/* * Transcriptional response of zinc homeostasis system in E. coli * * Model Status * * This CellML model runs in both PCEnv and COR to recreate the * published results. The units have been checked and they are * consistent. In this particular version of the model the zinc-buffering * effects of TPEN are considered. * * Model Structure * * ABSTRACT: BACKGROUND: The zinc homeostasis system in Escherichia * coli is one of the most intensively studied prokaryotic zinc * homeostasis systems. Its underlying regulatory machine consists * of repression on zinc influx through ZnuABC by Zur (Zn2+ uptake * regulator) and activation on zinc efflux via ZntA by ZntR (a * zinc-responsive regulator). Although these transcriptional regulations * seem to be well characterized, and there is an abundance of * detailed in vitro experimental data available, as yet there * is no mathematical model to help interpret these data. To our * knowledge, the work described here is the first attempt to use * a mathematical model to simulate these regulatory relations * and to help explain the in vitro experimental data. RESULTS: * We develop a unified mathematical model consisting of 14 reactions * to simulate the in vitro transcriptional response of the zinc * homeostasis system in E. coli. Firstly, we simulate the in vitro * Zur-DNA interaction by using two of these reactions, which are * expressed as 4 ordinary differential equations (ODEs). By imposing * the conservation restraints and solving the relevant steady * state equations, we find that the simulated sigmoidal curve * matches the corresponding experimental data. Secondly, by numerically * solving the ODEs for simulating the Zur and ZntR run-off transcription * experiments, and depicting the simulated concentrations of zntA * and znuC transcripts as a function of free zinc concentration, * we find that the simulated curves fit the corresponding in vitro * experimental data. Moreover, we also perform simulations, after * taking into consideration the competitive effects of ZntR with * the zinc buffer, and depict the simulated concentration of zntA * transcripts as a function of the total ZntR concentration, both * in the presence and absence of Zn(II). The obtained simulation * results are in general agreement with the corresponding experimental * data. CONCLUSION: Simulation results show that our model can * quantitatively reproduce the results of several of the in vitro * experiments conducted by Outten CE and her colleagues. Our model * provides a detailed insight into the dynamics of the regulatory * system and also provides a general framework for simulating * in vitro metal-binding and transcription experiments and interpreting * the relevant experimental data. * * In 1999-2001, Outten et al. published a series of papers presenting * data from several in vitro transcription and metal-binding competition * experiments in E. coli. These studies provide a detailed data * set on which a mathematical model can be based. In the study * described here, Cui and Kaandorp have developed such a mathematical * model, and they use it to simulate the in vitro transcriptional * response of zinc homeostasis system in E. coli. Cellerator, * an open source software, was used to automatically generate * the equations, and the model was subsequently translated into * CellML to facilitate future model exchange, reuse and implementation. * * model diagram * * [[Image file: cui_2008.png]] * * A schematic diagram of the reactions described in the model * of zinc homeostasis system in Escherichia coli. Extracellular * zinc enters the cytoplasm through ZnuABC and ZupT, where its * presence can cause Zur to bind to the znu operator and repress * the transcription of the znuACB gene cluster. Excess intracellular * zinc ions are exported by ZntA and ZitB, and cytosolic zinc * can bind with protein ZntR to form a strong transcriptional * activator of the zntA gene. Cytoplasmic zinc trafficking may * involve chaperone-like proteins. Abbreviations: Zur* : active * Zur; ZntR* : active ZntR; C? : (a possible zinc chaperone protein * whose existence is still under debate). * * The original paper reference is cited below: * * Jiangjun Cui, Jaap A. Kaandorp and Catherine M. Lloyd 2008, * BMC Systems Biology, 2:89. PubMed ID: 18950480 */ import nsrunit; unit conversion on; unit minute=60 second^1; // unit molar predefined unit per_molar=.001 meter^3*mole^(-1); // unit nanomolar predefined unit first_order_rate_constant=1 second^(-1); unit second_order_rate_constant=1E6 meter^3*second^(-1)*mole^(-1); unit third_order_rate_constant=1E12 meter^6*second^(-1)*mole^(-2); math main { realDomain time second; time.min=0; extern time.max; extern time.delta; real Py(time) nanomolar; when(time=time.min) Py=25.0; real Zn(time) nanomolar; when(time=time.min) Zn=10000.0; real Py1(time) nanomolar; when(time=time.min) Py1=0.0; real r3 third_order_rate_constant; r3=4.41E10; real r4 first_order_rate_constant; r4=9E-3; real Dw(time) nanomolar; when(time=time.min) Dw=4.0; real Qw2(time) nanomolar; when(time=time.min) Qw2=0.0; real k_1 first_order_rate_constant; k_1=0.9; real k1a second_order_rate_constant; k1a=1.0; real Qw1(time) nanomolar; when(time=time.min) Qw1=0.0; real Rw(time) nanomolar; when(time=time.min) Rw=50.0; real k2 second_order_rate_constant; k2=0.02; real k_2 first_order_rate_constant; k_2=0.3; real k3(time) first_order_rate_constant; real Mw(time) nanomolar; when(time=time.min) Mw=0.0; real Px(time) nanomolar; when(time=time.min) Px=50.0; real Px1(time) nanomolar; when(time=time.min) Px1=0.0; real Dz(time) nanomolar; when(time=time.min) Dz=2.0; real Qz4(time) nanomolar; when(time=time.min) Qz4=0.0; real r1 second_order_rate_constant; r1=2.73E2; real r2 first_order_rate_constant; r2=3.437E-4; real k1b second_order_rate_constant; k1b=1.253E-2; real Qz2(time) nanomolar; when(time=time.min) Qz2=0.0; real k1 second_order_rate_constant; k1=0.025; real Tp(time) nanomolar; when(time=time.min) Tp=10000.0; real Tp1(time) nanomolar; when(time=time.min) Tp1=0.0; real r5 second_order_rate_constant; r5=3E4; real r6 first_order_rate_constant; r6=1.506E-2; real Qz1(time) nanomolar; when(time=time.min) Qz1=0.0; real Rz(time) nanomolar; when(time=time.min) Rz=100.0; real k2a second_order_rate_constant; k2a=0.00005; real Qz3(time) nanomolar; when(time=time.min) Qz3=0.0; real Qz5(time) nanomolar; when(time=time.min) Qz5=0.0; real k2b second_order_rate_constant; k2b=0.0002; real k2c second_order_rate_constant; k2c=0.0037; real Mz(time) nanomolar; when(time=time.min) Mz=0.0; real td0 second; td0=1800.0; real td second; td=2700; // // Py:time=(r4*Py1-r3*Zn^2*Py); // Py1:time=(r3*Zn^2*Py+k_1*Qw2-(r4*Py1+k1a*Dw*Py1)); // Dw:time=(k_1*Qw2+k3*Qw1+k_2*Qw1-(k1a*Dw*Py1+k2*Dw*Rw)); // Rw:time=(k3*Qw1+k_2*Qw1-k2*Dw*Rw); // Qw1:time=(k2*Dw*Rw-(k3*Qw1+k_2*Qw1)); // Qw2:time=(k1a*Dw*Py1-k_1*Qw2); // Mw:time=(k3*Qw1); // Px:time=(r2*Px1+k_1*Qz4-(r1*Zn*Px+k1b*Dz*Px)); // Px1:time=(r1*Zn*Px+k_1*Qz2-(r2*Px1+k1*Dz*Px1)); // Tp:time=(r6*Tp1-r5*Zn*Tp); // Tp1:time=(r5*Zn*Tp-r6*Tp1); // Zn:time=(r2*Px1+r6*Tp1-(r1*Zn*Px+r5*Zn*Tp)); // Dz:time=(k_1*Qz2+k3*Qz1+k_2*Qz1+k_1*Qz4-(k1b*Dz*Px+k1*Dz*Px1+k2a*Dz*Rz)); // Rz:time=(k3*Qz1+k_2*Qz1+k3*Qz3+k_2*Qz3+k3*Qz5+k_2*Qz5-(k2a*Dz*Rz+k2b*Qz4*Rz+k2c*Qz2*Rz)); // Qz1:time=(k2a*Dz*Rz-(k3*Qz1+k_2*Qz1)); // Qz2:time=(k1*Dz*Px1+k3*Qz3+k_2*Qz3-(k_1*Qz2+k2c*Qz2*Rz)); // Qz3:time=(k2c*Qz2*Rz-(k3*Qz3+k_2*Qz3)); // Qz4:time=(k1b*Dz*Px+k3*Qz5+k_2*Qz5-(k_1*Qz4+k2b*Qz4*Rz)); // Qz5:time=(k2b*Qz4*Rz-(k3*Qz5+k_2*Qz5)); // Mz:time=(k3*Qz1+k3*Qz3+k3*Qz5); // k3=(if ((time>=(0 second)) and (time=td0) and (time