BACKGROUND
zinc is essential for life.
it serves as a structural or catalytic cofactor in a large number of proteins such as rna polymerase and zinc finger proteins  <cit> .
zinc also plays an important signalling role in various biological processes such as neurotransmission, cell proliferation, and apoptosis  <cit> .
however, due to the potential toxicity of zinc, intracellular zinc concentrations must be kept under tight control.
for example, a high intracellular zn2+ concentration can inhibit the aerobic respiratory chain in e. coli  <cit> .
e. coli achieves zinc homeostasis by regulating the uptake and efflux of zinc across the plasma membrane  <cit> .
as we can see in fig.
1a, extracellular zinc ions are transported into the cytoplasm through znuabc  and zupt , while the efflux of zinc is accomplished by znta  and zitb   <cit> .
within the cytoplasm, similar to copper, it is thought that zinc trafficking may involve chaperone-like proteins  <cit> .
however, despite considerable experimental effort, the zinc chaperone protein in e. coli has yet to be identified  <cit> .
the znuabc transporter  is composed of the periplasmic binding protein znua, the atpase znuc, and the integral membrane protein znub  <cit> .
this zinc uptake system is regulated by zur, a dimer protein which binds at least  <dig> zinc ions.
zur is sensitive to the intracellular zinc concentration, and zinc-bound zur  can compete with rna polymerase to bind to the znu operator and act as a repressor  <cit> .
in contrast to this mechanism, zinc efflux through znta is regulated by zntr, a zinc-responsive merr-like transcriptional regulator  <cit> .
zntr is a dimer protein which can bind one or two zinc ions per monomer depending on the buffer conditions  <cit> .
a metal occupancy assay of zntr, monitored by changes in tyrosine fluorescence, shows non-cooperative 1: <dig> binding of zn to the zntr dimer  <cit> .
zntr in its apo form only slightly activates znta transcription  means that zntr without the binding of zn)  <cit> .
the binding of zinc-bound zntr to the promoter introduces conformational changes in the dna, which apparently make the promoter a better substrate for rna polymerase, thus strongly activating the transcription of the znta gene and increasing the efflux of zinc from the cell  <cit> .
during 1999– <dig>  outten ce and her colleagues presented some results on in vitro transcription and metal-binding competition experiments of e. coli zinc homeostasis system and showed that both zntr and zur are extremely avid zinc sensors and are both saturated at femtomolar free zinc concentrations  <cit> .
in these experiments, the zn concentration was precisely controlled by using n,n,n',n'-tetrakis ethylenediamine  as a zinc buffer  <cit> .
the various assays relevant to this paper include the zur-dna interaction assay, zur transcription assay and two zntr transcription assays.
in the zur-dna interaction assay the dnase i footprinting technique was used and the zur-dna interaction was found to correlate with the concentration of free zn .
in the zur transcription assay, in vitro run-off transcription experiments with zur and the znu zn uptake system were conducted, and the levels of the znuc rna transcript were reported to correlate with the free zn concentration   <cit> .
in these real run-off transcription experiments, various reactants , rnap and heparin, etc.)
were added sequentially and allowed to equilibrate first .
then nucleoside triphosphates  were added and the reaction was stopped for  <dig> min .
similar run-off transcription experiments ) were conducted with zntr and the znta promoter and the levels of the znta rna transcript were reported to correlate with the free zn concentration   <cit> .
similarly in zntr transcription assay , the levels of the znta rna transcript were reported to correlate with the total zntr concentration, both with added zn and without zn .
moreover, it was also reported that the levels of the znta rna transcript correlated with the total zinc concentration   <cit> .
although the transcriptional regulation of the zinc homeostasis system in e. coli seems to be well characterized, and despite the fact that detailed in vitro experimental data on this system are also available, as yet there is no mathematical model to help interpret these data  <cit> .
the principal aim of this paper is to present a mathematical model which is capable of simulating this regulatory system and can be used to help interpret various experimental data.
we will present a unified mathematical model and use it to simulate the in vitro transcriptional response of the zinc homeostasis system in e. coli.
the construction of the model is based on biochemical principles and we use open source software  to automatically generate the equations  <cit> .
we validate our model by comparing the simulation results with the corresponding in vitro experimental data.
RESULTS
as shown in table  <dig>  we use  <dig> reactions to represent the four sub-processes involved in the zinc homeostasis system, namely:  zn2+-sensing by zur,  transcriptional repression of the znuc gene by zur,  zn2+-sensing by zntr and  transcriptional activation of the znta gene by zntr .
here we present our results for simulating various in vitro assays  and assay  are differences in the initial conditions and in that in assay  we take into consideration competition between zntr and tpen for zinc binding by including reaction  <dig>  whereas in assay , reaction  <dig> is not included).
note: abbreviations and synonyms used in this table are as follows: zn ; px ; px <dig> ; py ; py <dig> ; z ; dz ; rz ; mz ; qz <dig> ; qz <dig> ; qz <dig> ; qz <dig> ; qz <dig> ; w ;dw ; rw ; mw ; qw <dig> ; qw <dig> ; tp ; tp <dig> ; rnap .
zur-dna interaction
the zur-dna interaction assay involves only two reactions , which are expressed as  <dig> odes   <cit> .
by imposing the conservation restraints  + py <dig> + qw <dig> = pytot =  <dig> nm, dw + qw <dig> = d <dig> =  <dig> nm)  and solving the relevant steady state equations , we can depict the simulated ratio of steady state concentrations of the zn4zur-dna complex  and the total concentration of znuc dna  as a function of the logarithm of parameter zn as shown in fig.
<dig> .
from this figure, we can see that when the simulated free zinc concentration  ranges from 10- <dig> m to 10- <dig> m, the simulated protection ratio  rises from  <dig> % to  <dig> %.
this means that in the presence of higher free zinc concentrations, more zn2zur molecules become active and bind with znuc dna molecules to protect them from the binding of rna polymerase.
the simulated sigmoidal curve  seems to fit well with the corresponding experimental data   <cit> .
k
d
t
d
px
tot
py
tot
zn
tot
tpen
tot
note: k2b = 4*k2a.
moreover, according to the equilibrium theory of chemical reactions, r <dig> = kdr <dig>  r <dig> = r5/kzn−tpen' and k1b = k1kd/kd <dig>  the values of four parameters  are taken from hayot's model  <cit> .
these parameters origin from measured rate constants of the λ repressor gene ci in e. coli and are also quoted as physiologically reasonable values by ingram et al  <cit> .
kzn−tpen' is calculated in the same way as shown in  <cit>  .
we derive the same simulation results by directly solving the  <dig> relevant odes with py = pytot =  <dig> nm, dw = d <dig> =  <dig> nm, dw <dig> =  <dig>  py <dig>  =  <dig> as the initial conditions and depicting the simulated ratio of the final concentration of the zn4zur-dna complex  and d <dig>  this is because the system reaches equilibrium before t = td <dig> =  <dig> min.
zur transcription assay
as mentioned in the legend of fig.
1b, here we only simulate the transcription of the znuc gene.
we approximate the in vitro zur run-off transcription assay by a two-phase  sub-model.
in the first phase, the preliminary equilibrating process of reactants involves  <dig> reactions  which are expressed as  <dig> odes .
in the second phase, the run-off transcription involves  <dig> reactions  which are expressed as  <dig> odes .
by setting the initial conditions of the model simulation to be the same as those in the real experiment  = pytot =  <dig> nm, dw = d <dig> =  <dig> nm, rw = r <dig> =  <dig> nm and all the remaining initial concentrations are set to be 0) and numerically solving the  <dig> equations for the first phase and then solving the  <dig> odes for the second phase , we can depict the relevant transient curves for zn = 10- <dig> as shown in fig.
3a–c .
as shown in fig.
3a, due to the binding of free zinc, the simulated concentration of zn2zur , the zur dimer which contains two zinc ions per dimer and it is used in the corresponding real assay  <cit> ) quickly decreases from  <dig> nm to a steady state value of  <dig>  nm whereas the simulated concentration of active zur , the zur dimer which contains four zinc ions per dimer) quickly rises from  <dig> to  <dig>  nm.
the simulated concentration of rna polymerase ) decreases slightly from  <dig> nm to  <dig>  nm due to the effect of its binding with znuc dna.
as we can see from fig.
3b, the simulated free znuc dna concentration ) decreases rapidly  from  <dig> nm to a steady state of  <dig>  nm during the first phase due to the binding of active zur and rna polymerase.
the simulated concentration of the transcription initiation complex ) rapidly increases  from  <dig> to a steady state value of  <dig>  nm whereas the simulated concentration of zn4zur-dna complex ) quickly increases  from  <dig> to  <dig>  nm.
the initiation of the second phase seems to only have a small influence on the afore mentioned steady state values  and qw <dig> change from  <dig>  nm and  <dig>  nm at the end of first phase to  <dig>  nm and  <dig>  nm at the end of the second phase, respectively).
from fig.
3c, we can see that in the first  <dig> minutes, the concentration of mrna of znuc ) remains at  <dig> because the real transcription has not happened yet, and then in the subsequent  <dig> minutes it increases linearly from  <dig> to a final concentration of  <dig>  nm.
the rapid decrease in the concentration of free znuc dna ) shown in fig.
3b is due to the binding of znuc dna with active zur  and rna polymerase.
since in the whole process, the total increase in the simulated concentration of zn4zur-dna complex ) is  <dig>  nm, whereas the total decrease of the simulated free znuc dna concentration is about  <dig>  nm, we can conclude that when zn = 10- <dig> nm, the binding of active zur consumes the majority of the znuc dna to form the zn4zur-dna complex, which can not further bind with rna polymerase, and in this way the transcription of znuc is repressed.
we performed many simulations for various values of zn  and recorded the final values of the simulated mrna concentration ).
after normalizing these concentration values, depicting them as a function of zn , and smoothly connecting these simulated data points, we obtained the purple curve in fig.
3d .
zntr transcription assay
similarly, the zntr run-off transcription assay can also be simulated by a two-phase sub-model.
the first phase  involves  <dig> reactions , which are expressed as  <dig> odes .
the second phase  involves  <dig> reactions , which are expressed as  <dig> odes .
by setting the initial conditions of the simulation to be the same as those used in the real experiment  = pxtot =  <dig> nm, dz = d <dig> = <dig> nm, rz = r <dig> =  <dig> nm and setting all the remaining initial concentrations to be 0), and subsequently solving the relevant equations of the two-phase sub-model, we can depict the relevant transient curves for zn = 10- <dig> nm as shown in fig.
<dig> .
in this assay, reaction  <dig> is not included in the sub-model because the zntr concentration is too low to challenge the buffering capacity of tpen .
from fig.
4a we can see that due to the binding with free zinc, the simulated concentration of apo-zntr ) decreases from  <dig> nm to  <dig>  nm, whereas the simulated concentration of active zntr ) rises from  <dig> to  <dig>  nm, and the simulated concentration of rna polymerase ) decreases slightly from  <dig> nm to  <dig>  nm.
in the first phase, due to the binding with zntr and rna polymerase, the simulated unbound znta dna concentration ) decreases rapidly  from  <dig> nm to  <dig>  nm and then decreases gradually to  <dig>  nm at the end of the first phase ; in the second phase, the free znta dna concentration remains at roughly the same level .
the simulated znta mrna concentration ) remains at  <dig> nm during the first phase, as there is no transcription happening, and then increases seemingly linearly to a final concentration of  <dig>  nm during the second phase after ntps have been added .
the simulated transients curves in fig.
4c show that qz <dig> rapidly rises  from  <dig> to a peak value of  <dig>  nm and then gradually decreases to  <dig>  nm during the first  <dig> minutes whereas qz <dig> rapidly rises  from  <dig> to a peak value of  <dig>  nm and then gradually decreases to a  <dig>  nm during the first phase.
the initiation of the second phase causes a small decrease in the values of qz <dig> and qz <dig> .
as shown in fig.
4d, both qz <dig> and qz <dig> rise first  during the first phase whereas qz <dig> first dramatically increases up to a peak value of  <dig>  nm and then gradually decreases to its final value of  <dig>  nm.
the initiation of the second phase causes a small decrease in the value of qz <dig> and a slight increase in the value of qz <dig>  as judged by the small kinks in the corresponding two curves, whereas it has insignificant influence of the value of qz <dig>
using similar methods we can obtain the green curve in fig.
3d for the final values of the simulated znta mrna concentration ) as a function of the value of zn .
the results shown in fig.
3d indicate that when the simulated free zinc concentration ranges from 10- <dig> m to 10- <dig> m, the simulated normalized final concentrations of mrna of znuc ) decreases from 100% to  <dig> %, whereas the simulated normalized final concentration of mrna of znta ) increases from  <dig> % to 100%.
the half-maximal induction of znuc transcripts and the half maximal induction of znta transcripts occur at zn = 2*10- <dig> m and zn =  <dig> *10- <dig> m respectively, which are the same as previously reported values  <cit> .
the simulated purple curve  agrees with corresponding experimental data  extremely well.
similarly the simulated green curve ) also agrees with the corresponding experimental data , although to a slightly lesser degree  <cit> .
zntr transcription assay
in this assay, we take into consideration the competition between zntr and tpen for zinc binding by including reaction  <dig>  again, we will use a two-phase sub-model to simulate the real assay.
the first phase  of the assay  involves  <dig> reactions , which are expressed as  <dig> odes .
the second phase  involves  <dig> reactions , which are expressed as  <dig> odes .
by setting the initial conditions of the simulation equal to those used in the real experiment =tpentot=10μm,zn=zntot,px=pxtot,dz=d0=2nm,rz=r0=100nm, and all the remaining initial concentrations are set to 0) and solving the two-phase model, we depict the simulated final concentrations  of mrna of znta ) for zntot =  <dig> μm and zntot =  <dig> as a function of the logarithm of the doubled value of parameter pxtot  and we obtain the purple and green curves shown in fig.
5a <cit> .
we also perform many simulations under the following initial conditions: =tpentot=10μm,zn=zntot,px=pxtot=50nm,dz=d0=2nm,rz=r0=100nm, and all the remaining initial concentrations are set to 0) for various values of zntot within the range of  <dig> nm to  <dig> μm and eventually obtain the black curve shown in fig.
5b which describes the final values of the simulated mrna concentration ) as a function of the value of zntot .
discussion
the simulation results shown in fig.
<dig> indicate the complex interactions among three transcription processes of znta .
if we compare the dynamics of the simulated concentrations of three transcription initiation complexes involved in the zntr transcription assay , qz <dig> and qz5) as shown in fig.
4c and fig.
4d, we find that the dynamics of qz <dig> and qz <dig> are quite similar.
initially, they both increase rapidly, form low peaks , and then gradually decrease.
in contrast, the dynamics of qz <dig> only demonstrates a gradual increase to  <dig>  nm in the first  <dig> minutes.
the observation that the final steady state value of qz <dig>  is much higher than those of qz <dig> and qz <dig>  indicates that for zn = 10-5nm, when the system ) enters its final equilibrium, the dominating transcription process is znzntr activated transcription rather than the other two transcription processes .
to explain why the dynamics of qz <dig> shows a peak, we suggest that the initial increase of qz <dig> is due to the binding of znta dna with rna polymerase.
then following the conversion of apo-zntr to active zntr by zinc-binding , active zntr binds with znta dna to form the znzntr-dna complex .
this competitive binding of active zntr causes a sudden decrease in the free znta dna concentration  and the reversible reaction  <dig>  becomes dominated by its reverse side and qz <dig> begins to decrease after forming a small peak.
similarly, we can explain the dynamics of qz <dig>
by comparing the dynamics of the simulated zur and zntr transcription assays shown in fig.
3a–c and fig.
<dig>  we can see that when zn = 10-5nm, during the first phase, the simulated zur transcription system reaches its steady state in less than  <dig> seconds, much faster than the simulated zntr transcription system which takes more than  <dig> minutes.
as shown in fig.
3c and fig.
4b, the seemingly linear increase of the simulated concentrations of mrna  and mz) during the second phase indicates the progress of the relevant transcription processes.
if we calculate the slope of the linear curve in fig.
3c as follows:
mw/qw1s/td =  <dig>  nm/ <dig>  nm/ <dig> min =  <dig> s- <dig> where qz1s denotes the final steady state value of qz <dig>  we derive the same value as that of the transcription rate parameter k <dig>  obviously the simulated final concentrations of mrna  and mz) are generally proportional to td, which is in accordance with the experimental observation that the harvest of run-off transcription assay is related to the duration time of its transcription phase   <cit> .
the purple curve in fig.
5a indicates that for zntot =  <dig> μm, when the simulated total zntr monomer concentration  ranges from 10- <dig> m to 10- <dig> m, the simulated final concentrations of mrna of znta ) increases from  <dig>  nm to a peak value of  <dig>  nm when total = 10- <dig> m and then decreases to  <dig>  nm.
if we look at the corresponding experimental data , we can see that the relative induction of the znta transcripts increases, forms a peak , and eventually declines  <cit> .
thus our simulation successfully simulates the peak behaviour of the relative induction of the znta transcripts for increasing values of total in the presence of zinc.
further investigations show that if we perform the simulations excluding reaction  <dig>  then we can only reproduce the increasing behaviour rather than the peak behaviour.
thus one potential explanation for the peak behaviour is that, for low zntr concentrations, tpen is strong enough to buffer the zinc and more zntr will promote the transcription of znta gene; while for high zntr concentrations, the buffering capacity of tpen is exceeded and the free zinc concentration can not be maintained as a constant anymore and it subsequently decreases due to the binding of over-abundant zntr molecules, which in turn limits the transcription processes.
a similar comparison can be made for the case when zntot =  <dig> .
however, in the latter case, our model can only simulate the initial increase, but fails to reproduce the decline.
as described in detail in the methods section, in this model, we assume that the active form of zntr is znzntr because metal occupancy assays of zntr monitored by changes in tyrosine fluorescence show noncooperative 1: <dig> binding of zn to the zntr dimer  <cit> .
this assumption is valid only when the free zinc concentration and total zntr concentration are both extremely low .
when the total zntr concentration goes to the μm range, the binding kinetics of zn to the zntr dimer will be more complicated because zntr can bind one or two zinc ions per dimer depending on the buffer conditions  <cit> .
this explains why, as shown in fig.
5a in the case of with zn, there is a disagreement between the simulation results  and the corresponding experimental data  when zntr molecules are relatively abundant.
intuitively, we can think of it in this way: in the real case, the competitive ability of zntr for zn binding is stronger than the model prediction because at high zntr concentrations, zntr, on average, binds with more than one ion per dimer.
this results in a smaller and earlier peak because the buffering capacity of tpen is now easier to exceed.
in the absence of zn, the eventual abnormal decline in the experimental data  may be due to the normal deviations of the different experiments because the levels of znta transcript are very low in this case or perhaps this is due to some novel, unknown mechanisms )  <cit> .
as we can see from fig.
5b, when the simulated total zinc concentration  ranges from 10- <dig> m to 10- <dig> m, the simulated final concentration of mrna of znta ) increases from  <dig>  nm to  <dig>  nm  which means that more abundant free zinc ions bind with zntr to promote activation of the transcription of the znta gene.
the simulated curve  fits the experimental data  quite well  <cit> .
coupled feedback loops have been recently recognized as essential building blocks of cellular networks  <cit> .
the zinc homeostasis system in e. coli is a good example of such a building block because it follows from fig.
1a that zur and znuc form a 'negative circuit', since active zur represses znuc  while zinc influx via znuc leads to larger amounts of active zur molecules .
similar considerations point towards the negative circuit wiring between zntr and znta.
it is believed that such coupled negative feedback loops are quite helpful for enhancing homeostasis  <cit> .
as previously mentioned, cytoplasmic zinc trafficking in e. coli may involve chaperone-like proteins whose existence is still being debated  <cit> .
outten et al.
demonstrated in vitro that zntr and zur are sensitive to very low concentrations  of free zinc , therefore they proposed that free zinc in the cytosol of e. coli is not physiologically available under normal growth conditions  <cit> .
our simulation results further confirm their experimental data and support their proposal.
however, in order to better understand the in vivo transcriptional regulation mechanisms of zinc homeostasis, further investigations are required to simulate the in vivo transcription processes and their responses to various environmental conditions.
up until now, performing well-designed in vitro experiments has been one of the common ways used to infer the various characteristics of the corresponding in vivo systems.
the current work provides a good example of how to use a unified mathematical model to explain complicated datasets obtained from in vitro metal-binding and transcription experiments which have been widely performed for metal ion homeostasis and detoxification systems  <cit> .
the repression of zur on the transcription of znuacb gene cluster and the activation of zntr on the znta transcription constitute the critical parts of the regulatory mechanisms of the zinc homeostasis system in e. coli .
this means that if we want to make predictive and useful model for the in vivo zinc homeostasis system, we need to model these transcriptional regulations.
although the current model only simulates the in vitro kinetics, together with its fitted rate constants it can be used as a good basis and reference for the future modelling of the corresponding in vivo system.
moreover, the quantitative distinguishment of the three transcription processes of znta  in our model will be quite meaningful for modelling the in vivo system and it provides the possibility of including any additional regulations on these three processes which do happen in vivo  <cit> .
in order to further our understanding of the process of zinc homeostasis in e. coli, the most critical thing is to identify the intracellular zinc chaperone, which is very likely to exist.
recently, proteomics has progressed to such a stage that it can determine the cellular response to any perturbation at the level of protein activation  <cit> .
thus mass spectrometry-based proteomics can be used to search the possible molecular candidates in addition to genome-wide high-throughput screens  <cit> .
once the zinc chaperone has been identified, the next step will be to measure the interactions between the zinc chaperone and the membrane transport proteins  and the interactions between the zinc chaperone and the metalloregulatory proteins .
since similar work has already been done for the copper homeostasis system in e. hirae, the same equipments and experimental techniques used there  can also be used to measure the kinetics of these interactions in the zinc homeostasis system  <cit> .
furthermore, we need to further take into consideration zinc storage and zinc utilisation by proteins in e. coli and quantify the concentrations of the relevant proteins, dnas and mrnas.
finally, the subtle details of relevant regulatory processes  need to be further investigated  <cit> .
once we have characterised these processes in detail and have made corresponding sub-models for them, we can then integrate these sub-models together with the current model, in order to build a comprehensive model to describe the entire in vivo system.
further experiments determining the in vivo cellular response to various perturbations will be necessary for checking the validity of the model and also for model refinement.
in this way, step by step we will acquire a complete map of the zinc homeostasis system in e. coli and reach a full understanding of the system dynamics.
close cooperation between pioneering experimentalists and computational scientists through iterative systems biology procedure  will be necessary for achieving such ambitious goals  <cit> .
CONCLUSIONS
to summarize, we have built a mathematical model for simulating the in vitro transcriptional response of zinc homeostasis system in e. coli.
simulation results show that our model can quantitatively reproduce the various results of the in vitro experiments conducted by outten ce and her colleagues.
our model gives a detailed insight into the involved system dynamics and provides a general framework for simulating in vitro metal-binding and transcription experiments and interpreting relevant experimental data.

 zinc is essential for life. it serves as a structural or catalytic cofactor in a large number of proteins such as rna polymerase and zinc finger proteins. zinc also plays an important signalling role in various biological processes such as neurotransmission, cell proliferation, and apoptosis. however, due to the potential toxicity of zinc, intracellular zinc concentrations must be kept under tight control. for example, a high intracellular zn2+ concentration can inhibit the aerobic respiratory chain in e. coli. e. coli achieves zinc homeostasis by regulating the uptake and efflux of zinc across the plasma membrane. as we can see in fig. 1a, extracellular zinc ions are transported into the cytoplasm through znuabc and zupt, while the efflux of zinc is accomplished by znta and zitb. within the cytoplasm, similar to copper, it is thought that zinc trafficking may involve chaperone-like proteins. however, despite considerable experimental effort, the zinc chaperone protein in e. coli has yet to be identified. the znuabc transporter is composed of the periplasmic binding protein znua, the atpase znuc, and the integral membrane protein znub. this zinc uptake system is regulated by zur, a dimer protein which binds at least zinc ions. zur is sensitive to the intracellular zinc concentration, and zinc-bound zur can compete with rna polymerase to bind to the znu operator and act as a repressor. in contrast to this mechanism, zinc efflux through znta is regulated by zntr, a zinc-responsive merr-like transcriptional regulator. zntr is a dimer protein which can bind one or two zinc ions per monomer depending on the buffer conditions. a metal occupancy assay of zntr, monitored by changes in tyrosine fluorescence, shows non-cooperative 1: binding of zn to the zntr dimer. zntr in its apo form only slightly activates znta transcription means that zntr without the binding of zn). the binding of zinc-bound zntr to the promoter introduces conformational changes in the dna, which apparently make the promoter a better substrate for rna polymerase, thus strongly activating the transcription of the znta gene and increasing the efflux of zinc from the cell. during 1999– outten ce and her colleagues presented some results on in vitro transcription and metal-binding competition experiments of e. coli zinc homeostasis system and showed that both zntr and zur are extremely avid zinc sensors and are both saturated at femtomolar free zinc concentrations. in these experiments, the zn concentration was precisely controlled by using n,n,n',n'-tetrakis ethylenediamine as a zinc buffer. the various assays relevant to this paper include the zur-dna interaction assay, zur transcription assay and two zntr transcription assays. in the zur-dna interaction assay the dnase i footprinting technique was used and the zur-dna interaction was found to correlate with the concentration of free zn. in the zur transcription assay, in vitro run-off transcription experiments with zur and the znu zn uptake system were conducted, and the levels of the znuc rna transcript were reported to correlate with the free zn concentration. in these real run-off transcription experiments, various reactants, rnap and heparin, etc.) were added sequentially and allowed to equilibrate first. then nucleoside triphosphates were added and the reaction was stopped for min. similar run-off transcription experiments ) were conducted with zntr and the znta promoter and the levels of the znta rna transcript were reported to correlate with the free zn concentration. similarly in zntr transcription assay, the levels of the znta rna transcript were reported to correlate with the total zntr concentration, both with added zn and without zn. moreover, it was also reported that the levels of the znta rna transcript correlated with the total zinc concentration. although the transcriptional regulation of the zinc homeostasis system in e. coli seems to be well characterized, and despite the fact that detailed in vitro experimental data on this system are also available, as yet there is no mathematical model to help interpret these data. the principal aim of this paper is to present a mathematical model which is capable of simulating this regulatory system and can be used to help interpret various experimental data. we will present a unified mathematical model and use it to simulate the in vitro transcriptional response of the zinc homeostasis system in e. coli. the construction of the model is based on biochemical principles and we use open source software to automatically generate the equations. we validate our model by comparing the simulation results with the corresponding in vitro experimental data.  as shown in table we use reactions to represent the four sub-processes involved in the zinc homeostasis system, namely: zn2+-sensing by zur, transcriptional repression of the znuc gene by zur, zn2+-sensing by zntr and transcriptional activation of the znta gene by zntr. here we present our results for simulating various in vitro assays and assay are differences in the initial conditions and in that in assay we take into consideration competition between zntr and tpen for zinc binding by including reaction whereas in assay, reaction is not included). note: abbreviations and synonyms used in this table are as follows: zn; px; px; py; py; z; dz; rz; mz; qz; qz; qz; qz; qz; w;dw; rw; mw; qw; qw; tp; tp; rnap. zur-dna interaction the zur-dna interaction assay involves only two reactions, which are expressed as odes. by imposing the conservation restraints + py + qw = pytot = nm, dw + qw = d = nm) and solving the relevant steady state equations, we can depict the simulated ratio of steady state concentrations of the zn4zur-dna complex and the total concentration of znuc dna as a function of the logarithm of parameter zn as shown in fig.. from this figure, we can see that when the simulated free zinc concentration ranges from 10- m to 10- m, the simulated protection ratio rises from % to %. this means that in the presence of higher free zinc concentrations, more zn2zur molecules become active and bind with znuc dna molecules to protect them from the binding of rna polymerase. the simulated sigmoidal curve seems to fit well with the corresponding experimental data. k d t d px tot py tot zn tot tpen tot note: k2b = 4*k2a. moreover, according to the equilibrium theory of chemical reactions, r = kdr r = r5/kzn−tpen' and k1b = k1kd/kd the values of four parameters are taken from hayot's model. these parameters origin from measured rate constants of the λ repressor gene ci in e. coli and are also quoted as physiologically reasonable values by ingram et al. kzn−tpen' is calculated in the same way as shown in. we derive the same simulation results by directly solving the relevant odes with py = pytot = nm, dw = d = nm, dw = py = as the initial conditions and depicting the simulated ratio of the final concentration of the zn4zur-dna complex and d this is because the system reaches equilibrium before t = td = min. zur transcription assay as mentioned in the legend of fig. 1b, here we only simulate the transcription of the znuc gene. we approximate the in vitro zur run-off transcription assay by a two-phase sub-model. in the first phase, the preliminary equilibrating process of reactants involves reactions which are expressed as odes. in the second phase, the run-off transcription involves reactions which are expressed as odes. by setting the initial conditions of the model simulation to be the same as those in the real experiment = pytot = nm, dw = d = nm, rw = r = nm and all the remaining initial concentrations are set to be 0) and numerically solving the equations for the first phase and then solving the odes for the second phase, we can depict the relevant transient curves for zn = 10- as shown in fig. 3a–c. as shown in fig. 3a, due to the binding of free zinc, the simulated concentration of zn2zur, the zur dimer which contains two zinc ions per dimer and it is used in the corresponding real assay ) quickly decreases from nm to a steady state value of nm whereas the simulated concentration of active zur, the zur dimer which contains four zinc ions per dimer) quickly rises from to nm. the simulated concentration of rna polymerase ) decreases slightly from nm to nm due to the effect of its binding with znuc dna. as we can see from fig. 3b, the simulated free znuc dna concentration ) decreases rapidly from nm to a steady state of nm during the first phase due to the binding of active zur and rna polymerase. the simulated concentration of the transcription initiation complex ) rapidly increases from to a steady state value of nm whereas the simulated concentration of zn4zur-dna complex ) quickly increases from to nm. the initiation of the second phase seems to only have a small influence on the afore mentioned steady state values and qw change from nm and nm at the end of first phase to nm and nm at the end of the second phase, respectively). from fig. 3c, we can see that in the first minutes, the concentration of mrna of znuc ) remains at because the real transcription has not happened yet, and then in the subsequent minutes it increases linearly from to a final concentration of nm. the rapid decrease in the concentration of free znuc dna ) shown in fig. 3b is due to the binding of znuc dna with active zur and rna polymerase. since in the whole process, the total increase in the simulated concentration of zn4zur-dna complex ) is nm, whereas the total decrease of the simulated free znuc dna concentration is about nm, we can conclude that when zn = 10- nm, the binding of active zur consumes the majority of the znuc dna to form the zn4zur-dna complex, which can not further bind with rna polymerase, and in this way the transcription of znuc is repressed. we performed many simulations for various values of zn and recorded the final values of the simulated mrna concentration ). after normalizing these concentration values, depicting them as a function of zn, and smoothly connecting these simulated data points, we obtained the purple curve in fig. 3d. zntr transcription assay similarly, the zntr run-off transcription assay can also be simulated by a two-phase sub-model. the first phase involves reactions, which are expressed as odes. the second phase involves reactions, which are expressed as odes. by setting the initial conditions of the simulation to be the same as those used in the real experiment = pxtot = nm, dz = d = nm, rz = r = nm and setting all the remaining initial concentrations to be 0), and subsequently solving the relevant equations of the two-phase sub-model, we can depict the relevant transient curves for zn = 10- nm as shown in fig.. in this assay, reaction is not included in the sub-model because the zntr concentration is too low to challenge the buffering capacity of tpen. from fig. 4a we can see that due to the binding with free zinc, the simulated concentration of apo-zntr ) decreases from nm to nm, whereas the simulated concentration of active zntr ) rises from to nm, and the simulated concentration of rna polymerase ) decreases slightly from nm to nm. in the first phase, due to the binding with zntr and rna polymerase, the simulated unbound znta dna concentration ) decreases rapidly from nm to nm and then decreases gradually to nm at the end of the first phase; in the second phase, the free znta dna concentration remains at roughly the same level. the simulated znta mrna concentration ) remains at nm during the first phase, as there is no transcription happening, and then increases seemingly linearly to a final concentration of nm during the second phase after ntps have been added. the simulated transients curves in fig. 4c show that qz rapidly rises from to a peak value of nm and then gradually decreases to nm during the first minutes whereas qz rapidly rises from to a peak value of nm and then gradually decreases to a nm during the first phase. the initiation of the second phase causes a small decrease in the values of qz and qz. as shown in fig. 4d, both qz and qz rise first during the first phase whereas qz first dramatically increases up to a peak value of nm and then gradually decreases to its final value of nm. the initiation of the second phase causes a small decrease in the value of qz and a slight increase in the value of qz as judged by the small kinks in the corresponding two curves, whereas it has insignificant influence of the value of qz using similar methods we can obtain the green curve in fig. 3d for the final values of the simulated znta mrna concentration ) as a function of the value of zn. the results shown in fig. 3d indicate that when the simulated free zinc concentration ranges from 10- m to 10- m, the simulated normalized final concentrations of mrna of znuc ) decreases from 100% to %, whereas the simulated normalized final concentration of mrna of znta ) increases from % to 100%. the half-maximal induction of znuc transcripts and the half maximal induction of znta transcripts occur at zn = 2*10- m and zn = *10- m respectively, which are the same as previously reported values. the simulated purple curve agrees with corresponding experimental data extremely well. similarly the simulated green curve ) also agrees with the corresponding experimental data, although to a slightly lesser degree. zntr transcription assay in this assay, we take into consideration the competition between zntr and tpen for zinc binding by including reaction again, we will use a two-phase sub-model to simulate the real assay. the first phase of the assay involves reactions, which are expressed as odes. the second phase involves reactions, which are expressed as odes. by setting the initial conditions of the simulation equal to those used in the real experiment =tpentot=10μm,zn=zntot,px=pxtot,dz=d0=2nm,rz=r0=100nm, and all the remaining initial concentrations are set to 0) and solving the two-phase model, we depict the simulated final concentrations of mrna of znta ) for zntot = μm and zntot = as a function of the logarithm of the doubled value of parameter pxtot and we obtain the purple and green curves shown in fig. 5a. we also perform many simulations under the following initial conditions: =tpentot=10μm,zn=zntot,px=pxtot=50nm,dz=d0=2nm,rz=r0=100nm, and all the remaining initial concentrations are set to 0) for various values of zntot within the range of nm to μm and eventually obtain the black curve shown in fig. 5b which describes the final values of the simulated mrna concentration ) as a function of the value of zntot. discussion the simulation results shown in fig. indicate the complex interactions among three transcription processes of znta. if we compare the dynamics of the simulated concentrations of three transcription initiation complexes involved in the zntr transcription assay, qz and qz5) as shown in fig. 4c and fig. 4d, we find that the dynamics of qz and qz are quite similar. initially, they both increase rapidly, form low peaks, and then gradually decrease. in contrast, the dynamics of qz only demonstrates a gradual increase to nm in the first minutes. the observation that the final steady state value of qz is much higher than those of qz and qz indicates that for zn = 10-5nm, when the system ) enters its final equilibrium, the dominating transcription process is znzntr activated transcription rather than the other two transcription processes. to explain why the dynamics of qz shows a peak, we suggest that the initial increase of qz is due to the binding of znta dna with rna polymerase. then following the conversion of apo-zntr to active zntr by zinc-binding, active zntr binds with znta dna to form the znzntr-dna complex. this competitive binding of active zntr causes a sudden decrease in the free znta dna concentration and the reversible reaction becomes dominated by its reverse side and qz begins to decrease after forming a small peak. similarly, we can explain the dynamics of qz by comparing the dynamics of the simulated zur and zntr transcription assays shown in fig. 3a–c and fig. we can see that when zn = 10-5nm, during the first phase, the simulated zur transcription system reaches its steady state in less than seconds, much faster than the simulated zntr transcription system which takes more than minutes. as shown in fig. 3c and fig. 4b, the seemingly linear increase of the simulated concentrations of mrna and mz) during the second phase indicates the progress of the relevant transcription processes. if we calculate the slope of the linear curve in fig. 3c as follows: mw/qw1s/td = nm/ nm/ min = s- where qz1s denotes the final steady state value of qz we derive the same value as that of the transcription rate parameter k obviously the simulated final concentrations of mrna and mz) are generally proportional to td, which is in accordance with the experimental observation that the harvest of run-off transcription assay is related to the duration time of its transcription phase. the purple curve in fig. 5a indicates that for zntot = μm, when the simulated total zntr monomer concentration ranges from 10- m to 10- m, the simulated final concentrations of mrna of znta ) increases from nm to a peak value of nm when total = 10- m and then decreases to nm. if we look at the corresponding experimental data, we can see that the relative induction of the znta transcripts increases, forms a peak, and eventually declines. thus our simulation successfully simulates the peak behaviour of the relative induction of the znta transcripts for increasing values of total in the presence of zinc. further investigations show that if we perform the simulations excluding reaction then we can only reproduce the increasing behaviour rather than the peak behaviour. thus one potential explanation for the peak behaviour is that, for low zntr concentrations, tpen is strong enough to buffer the zinc and more zntr will promote the transcription of znta gene; while for high zntr concentrations, the buffering capacity of tpen is exceeded and the free zinc concentration can not be maintained as a constant anymore and it subsequently decreases due to the binding of over-abundant zntr molecules, which in turn limits the transcription processes. a similar comparison can be made for the case when zntot =. however, in the latter case, our model can only simulate the initial increase, but fails to reproduce the decline. as described in detail in the methods section, in this model, we assume that the active form of zntr is znzntr because metal occupancy assays of zntr monitored by changes in tyrosine fluorescence show noncooperative 1: binding of zn to the zntr dimer. this assumption is valid only when the free zinc concentration and total zntr concentration are both extremely low. when the total zntr concentration goes to the μm range, the binding kinetics of zn to the zntr dimer will be more complicated because zntr can bind one or two zinc ions per dimer depending on the buffer conditions. this explains why, as shown in fig. 5a in the case of with zn, there is a disagreement between the simulation results and the corresponding experimental data when zntr molecules are relatively abundant. intuitively, we can think of it in this way: in the real case, the competitive ability of zntr for zn binding is stronger than the model prediction because at high zntr concentrations, zntr, on average, binds with more than one ion per dimer. this results in a smaller and earlier peak because the buffering capacity of tpen is now easier to exceed. in the absence of zn, the eventual abnormal decline in the experimental data may be due to the normal deviations of the different experiments because the levels of znta transcript are very low in this case or perhaps this is due to some novel, unknown mechanisms ). as we can see from fig. 5b, when the simulated total zinc concentration ranges from 10- m to 10- m, the simulated final concentration of mrna of znta ) increases from nm to nm which means that more abundant free zinc ions bind with zntr to promote activation of the transcription of the znta gene. the simulated curve fits the experimental data quite well. coupled feedback loops have been recently recognized as essential building blocks of cellular networks. the zinc homeostasis system in e. coli is a good example of such a building block because it follows from fig. 1a that zur and znuc form a'negative circuit', since active zur represses znuc while zinc influx via znuc leads to larger amounts of active zur molecules. similar considerations point towards the negative circuit wiring between zntr and znta. it is believed that such coupled negative feedback loops are quite helpful for enhancing homeostasis. as previously mentioned, cytoplasmic zinc trafficking in e. coli may involve chaperone-like proteins whose existence is still being debated. outten et al. demonstrated in vitro that zntr and zur are sensitive to very low concentrations of free zinc, therefore they proposed that free zinc in the cytosol of e. coli is not physiologically available under normal growth conditions. our simulation results further confirm their experimental data and support their proposal. however, in order to better understand the in vivo transcriptional regulation mechanisms of zinc homeostasis, further investigations are required to simulate the in vivo transcription processes and their responses to various environmental conditions. up until now, performing well-designed in vitro experiments has been one of the common ways used to infer the various characteristics of the corresponding in vivo systems. the current work provides a good example of how to use a unified mathematical model to explain complicated datasets obtained from in vitro metal-binding and transcription experiments which have been widely performed for metal ion homeostasis and detoxification systems. the repression of zur on the transcription of znuacb gene cluster and the activation of zntr on the znta transcription constitute the critical parts of the regulatory mechanisms of the zinc homeostasis system in e. coli. this means that if we want to make predictive and useful model for the in vivo zinc homeostasis system, we need to model these transcriptional regulations. although the current model only simulates the in vitro kinetics, together with its fitted rate constants it can be used as a good basis and reference for the future modelling of the corresponding in vivo system. moreover, the quantitative distinguishment of the three transcription processes of znta in our model will be quite meaningful for modelling the in vivo system and it provides the possibility of including any additional regulations on these three processes which do happen in vivo. in order to further our understanding of the process of zinc homeostasis in e. coli, the most critical thing is to identify the intracellular zinc chaperone, which is very likely to exist. recently, proteomics has progressed to such a stage that it can determine the cellular response to any perturbation at the level of protein activation. thus mass spectrometry-based proteomics can be used to search the possible molecular candidates in addition to genome-wide high-throughput screens. once the zinc chaperone has been identified, the next step will be to measure the interactions between the zinc chaperone and the membrane transport proteins and the interactions between the zinc chaperone and the metalloregulatory proteins. since similar work has already been done for the copper homeostasis system in e. hirae, the same equipments and experimental techniques used there can also be used to measure the kinetics of these interactions in the zinc homeostasis system. furthermore, we need to further take into consideration zinc storage and zinc utilisation by proteins in e. coli and quantify the concentrations of the relevant proteins, dnas and mrnas. finally, the subtle details of relevant regulatory processes need to be further investigated. once we have characterised these processes in detail and have made corresponding sub-models for them, we can then integrate these sub-models together with the current model, in order to build a comprehensive model to describe the entire in vivo system. further experiments determining the in vivo cellular response to various perturbations will be necessary for checking the validity of the model and also for model refinement. in this way, step by step we will acquire a complete map of the zinc homeostasis system in e. coli and reach a full understanding of the system dynamics. close cooperation between pioneering experimentalists and computational scientists through iterative systems biology procedure will be necessary for achieving such ambitious goals.  to summarize, we have built a mathematical model for simulating the in vitro transcriptional response of zinc homeostasis system in e. coli. simulation results show that our model can quantitatively reproduce the various results of the in vitro experiments conducted by outten ce and her colleagues. our model gives a detailed insight into the involved system dynamics and provides a general framework for simulating in vitro metal-binding and transcription experiments and interpreting relevant experimental data. 
its underlying regulatory machine consists of repression on zinc influx through znuabc by zur and activation on zinc efflux via znta by zntr. 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. the obtained simulation results are in general agreement with the corresponding experimental data. 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. we develop a unified mathematical model consisting of reactions to simulate the in vitro transcriptional response of the zinc homeostasis system in e. coli. 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. 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. by imposing the conservation restraints and solving the relevant steady state equations, we find that the simulated sigmoidal curve matches the corresponding experimental data. 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. the zinc homeostasis system in escherichia coli is one of the most intensively studied prokaryotic zinc homeostasis systems.
 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 and activation on zinc efflux via znta by zntr. 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.  we develop a unified mathematical model consisting of 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 ordinary differential equations. 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. the obtained simulation results are in general agreement with the corresponding experimental data.  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. 