sie of infecion and perform heir funcion by releasing lymphooxins ha form pores in he arge cell s plasma membrane and cause i o burs or lyse. On he oher hand, T helper cells (CD4+) inerac wih MHC class II molecules and produce pro-inflammaory cyokines such as inerferongamma (IFN-γ) ha inhibi chlamydial growh, and acivae macrophages and humoral cells in order o accelerae clearance of infecion [12, 13, 9]. 3 Mahemaical model To model chlamydial infecion in he genial rac, we use a recangular domain Ω = [0,L x ] [0,L y ], where he side boundaries are subjec o a wrapping condiion. This resuls in a domain shaped like he exerior surface of a cylinder an oversimplified picure of he surface of he genial rac (see Fig. 5). L y wrapped boundary L x Figure 5: Compuaionally, he genial rac is modelled using a recangular grid ha is hen subjeced o wrapped boundary condiions for x = 0 and x = L x. In order o simulae he infecion process and he effeciveness of immune responses during hree differen phases of he female mensrual cycle, we will solve he model equaions over a ime range of 28 days, he average lengh of he female mensrual cycle. As we discussed in he previous secion, free chlamydial paricles and inracellular Chlamydia are subjeced o he effecs of several differen elemens of he immune sysem including circulaing cells of he innae immune sysem and he adapive immune sysem. Moreover, cyokines and chemokines are produced in response o he infecion caused by C. rachomais. In his paper, K represens he concenraion of cyokines and chemokines. In order o simplify he complexiy induced by considering a large number of immune cells involved in he clearance of C. rachomais, here we denoe by H hose immune cells ha clear free exracellular chlamydial paricles and

4 , Ocober 20-22, 2010, San Francisco, USA by M he componens of he immune sysem ha aack he inracellular Chlamydia. The free chlamydial paricles, uninfeced epihelial cells, and infeced cells are represened by C, E and I respecively. To model he evens described in he previous secion, we assume ha iniially here are some free chlamydial paricles a he lower region of he genial rac and represen his using he iniial condiion C(x, y, 0) = 10e (y 1)2. Iniially here are no infeced cells (all chlamydial paricles are iniially ouside of he hos cells) and epihelial cells are a heir normal concenraion levels hroughou he domain. Furher, here is iniially an absence of chemokines/cyokines and of componens of he immune sysem ha aack Chlamydia when i is inside hos cells. Immune cells ha clear free Chlamydia paricles are iniially circulaing in he sysem a some normal level given. Hence, iniially we have E(x, y, 0) = E max, I(x, y, 0) = K(x, y, 0) = M(x, y, 0)=0, H(x, y, 0) = H 0. We make he assumpion ha each of he species of ineres in our model is subjec o a no-flux condiion a he upper and lower ends of he domain; in he oher words, hey do no leave or ener he sysem via hose end regions. Therefore, we have boundary condiions C y = E y = I y = H y = M y = K y = 0 (1) a he boundaries y = 0 and y = L y. For free chlamydial paricles, we assume ha he paricles diffuse in he Fickian sense, wih diffusion coefficien D C. When infeced cells burs a rae κ, P new EBs are released from each infeced cell. The inernalisaion of EBs ino he epihelial cells during he infecion process occurs a rae g() and varies depending on he curren phase of he female mensrual cycle. Chlamydial paricles are cleared due o he humoral immune response, however as previously menioned, he immune response o Chlamydia is delayed and his is modelled using he expression h(). Combining hese elemens, we obain he free paricle conservaion equaion C = D C 2 C + PκI g()ce h()ch. (2) By day 5 o 7 of he mensrual cycle, shedding of he endomerial lining of he uerus resuls in a drop in he rae of EB inernalisaion. Vaginal epihelial hickness is minimal during he firs phase, and increases in hickness hroughou he reproducive phase [18]. Therefore, he rae of EB inernalisaion gradually increases righ afer follicular phase unil reaching a seady sae level in he high progesrone environmen. Thus we model he rae of EB inernalisaion using (exp () 1) g() = < oherwise. The delay of he innae and humoral immune response is imposed using he funcion { (exp() 1) h() = 0.1 oherwise. The populaion of uninfeced epihelial cells is also allowed o undergo Fickian diffusion, wih coefficien D E. Epihelial cells are also assumed o be produced according o a logisic growh law o he normal/maximum cell level of E max and o die naurally a rae μ 1. The cells are also los o he infeced populaion when hey come in conac wih free Chlamydia paricles a rae g(). Combining hese we obain he conservaion equaion E = D E 2 E + E(E max E) g()ce μ 1 E. (3) Infeced cells also diffuse randomly wih coefficien D I. The inernalisaion of EBs ino he epihelial cells causes an increase in he number of infeced cells a rae g(). The infeced cell populaion decreases due o he lysis of cells when he inracellular developmenal cycle is complee, wih a lysis rae of κ. Finally, he cell mediaed immune sysem, wih a delayed response modelled by s(), decreases he number of infeced cells. This gives I = D I 2 I + g()ce κi s()im (4) for he populaion of infeced cells. Sudies [6] show ha CD4+ and CD8+ cells have a significan role in he clearance of Chlamydia compared o he innae immune response and ha he rae of clearance by cell mediaed immuniy is higher han humoral clearance rae. Similarly, i akes 4 o 7 days for cell mediaed response o be induced. The delay of he cell mediaed immune response is hus modelled by he funcion { (exp() 1) s() = 0.3 oherwise The effec of humoral and cell mediaed immuniy are described in equaions (5) and (6). The firs erm in each of hese equaions represens he random movemen of humoral and cell mediaed immune cells wih consan

5 , Ocober 20-22, 2010, San Francisco, USA coefficiens D H and D M respecively. The second erm in equaions (5) and (6) represens he acivaion of humoral immuniy in response o free chlamydial paricles and cell mediaed immuniy in response o infeced cells respecively. Naural deah for elemens of he humoral and cell mediaed immune response has been characerised by he hird erms in he corresponding equaions, a raes of μ 2 and μ 3 respecively. The final erms in equaions (5) and (6), model chemoacic responses. We propose ha humoral and cell mediaed immune cells direc heir movemen oward higher concenraions of chemokines and cyokines produced in response o eiher free chlamydial paricles ouside he hos cell or infeced cells, wih chemoacic coefficiens of χ H and χ M respecively. Togeher, hese componens give he conservaion equaions H = D H 2 H + γ 1 CH μ 2 H (χ H (H K)), (5) M = D M 2 M + γ 2 IM μ 3 M (χ M (M K)). (6) D E = D I = D H = D M = D K =10 5. Similarly, we se he chemoacic coefficiens o χ H = χ M = Figure 6: Number of free paricles C over ime (days) Cyokines and chemokines are also assumed o undergo random Fickian diffusion wih coefficien D K. Chemokines produced by epihelial cells and some cyokines such as IFN-γ secreed by macrophages simulae humoral immune cells o defend agains free exracellular chlamydial paricles a rae β 1. Addiionally, chemokines secreed by infeced cells a he RB sage of chlamydial developmenal cycle, rigger T cells o produce cyokines in order o inhibi chlamydial growh or clear/conrol infecion. In shor, once chlamydial paricles are inside he hos cells, cell mediaed immuniy may preven infecion. These componens combine o give he conservaion equaion K = D K 2 K + β 1 CH + β 2 IM. (7) 4 Numerical resuls In his preliminary sudy, we simulae chlamydial paricles growing on a square domain Ω = [0, 5] [0, 5]. An explici finie difference mehod was used o solve he model equaions (2 7) subjec o he iniial and boundary condiions given in Secion 3. In his paper, we assume he firs day of Chlamydia infecion corresponds o he firs day of mensruaion and hen coninue o simulae he following 28 days of infecion. Wilson [15] provides a number of esimaes for rae consans and oher parameers, including P = 350, κ = 0.45, μ 1 = 0.3. We esimae ha μ 2 = μ 3 = 0.01, β 1 = 0.4 and β 2 = 0.1. Wihou access o furher experimenally-informed parameers, we assume ha he diffusion coefficiens for cell species are an order of magniude smaller han he coefficiens for paricles (simply due o relaive sizes). Hence, we se D C = 10 4 and Figure 7: Number of infeced cells I over ime (days). Fig. 6 shows an iniially sable populaion of infecious paricles (0-8 days) due o he low rae of cell infecion caused by shedding of he uerus lining and he delayed immune clearance. Then afer abou 7-8 days, an increase in he amoun of progeserone leads o he high rae of cell infecion causing a rise in he populaion of chlamydial paricles. Ashkan [2] sae experimenally ha here is a direc connecion beween he number of chlamydial paricles and non-infeced cells and as he number of infecious paricles increase he number of non-infeced cells decrease. This experimenal resul is observed wih his model, bu no shown here due o space resricions. The number of infeced cells (see Fig. 7) increases as he populaion of EBs increases. The level of humoral and cell mediaed immuniy also rises as he populaion of chlamydial paricles and infeced cells increase respecively (see for example, Fig. 8). Finally he model also predics an increase in he level of secreed cyokines and chemokines in response o he free chlamydial paricles and infeced cells respecively (no shown). These resuls also confirmed by experimenal resuls of Ashkan [2] ha he level of humoral, cell mediaed immuniy and cyokine is a he lowes level in he firs 8 days and increases dramaically aferwards.

6 , Ocober 20-22, 2010, San Francisco, USA adjuvan proecs agains Chlamydia genial and respiraory infecions, Vaccine, V27, N44, pp , [6] Su, H., Caldwell, H.D., CD4 + T Cells Play a Significan Role in Adopive Immuniy o Chlamydia rachomais Infecion of he Mouse Genial Trac, Infec. Immun., V63, N9, pp , Figure 8: (days). 5 Conclusion Humoral immuniy measure H over ime In his research we have exended he ordinary differenial equaion mahemaical model of Wilson [15] o a parial differenial equaion model ha allows for he consideraion of spaio-emporal variaion in he relevan biological species. This includes he incorporaion of random moion of paricles and cells, as well as he chemoacic response of componens of he immune sysem. We have also inroduced more explici racking of facors such hormones and he delayed immune response ha have a significan role in he growh of Chlamydia. The model presened here has successfully reproduced experimenal resuls observed in he laboraory by Amirshahi and ohers and as such his new model provides a framework for furher invesigaion of Chlamydial infecion hrough exension of he mahemaical descripion. Such exensions are a opic of curren research of he auhors. References [1] Abdelrahman, Y.M., Belland, R.J., The chlamydial developmenal cycle, FEMS Microbiology Reviews, V29, N5, , [2] Amirshahi, A., Effec of female sex hormones on Chlamydia rachomais growh and gene expression, Maser of Applied Science (Research) Thesis, Queensland Universiy of Technology, [3] Belland, R.J., e al., Genomic ranscripional profiling of he developmenal cycle of Chlamydia rachomais, PNAS, V100, N14, pp , [4] Dixon, R.E., e al., Chlamydia infecion causes loss of oviduc pacemaker cells and inhibis oocye ranspor in he mouse oviduc, Biol Reprod., V80, N4, pp , [5] Hickey, D.K., Aldwell, F.E., Beagley, K.W., Transcuaneous immunizaion wih a novel lipid-based [7] Kaushic, C., Zhou, F., Murdin, A.D., Wira, C.R., Effecs of Esradiol and Progeserone on Suscepibiliy and Early Immune Responses o Chlamydia rachomais Infecion in he Female Reproducive Trac, Infec. Immun., V68, N7, pp , [8] Kaushic, C., Ashkar, A.A., Reid, L.A., Rosenhal, K.L., Progeserone Increases Suscepibiliy and Decreases Immune Responses o Genial Herpes Infecion, J Virol., V77, N8, pp , [9] Morrison, S.G., e al., Immuniy o Murine Chlamydia rachomais Genial Trac Reinfecion Involves B Cells and CD4+ T Cells bu No CD8+ T Cells, Infec. Immun., V68, N12, pp , [10] Paul, W.E., ed. Fundamenal immunology, Lippinco Williams & Wilkins, Philadelphia, [11] Revillard, J.P., ed. Cell-mediaed immuniy: in viro correlaes, Karger Basel, [12] Sarnbach, M.N., e al., An inclusion membrane proein from Chlamydia rachomais eners he MHC class I pahway and simulaes a CD8 + T cell response, J Immunol., V171, N9, pp , [13] Sephens, R.S., ed. Chlamydia: inracellular biology, pahogenesis, and immuniy, ASM Press: Washingon, DC, [14] Wilson, D.P., Timms, P., McElwain, D.L.S., A mahemaical model for he invesigaion of he Th1 immune response o Chlamydia rachomais, Mah Biosci., V182, N1, pp , [15] Wilson, D.P., Mahemaical modelling of Chlamydia, ANZIAM J, V45(E), pp. C201-C214, [16] Wilson, D.P, e al., Type III Secreion, Conacdependen Model for he Inracellular Developmen of Chlamydia, Bull Mah Biol., V68, N1, pp , [17] Wira, C.R., Kaushic, C., Mucosal immuniy in he female reproducive rac: effec of sex hormones on immune recogniion and responses. In: Kiyono, H., Ogra, P.L., McGhee, J.R., ediors. Mucosal vaccines: new rends in immunizaion. New York, N.Y: Academic Press, pp , 1996.

7 , Ocober 20-22, 2010, San Francisco, USA [18] Wira, C.R., Rossoll, R.M., Young, R.C., Polarized Uerine Epihelial Preferenially Presen Anigen a he Basolaeral Surface: Role of Sromal Cells in Regulaing class II- Mediaed Epihelial Cell anigen Presenaion, J Immunol. V175, pp , 2005.