A conceptual cellular interaction model of left ventricular remodelling post-mi: dynamic network with exit-entry competition strategy

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1 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 RESEARCH Open Aess A oneptual ellular interation moel of left ventriular remoelling post-m: ynami network with exit-entry ompetition strategy Yunji Wang 1, Hai-Chao Han, Jak Y Yang 3, Merry L Linsey 4, Yufang Jin 1* From The SBM nternational Joint Conferenes on Bioinformatis, Systems Biology an ntelligent Computing (JCBS) Shanghai, China. 3-8 August 009 Abstrat Bakgroun: Progressive remoelling of the left ventrile (LV) following myoarial infartion (M) is an outome of spatial-temporal ellular interations among ifferent ell types that leas to heart failure for a signifiant number of patients. Cellular populations emonstrate temporal profiles of flux post-m. However, little is known about the relationship between ell populations an the interation strength among ells post-m. The objetive of this stuy was to establish a oneptual ellular interation moel base on a reently establishe graph network to esribe the interation between two types of ells. Results: We performe stability analysis to investigate the effets of the interation strengths, the initial status, an the number of links between ells on the ellular population in the ynami network. Our analysis generate a set of onitions on interation strength, struture of the network, an initial status of the network to preit the evolutionary profiles of the network. Computer simulations of our oneptual moel verifie our analysis. Conlusions: Our stuy introues a ynami network to moel ellular interations between two ifferent ell types whih an be use to moel the ellular population hanges post-m. The results on stability analysis an be use as a tool to preit the responses of partiular ell populations. Bakgroun Progressive remoelling of the left ventrile (LV) following myoarial infartion (M) involves spatial-temporal ellular interations among ifferent ell types [1]. Apoptosis of myoytes, infiltration of neutrophils, ativation of marophages, ativation of enothelial ells, an proliferation of fibroblasts are LV remoelling omponents [-5]. These events are aompanie with a temporal flux in the ellular population profiles post-m [6-13]. Among these ells, marophages play a pivotal role by oorinating phagoytosis of ellular ebris at the M site an sereting ytokines interleukin-1b, interleukin-6, an tumor nerosis fator a, matrix metalloproteinases (MMPs), tissue inhibitor of metalloproteinases * Corresponene: yufang.jin@utsa.eu 1 Department of Eletrial an Computer Engineering, University of Texas at San Antonio, San Antonio, USA (TMPs), an growth fators [14-17]. Marophages are known to unergo a lassial ativation haraterize by pro-inflammatory gene expression in the early stage post-m. n the later stage post-m, marophages unergo an alternative ativation haraterize by the seretion of fators that promote fibrosis, woul healing, neovasularization an granuloma formation. While researh has been arrie out to investigate the populations of marophages through ifferent ativation shemes [18], the relationship an interations between these two ativate marophage ellular populations post-m remains unlear. Marophages are believe to first unergo lassial ativation, an then proee through the alternative ativation pathway [19,0]. Marophages o not ie loally in the sar tissue, but emigrate from sar tissue to the lymph noe system [1]. Thus, the M site 010 Jin et al; liensee BioMe Central Lt. This is an open aess artile istribute uner the terms of the Creative Commons Attribution Liense ( whih permits unrestrite use, istribution, an reproution in any meium, provie the original work is properly ite.

2 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page of 10 behaves as a network that regulates the exit an entry of marophages, an the loal ytokine environment etermines the populations of lassially an alternatively ativate marophages. Aoringly, the purpose of this stuy was to investigate the mathematial relationship among marophage populations an interations in a ynami network. The evolution of a ynami network has been arrie out in game theory, soial networks, an other biologial systems [-6]. Existing stuies have emonstrate that outomes of tumor growth are etermine by the ellular interations, an these interations inlue both ooperation an ompetition among these ells through a ynami network [7]. n our researh, we have generate stability onitions of a LV network ontaining two types of marophages an introue a new approah to moel the temporal ativation of marophages post-m. Results We evelope a ynami network inluing two types of marophages base on a previous graphi moel publishe by Nowak an olleagues [8]. To eluiate the unerlying mehanisms of the ynamial evolution, theoretial analysis was arrie out an onitions for ifferent evolutionary profiles were obtaine. Computer simulations illustrate the ynami evolution of the network with interations among two types of marophages. Mathematial moel of exit-entry upating law in a ynami network A total of N well-mixe ells are istribute over the network. Eah ell oupies a vertex of the struture network an links to k other ajaent ells. A linkage between two ells is the ege of the network, enoting the interation strength between ells. A general interation matrix an be written as A = A a C C b where A an C enote the type of ells in the network (A is the alternative ativate marophage an C is the lassial ativate marophage), parameters a, b,, an enote the interation strength between type A an C. Cells. Speifially, a type A ell provies energy a to an interate type A ell an provies energy b to an interate type C ell. A type C ell provies energy to an interate type A ell an to an interate type C ell. n the interation matrix, ifferent parameter settings of a, b,, an represent ifferent interation strengths among ells. Within the network, eah ell has an energy funtion ε base on the interations with all of its linke ells as shown in Figure 1. Fitness funtion of a ell, F, is etermine by equation F=1 ω + ωε (1) where ω is a variable between (0, 1), enoting seletion strength. The larger the intensity of seletion is, the larger the ontribution of payoff to the fitness funtion is. A strong seletion is iniate as ω = 1 an a weak seletion is iniate as ω << 1. n this stuy, an exit-entry strategy was hosen for the oneptual moel, sine exit-entry is a funamental ellular migration sheme for ellular interation post-m. n the exit-entry strategy, eah iterative step in the exitentry evolutionary proess is alle a generation. During theevolutionaryproess,aellishosenranomlyto exit in eah generation. Assuming a vaate vertex ause by ellular exiting will be only replae with either a new type A or type C ell, a probability of replaing by a type A ell is etermine by F A /(F A +F C ), where F A an F C are fitness funtions of all ajaent ells linke to the vaate vertex. To be speifi, the fitness funtion ontribute by all the neighbouring type A ells onnete to the exiting ell, is alulate as F A K A ( 1 ) A i= 1 i = +,whereωis the intensity of seletion, K A is the number of type A ells linke to the exiting ell. For phenotypes with weak seletion, the primary ifferential equations were set up as p w k = 1 N p ( a + b ) + O ( w ), () p q A AC a b k = 1 k q q q 1 ( + ) + k q, a A A A A CC A A k = 1 k q q q 1 ( + ) + k q, b C A A A CC CC k = 1 k q q q 1 ( + ) + k q, (3) AC A A CC A A k = 1 k q q q 1 ( + ) + k q, CC A A CC CC = kn p [ 1+ ( k 1)( q q )] + O ( w ), (4) AA AC A C A A paa pac = ( ) = [ 1+ ( k 1 )( qac qaa)] + O( w), (5) t p kn p AA A A

3 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 3 of 10 Figure 1 The struture of the ynami network in the exit-entry upating proess During the exit-entry upating proess, a vaate vertex is replae by a new marophage aoring to the fitness funtion etermine by its neighbouring ells. An original network is shown in the left part of Figure 1 (Left). f a ell, marke in gray, exits the network, the possibility of replaing this ell with either a type C (lassially ativate marophage) or type A (alternatively ativate marophage) ell is etermine by the fitness funtion of the neighbouring ells. Cost funtion of eah ell linke to the vaate ell is shown in the mile of the figure (Mile). n this ase, the fitness funtion of all type C ells is alulate as F A = 4(1 ω)+ ω(10a +b). Similarly, the fitness funtion etermine by the onnete type C ells an be alulate as FA F C = 4(1 ω)+ ω(3 +5). The gray vertex may be replae by a type A ell with probability, or a type C ell with probability of F FA + FC C aoring to the exit-entry evolutionary strategy, whih is shown in the right part of Figure 1 (Right). FA + FC where O( ) enotes higher orer terms of a variable. We efine P A an P C as the ratio of type A an C ellular population over the total population. Variables P AA, P AC, P CA, anp CC enote the frequenies of eges between AA, AC, CA, an CC interative ellular pairs. n aition, let q X Y enote the onitional probability of a ell type X given the ajaent vertex as ell type Y, where X an Y represent ellular type of A or C. Aoring to the esribe exit-entry strategy an the physial meaning of the efine variables, the following ientities hol in the struture ynami network pa + pc = 1, qax + qcx = 1, pxy = qx Y py, p = p. AC CA (6) Sine P AA =q A A P A, P A an q A A are two inepenent variables, equations (), (3), an (5) are hosen to esribe the evolution of the network. Theoretial analysis n the ase of weak seletion, ω << 1 hols. Therefore, equation (5) represents a fast manifol an equation () represents a slow manifol of the ynamis. Our analysis has le to three equilibriums, P A =0,1,or ( k+ 1) a+ ( k k 1) b ( k 1) root =. of the ( a b + )(( k+ 1)( k )) ynami network base on the onstraints of the interation strength parameters a, b,,, an the number of links k in the situation of weak seletion. We summarize the following onitions for the three equilibriums. Case 1: Stable equilibrium at P A =1

4 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 4 of 10 A stable equilibrium, P A = 1, of the system exists if the interation strength satisfies the following onitions, a+ < b+ { ( k+ 1) a+ ( k k 1) b > + ( k 1) ( k 1) a+ b ( k k 1) + ( k + k+ 3) > 0. a+ > b+ or { ( + ) + ( 1 1 ) > + ( k a k k b k 1 ) a+ > b+ ( k+ 1) a+ ( k k 1) b < + ( k 1) or { ( k 1) a+ b ( k k 1) + ( k + k+ 3) > 0 ( k+ 1) a+ ( k k 1) b ( k 1) PA0 > root = ( a b + )((k+ 1)( k )) where P A0 is the initial position of P A. Case : Stable equilibrium at P A = 0 exists while the onitions shown in equation (8) are satisfie. or a+ < b+ { ( k ) a ( k k ) b ( < + k 1 ) a+ > b+ { ( k+ 1) a+ ( k k 1) b < + ( k 1) ( k 1) a+ b ( k k 1) + ( k + k+ 3) < 0 Case 3: Stable equilibrium of P A Î (0, 1) exists with the onstraints on interation strength an the number of links k satisfies the onition a+ < b+ ( k+ 1) a+ ( k k 1) b > + ( k 1) ( k 1) a+ b ( k k 1) + ( k + k+ 3) < 0 Computational simulations Base on the theoretial analysis, we preite three types of evolutionary profiles: 1) population of type A ells (alternatively ativate marophages) ominates the total ellular population, ) population of type C ells (lassially ativate marophages) ominates the total ellular population, or 3) populations of type A an type C ells reah a ynami balane in the total ellular population. n aition, the simulations also showe that the evolutionary profiles are relate with the (7) (8) (9) interation strengths, the number of links between ells, an the initial status of the ellular population. To verify the preition, we esigne omputer simulations base on the onitions given from equations (7-9) to explore the evolutionary profiles of the network. Effets of the interation strengths on the evolutionary profiles We have run three sets of omputer simulations with interation matrix in the form of = 3. = = ,, an.thenumber of links between two ells was set as k = 4, an the 1 0 initial population of type A an type C ells were set as 9900 an 100, respetively. These values were selete base on alulations of the amount of lassially versus alternatively ativate marophages at ay 3 post-m. The simulation results emonstrate a ominant population of type A ells (P A! 1) with interation matrix form 1 in Figure, a ominant population with type C ells (P A! 0) with interation matrix form in Figure 3, an a balane population of both type A an type C ells (P A! 0.5) with interation matrix 3 in Figure 4. The omination was ahieve within finite generations. These 3 sets of simulations emonstrate three ifferent profiles while sharing the same initial status an the number of links among ells, iniating strong interations among type A an type C ells to rive the network eviating from the initial ominant type A population in these three simulations. These results have been shown aompanying with the analysis on the evolutionary spee of type A ellular population. Effets of initial status on the evolutionary profiles of a ynami network To investigate the effets of the initial status on the evolutionary profiles, we run three more omputer simulations. While sharing the same interation matrix, the number of links, initial status of P A was set to 0.99 an 0.4 in the simulations shown in Figures 4 an 5, respetively. n these two simulations, population of type A ells an type C ells reah a ynami balane espite the signifiant ifferenes in the initial status. n simulation pairs shown in Figures 6 an 7, the simulations share the same interation matries (0-1; - 0), the number of link (k=4), an the seletion strength ω = 0.01, but initial status P A = 0.99 le to a ominant population of type A ells in figure 6, an P A =0.letoa ominant population of type C ells in figure 7. Comparisons of these simulations emonstrate that initial status, together with interation strengths etermine the evolutionary profiles of the network.

5 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 5 of 10 Figure Effets of interation strengths on the stability of the ynami network with ifferent interation matrix (P A! 1) Figure emonstrates the evolutionary ynamis of a network with interation matrix [1 0; 0 1], k=4, an initial populations of type A an type C ells are set as 9900 an 100 in a total population of ells, base on previously publishe experimental results. The intensity of seletion ω equals to n the left part of Figure (Left), X axis represents the variable P A, the ratio of type A ellular population over the total population. The Y axis represents the evolutionary rate of type A ellular population enote as Ṗ A. The re star is the initial status of P A. At the initial status, Ṗ A is positive, making P A inrease until P A reahes 1 where Ṗ A ereases to 0. n the right part of Figure (Right), the simulation results emonstrate that population of type A ells inreases until it ominates the whole population within 300 generations. Figure 3 Effets of interation strengths on the stability of the ynami network with ifferent interation matrix (P A! 1) Figure 3 emonstrates the evolutionary ynamis with interation matrix as [0-0.5; 1 0], k=4, an initial populations of type A an C ells are set as 9900 an 100 in a total population of ells. The intensity of seletion ω equals to As shown in the left part of figure (Left), Ṗ A is negative in the region of P A Î (0, 1). t means the population of type A ells ereases in the interval of P A Î (0, 1) until all type A ells exit the system, an Ṗ A = 0. n the right part of the figure (Right), the simulation results emonstrate that population of type A ells ereases until all type A ells exit the system within 600 generations. Effets of the number of links on the evolutionary profiles of a ynami network We also esigne omputer simulations as shown in Figure 8 to illustrate the effets of the number of links on the evolutionary profiles of a network by perturbing the value of k an parameters b an in the interation matrix. Given a fixe number of links k, inreasing b or ereasing benefits the inreasing population of type C ells. Dereasing b or inreasing will lea to ereasing of type C population. n aition, given a fixe parameter b or, variations of k gave the same tren of evolution but hange evolutionary spee.

6 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 6 of 10 Figure 4 Effets of interation strength on the stability of the ynami network with ifferent interation matrix (P A Î(0,1))Figure 4 shows the evolutionary ynamis with interation matrix as [0 1; 1 0], k=4, an initial populations of type A an C ells are set as 9900 an 100 in a total population of 10000, respetively. The intensity of seletion ω equals to Variable Ṗ A is negative when the initial status of P A stays in the region between the marke root (re square) an 1 in the left part of the figure(left). When P A goes to the marke root, Ṗ A reahes 0. Aoringly, the population of type A ells ereases in the interval of [root, 1] until P A goes to the root enote at 0.5 in the simulation. The simulation results shown in the right part of the figure (Right) emonstrate that the population of type A ells approahe 5000 within 1000 generations. Figure 5 Effets of initial status on the stability of the ynami network with ifferent interation matrix (P A Î (0, 1)) Figure 5 shows the evolutionary ynamis with interation matrix as [0 1; 1 0], k=4, an initial populations of A an B are as 4000 an 6000 in a total population of ells. The intensity of seletion equals to Ṗ A is positive when the initial status of P A stays between 0 an the marke root (re square) in the left part of figure (Left). Aoringly, the population of type A ells inreases from the initial status in the interval of [0, root] until P A goes to the root enote at 0.5 in this simulation. The simulation results shown in the right part of Figure 5 (Right) emonstrate that the population of type A ells approahes 5000 within 800 generations. All the initial onitions, interation strength, an the number of links liste in the simulations satisfie the onition assoiate with the speifie equilibrium. The simulations verifie preitions on the evolutionary profiles of the network base on our theoretial analysis. Disussion We have use a ynami network moel to stuy the ellular interations with an exit-entry strategy. Our results emonstrate that evolutionary profiles of a ynami network oul be stabilize at ifferent states by perturbing the interation strength matrix, the number of links, an

7 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 7 of 10 Figure 6 Effets of initial status on the stability of the ynami network Figure 6 shows the evolutionary ynamis with interation matrix as [0-1; - 0], k=4, an initial populations of A an B are set as 9900 an100, respetively, in a total population of The intensity of seletion ω equals to As shown in the left part of Figure 6, in the ase that initial status of P A stays between the root an 1, Ṗ A is positive. Therefore, P A inreases from the initial status until it goes to 1 an Ṗ A goes to 0. Simulation results emonstrate that population of type A ells reah within 300 generations as shown the right part of Figure 6 (Right). Figure 7 Effets of initial status on the stability of the ynami network Figure 7 shows the evolutionary ynamis with interation matrix as [0-1; - 0], k=4, an initial populations of A an B are set as 000 an 8000, respetively, in a total population of The intensity of seletion ω equals to As shown in the left part of Figure 7, in the ase that initial status of P A stays between 0 an the root (re square), Ṗ A is negative. Thus, P A ereases from the initial status until it goes to 0 an Ṗ A goes to 0. Simulation results emonstrate the type A ells totally exit the system within 400 generations as shown the right part of Figure 7 (Right). initial status of the network. We have quantifie onitions for stable states in terms of interation strengths, the initial status, an the number of links in the network. Our omputer simulations verifie preitions of our analyti results. While we use an exit-entry strategy presente by game theory [8,9], our stability analysis provie not only the stability property but also the onvergene states of the system, whih is broaer than the previous evaluations [8]. We extene analytial stability to the urrent analysis methos that quantify results using graph theory [4,6,30,31]. Here we have two remarks of our methos. First, we only onsiere an exit-entry strategy in a struture ynami network. The exit-entry strategy was hosen beauseitwasthemostfunamentalanlogialellular funtion for an initial investigation of the interations between populations of lassially an alternatively ativate marophages post-m. There exist other

8 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 8 of The population of type C ells k=4 Green Star k=5 Yellow Square k=6 Re Cirle The population of type C ells k=4 Green Star k=5 Yellow Square k=6 Re Cirle b (Left) (Right) Figure 8 Effets of number of links on the stability of the ynami network Figure 8 shows evolutionary profiles of two ynami networks with exit-entry interation matrix [0 b; 1 0] (Left) an [1 0; 0 ] (Right) were simulate with ifferent number of links (k=4, Green stars, k=5 Yellow square, an k=6 Re irles. Population of initial type C ells was set as 100 with a total population of 10,000 ells in the system. The intensity of seletion ω was set as The markers are the average populations of type C ells after five generations over 10 4 runs. n the left part of the figure (Left), the horizontal oorinate represents the hanges of b value an the vertial oorinate represents the population of type C ells. n the right figure, the horizontal oorinate represents the hanges of value an the vertial oorinate represents the population of type C ells. evolutionary strategies suh as entry-exit, mutation, an imitation. These strategies will nee to be onsiere an potentially inorporate in future moels. Seonly, the struture of the ynami network is fixe by assuming a weak seletion, ω << 1, an a onstant interation strength matrix. However, stability analysis of the evolutionary strategies with varying strutures has more realisti appliations to biologial systems an has attrate lots of researh interest to game theory reently [,3]. Stability analysis of ynami networks with varying struture nees to be inlue in future researh moels. We provie here the first appliation of a ynami network moel to esribe marophage interations. We have obtaine expliit onitions that etermine interation strength an have establishe a struture of the network that allows us to preit the stability an equilibrium of the post-m ynami network. Our simulation results onfirme the preition of the stability an the equilibriums of the network. Conlusions We use a new approah to moel the ellular interations between marophage ativation types in the post- M setting. The results on stability analysis an be use as a useful tool to preit the responses of speifi ellular populations. Methos Stability analysis of the exit-entry ynami network The establishe mathematial moel in equation () an (5) is a high orer nonlinear system. n a weak seletion, ω << 1, equation (5) represents the fast manifol of the evolution an equation () represents a slow manifol of the system. Thereby, the equilibrium of q A A an be approximate from equation (5) by ignoring ω epenent terms as 1 1 qaa = + qac = pa+ ( pa) k 1 k 1 1 (10) The onitional probabilities an then be rewritten as q q q AC ( 1 qaa ) = p 1 pa = 1 qaa = 1 q CA CC AC A (11) n a weak seletion, equations (-3) an be simplifie as a b ( k+ 1)( k ) k + 1 = pa + kk ( 1) kk ( 1) ( k+ 1)( k ) = p kk ( 1) A k k 1 + kk ( 1) ( k+ 1)( k ) 1 = pa + kk ( 1) kk ( 1) ( k+ 1)( k ) k k = pa kk ( 1) 1 kk ( 1) (1)

9 Wang et al. BMC Systems Biology 010, 4(Suppl 1):S5 Page 9 of 10 p w k A = pa k+ 1 a+ k k 1 b k 1 Nk( k 1) { [( ) ( ) ( ) ] + pa[( k+ 1)( k 3) a ( k k 3) b ( k k 3) ( k+ 1) ] 3 pa ( k+ 1)( k )( a b + )}. (13) Stability of P A is analyze by hoosing a positive efinite Lyapunov funtion as V = Nk( k 1) k w p A.ferivative of V is negative semi-efinite, P A is Lyapunov ( ) stable, an the erivative of V an be written as equation (14). Nk( k 1) V = k w p A p A ( ) = pa{[( k+ 1) a+ ( k k 1) b ( k 1) ] + pa[( k+ 1)( k 3) a ( k k 3) b ( k k 3) ( k+ 1) ] p A( k+ 1)( k )( a b + )} + O( ) = p [ p + p + ] + O( ) A A 1 A 0 (14) Define the parameters 0 =(k +1)a +(k k 1)b +(k 1) 1 =(k +1)(k 3)a (k k 3)b (k k 3) (k +1) an = (k +1)(k )(a b + ) in the ase of weak seletion(ω << 1), stability of P A is etermine by heking the sign of polynomial p A + p 1 A + 0.There exist three equilibriums for V = 0 i.e., p A =0,1or ( k+ 1) a+ ( k k 1) b ( k 1) = root,. ( a b + )(( k+ 1)( k )) Stability of the system an be heke with 3 ases base on the position of the thir root an sign of the parameter. The relations of the three equilibriums have been shown in Figures (,3,4,5,6,7). Aknowlegements The authors aknowlege grant support from NH R01 HL75360, AHA Grantin-Ai F, an the Morrison Fun (to MLL), grant support from NSF an (to HCH), an grant support from NSF an NH 1SC HL (to YJ). This artile has been publishe as part of BMC Systems Biology Volume 4 Supplement 1, 010: Proeeings of the SBM nternational Joint Conferenes on Bioinformatis, Systems Biology an ntelligent Computing (JCBS). The full ontents of the supplement are available online at Author etails 1 Department of Eletrial an Computer Engineering, University of Texas at San Antonio, San Antonio, USA. Department of Mehanial Engineering, University of Texas at San Antonio, San Antonio, USA. 3 Center for Researh in Biologial Systems, University of California at San Diego, La Jolla, California , USA. 4 Department of Meiine, University of Texas Health Siene Center at San Antonio, San Antonio, USA. Authors ontributions Y.J an M.L.L esigne the researh; Y.J an Y.W performe omputational analysis an simulation. J.Y.Y involve in the analysis an provie useful insights in the appliation to ellular funtions. Y.J, Y.W, H.C.H, an M.L.L analyze the results an wrote the manusript. Competing interests The authors elare that they have no ompeting interests. Publishe: 8 May 010 Referenes 1. Cohn JN, Ferrari R, Sharpe N: Caria remoeling onepts an linial impliations: a onsensus paper from an international forum on aria remoeling. Behalf of an nternational Forum on Caria Remoeling. J Am Coll Cariol 000, 35(3): Pfeffer MA, Braunwal E: Ventriular Remoeling After Myoarial nfartion. 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