BACKGROUND
biochemical systems in living cells are robust and flexible.
investigating the responses of cells  to environmental changes typically requires a system-level analysis of the interactions between the various molecular elements  that comprise the system.
a key step to analyze system responses to environmental changes is identifying large state changes or "transitions".
a statistical method that could detect such transitions would be a powerful analytical tool for finding important factors in large-scale profiles, such as variations in gene expression.
previous analyses of gene expression profiles have often made use of graphical models, such as bayesian networks  <cit> , graphical gaussian modelling  <cit> , boolean networks  <cit> , and auto-regressive models  <cit> .
however, not many approaches have explicitly modelled observational noise.
in auto-regressive analyses, for example, observational vectors yt are recursively defined by the following equation  <cit> :
yt = ayt- <dig> + εt
where yt is an observational vector of genes or metabolites at time t, a is an observational transition matrix, and εt is gaussian noise.
because this model does not distinguish observational and inherent  noises, identification of transition states becomes difficult in the presence of substantial noise.
here we propose an extension of the auto-regressive model  <cit> , which has been modified by the addition of reduced set of internal states, as explained in the results and discussion section.
we chose a mathematical model, the linear dynamical system , as the basis of our method because it does not impose any specific requirements on the data used.
lds is expected to eliminate the confounding influence of observational noise in time series data.
the model was applied to detect cellular state transitions in transcriptome and metabolome time series datasets from bacillus subtilis and arabidopsis thaliana maintained under stress conditions.
RESULTS
overview of our method
our method has two steps.
first, transition time points for each time series are detected using lds, which mathematically distinguishes transitional fluctuations from experimental noise.
the transition point is detected by the logarithm of the likelihood values.
here "likelihood value" means the generative probability of current data based on the condition of the past datasets.
if this value is low, then the current data cannot be adequately explained by past datasets.
in other words, a transition has occurred.
in the second step, relevant factors such as genes and/or metabolites related to the transitions are extracted by batch-learning self organizing mapping  using changes in expression levels  <cit> .
in summary, the lds uses compressed information called "internal states", defined as the degenerate parameters of gene expression/metabolite accumulation profiles, to detect transitions, and then bl-som generates a 'feature map', which is a two-dimensional lattice reflecting the similarity among clusters, based on the gene expression/metabolite accumulation profiles in order to visually characterize each state.
linear dynamical system  for time series analyses
lds uses internal state variables in the generative model for cellular internal state changes.
these internal states correspond to the compressed description of the observed biological system prior to adding noise factors.
the total experimental dataset of the time series and the corresponding internal state are denoted by y1:t = {y <dig>  y <dig> ..., yt} and x1:t = {x <dig>  x <dig> ..., xt}, respectively.
each element in these vectors is defined as:
yt = ' ∈ rd
xt = ' ∈ rn
where t =  <dig>   <dig> ..., t is the measurement order of the time series, d is the dimension of vector yt representing expression levels of d genes or metabolites, and n is the dimension of vector xt representing internal states.
to distinguish observational noise from true information on cellular transitions, we focus on two probability densities: the density between internal state variables p, and the density for evaluation of observational noise p. the proposed model is further defined as follows:
observational equation: yt = vxt + ηt
transition equation: xt = wxt- <dig> + εt
where v is a d × n observational matrix, w is an n × n internal state transition matrix, d-dimensional vector ηt is observational noise, n-dimensional vector εt is transition noise.
the vectors x <dig>  εt, ηt are generated according to the following equations:
x <dig> ~ nn
εt ~ nn
ηt ~ nd
the next step is to define the relevant probability densities.
np is a probability density function when a p-dimensional probabilistic vector x obeys a gaussian distribution whose mean vector is m, and covariance matrix Σ .
np≡−p/2|∑|−1/2exp⁡
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgobgtdawgaawcbagaemicaahabeaakiabcicaoiabhiha4naaeeaabagaecyba0maeiilawiaeyyeiuuaeiykakcacaglhwoacqghhjiucqggoaakcqaiyagmiigacqwfapaccqggpaqkdaahaawcbeqaaiabgkhitiabdchawjabc+caviabikdayaaakmaaemaabagaeyyeiuoacaglhwuaayjcsdwaawbaasqabeaacqghsislcqaixaqmcqggvawlcqaiyagmaagccygglbqzcqgg4baecqggwbaccqggbbwwcqghsisldawcaaqaaiabigdaxaqaaiabikdayaaacqggoaakcqwh4baecqghsislcqwhtbqbcqggpaqkcqggnawjcqqhjowucqggoaakcqwh4baecqghsislcqwhtbqbcqggpaqkcqggdbqxaaa@6259@
we assume that the observational and internal transition noises are both gaussian, and therefore the relationship is a first-order markov process .
p = pp
the model parameter of – is defined as the parameter set θ:
θ = {μ <dig>  σ <dig>  w, σε, v, ση}
note that the model corresponds to a kalman filter when θ is known   <cit> .
the initial state x <dig> is defined as:
p = nn
from equations  and , the following function is obtained:
p = nn
from equations  and , the following function is obtained:
p = nd
using these results, the following joint probability is obtained:
p=p{∏t=2tp}{∏t=1tp}
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbaccqggoaakcqwgzbqwdawgaawcbagaegymaejaeiooaojaemivaqfabeaakiabcycasiabdifaynaabaaaleaacqaixaqmcqgg6agocqwgubavaeqaaowaaqqaaeaaiigacqwf4oqcaiaawea7aiabcmcapiabg2da9iabdchawjabcicaoiabhiha4naabaaaleaacqaixaqmaeqaaowaaqqaaeaacqwf4oqcaiaawea7aiabcmcapmaacmaabawaaybcaeqaleaacqwg0badcqgh9aqpcqaiyagmaeaacqwgubava0qaaiabg+givdaakiabdchawjabcicaoiabhiha4naabaaaleaacqwg0badaeqaaowaaqqaaeaacqwh4baedawgaawcbagaemidaqnaeyoei0iaegymaedabeaakiabcycasiab=h7axbgaay5bsdgaeiykakcacagl7bgaayzfaawaaiwaaeaadagfwbqabsqaaiabdsha0jabg2da9iabigdaxaqaaiabdsfaubqdbagaey4diunaaogaemicaanaeiikagiaecyeak3aasbaasqaaiabdsha0bqabagcdaabbaqaaiabhiha4naabaaaleaacqwg0badaeqaaogaeiilawiae8hudehacaglhwoacqggpaqkaiaawuhacagl9baaaaa@7622@
the parameter optimization follows a standard em algorithm .
criterion for detecting internal state transitions
using the resulting estimated parameter, the log-likelihood with respect to the present time point t when all time points are given is defined by equation :
log lt = log p
this is calculated using the e-step formula  after parameter estimation using the kalman filter.
when the log-likelihood value log lt becomes much lower than log lt- <dig>  then yt cannot be explained by y1:t- <dig>  i.e., the cellular internal state has changed at time t. in this study, the point at which the log-likelihood value becomes relatively low between whole time points is defined as the state transition point.
if the log-likelihood value remains low over a certain period, then the cells are changing their states continuously during that period.
analysis of the bacillus subtilis data
we first analyzed the relationship of cell population to state transition time on transcriptome data of bacillus subtilis .
here, the exponential growth phase and stationary phase are commonly used microbiology terms referring to the state of the cellular population, as measured by the optical density .
the transition from exponential growth to the stationary phase was observed in  <dig> culture media: lysogeny broth , minimum glucose medium , glucose starvation , phosphate starvation , competence medium , difco sporulation medium , competence sporulation medium  and dsm plus glucose glutamine .
we confirmed that the log-likelihood index produced by lds was smaller at the transition time between two phases.
next, we fitted the index calculated by the model to the phase transition data.
for cell populations growing under two culture conditions, namely lb  and mgm , we found that bl-som yielded different classification results for gene expression .
this result indicates that expression of the genes responsible for the transition varied between the different environmental conditions, although their transitions appeared similar.
for cells grown in either csm or dsm, two transition points for sporulation were detected.
the first was the well-known transition from exponential growth to the stationary phase.
however, the second was a novel transition detected by this approach.
at the first transition in csm at around time point  <dig>  log-likelihood values show a sustained drop.
the analysis suggests that cells take a long time to adapt to the csm culture environment.
the second transition point in the sporulation media was further investigated by analysis of feature maps generated by bl-som  <cit> .
the candidate genes for the second transition were those activated just before the transition point and repressed soon after the transition point .
these genes are listed in table  <dig> and include those related to lysis of the mother cell, such as cwlh.
thus, the second transition corresponded to mother cell lysis  <cit> , a type of apoptosis.
using the analytical approach described here, we not only succeeded in detecting the well-known transition from exponential growth to the stationary phase, but also identified another, novel transition point.
this result suggests the possibility that, even in periods that are assumed to be eventless, cells may be invoking their adaptive systems.
analysis of the arabidopsis data
as described in the methods section, we analysed changes in gene expression and metabolite accumulation in arabidopsis plants following their transfer to sulfur-deficient conditions.
we detected a transition between  <dig> and  <dig> hours in both gene expression and metabolite accumulation profiles in both leaves and roots.
in addition, we detected a second transition at the final time point  in the metabolite accumulation profile in roots .
at the transition point of 12– <dig> hr, glucosinolate biosynthesis was decreased in leaves and anthocyanin biosynthesis was initiated in roots.
the predicted transitions obtained by this analysis are consistent with those identified previously  <cit> , indicating that our method can reliably identify candidate genes and metabolites involved in transition points.
the transition time point detected for root metabolites at the end of the experiment  showed that even after this period of time roots of a. thaliana continued to change in response to sulfur deprivation, at least in terms of metabolite accumulation.
on the basis of the estimated transition results, coupled with prior knowledge and the feature map subtraction obtained by bl-som, we identified metabolites whose accumulation profiles showed changes that coordinated with the predicted transition point.
these metabolites were found to be involved in biochemical pathways that are critical for the response to sulfur deprivation stress, for example, glucosinolate biosynthesis in leaf and anthocyanin biosynthesis in root  <cit> .
our results also suggested the presence of lipid metabolic responses in arabidopsis to sulfur stress.
the accumulation patterns of detected ion peaks whose mass-to-charge ratio  values corresponded to molecular species with various acyl groups, such as phosphatidylglycerol, phosphatidylethanolamine, phosphatidylcholine, phosphatidic acid, and sulfoquinovosyl diacylglycerol, are shown in figure  <dig>  because the accumulation profiles of these compounds showed similar patterns, we predict that lipid biosynthesis was also co-ordinately repressed at the transition at  <dig> hr.
a profile of sulfate accumulation was generated using capillary electrophoresis .
in comparison with the control condition, the accumulation of sulfate was strongly repressed immediately after the shift to sulfur deficiency.
under sulfur deprivation, it was believed that sulfate levels  would only decrease.
during the transition period from  <dig> to  <dig> hr after the shift to sulfur deficiency, however, sulfate levels temporarily ceased declining and stayed relatively constant as compared to the control.
from these results, we hypothesize that sulfate is in an active form and is distributed throughout the plant at the transition time.
during this period, in order to maintain the intracellular environment, membrane lipids are temporarily degraded and re-synthesized after the transition.
this suggestion is consistent with the reported decrease in lipids under conditions of sulfur starvation  <cit> .
CONCLUSIONS
in summary, by using a linear dynamical system, we have identified transition times in the adaptation processes of bacillus subtilis and arabidopsis thaliana to environmental stresses.
by focusing on transition information based on a well-defined probabilistic index, we obtained novel observations on apoptosis in bacillus subtilis and the regulation of lipid metabolism connected with sulfur-stress responses in arabidopsis thaliana.
as this approach uses probabilistic values to detect the transitions, the results are objectively supported without the risk of misinterpretation due to experimental noise.
the results of this approach will enable us to more effectively design experiments specifically tailored for functional identification of genes and metabolites.
by obtaining time series data with higher temporal resolution around the transition time points, we can obtain more precise information on the details of the responses.
the strategy described here was successful in identifying a small number of candidate genes and metabolites, from the vast number of genes and metabolites in comprehensive "omics" databases.

 biochemical systems in living cells are robust and flexible. investigating the responses of cells to environmental changes typically requires a system-level analysis of the interactions between the various molecular elements that comprise the system. a key step to analyze system responses to environmental changes is identifying large state changes or "transitions". a statistical method that could detect such transitions would be a powerful analytical tool for finding important factors in large-scale profiles, such as variations in gene expression. previous analyses of gene expression profiles have often made use of graphical models, such as bayesian networks, graphical gaussian modelling, boolean networks, and auto-regressive models. however, not many approaches have explicitly modelled observational noise. in auto-regressive analyses, for example, observational vectors yt are recursively defined by the following equation: yt = ayt- + εt where yt is an observational vector of genes or metabolites at time t, a is an observational transition matrix, and εt is gaussian noise. because this model does not distinguish observational and inherent noises, identification of transition states becomes difficult in the presence of substantial noise. here we propose an extension of the auto-regressive model, which has been modified by the addition of reduced set of internal states, as explained in the results and discussion section. we chose a mathematical model, the linear dynamical system, as the basis of our method because it does not impose any specific requirements on the data used. lds is expected to eliminate the confounding influence of observational noise in time series data. the model was applied to detect cellular state transitions in transcriptome and metabolome time series datasets from bacillus subtilis and arabidopsis thaliana maintained under stress conditions.  overview of our method our method has two steps. first, transition time points for each time series are detected using lds, which mathematically distinguishes transitional fluctuations from experimental noise. the transition point is detected by the logarithm of the likelihood values. here "likelihood value" means the generative probability of current data based on the condition of the past datasets. if this value is low, then the current data cannot be adequately explained by past datasets. in other words, a transition has occurred. in the second step, relevant factors such as genes and/or metabolites related to the transitions are extracted by batch-learning self organizing mapping using changes in expression levels. in summary, the lds uses compressed information called "internal states", defined as the degenerate parameters of gene expression/metabolite accumulation profiles, to detect transitions, and then bl-som generates a'feature map', which is a two-dimensional lattice reflecting the similarity among clusters, based on the gene expression/metabolite accumulation profiles in order to visually characterize each state. linear dynamical system for time series analyses lds uses internal state variables in the generative model for cellular internal state changes. these internal states correspond to the compressed description of the observed biological system prior to adding noise factors. the total experimental dataset of the time series and the corresponding internal state are denoted by y1:t = {y y..., yt} and x1:t = {x x..., xt}, respectively. each element in these vectors is defined as: yt =' ∈ rd xt =' ∈ rn where t =..., t is the measurement order of the time series, d is the dimension of vector yt representing expression levels of d genes or metabolites, and n is the dimension of vector xt representing internal states. to distinguish observational noise from true information on cellular transitions, we focus on two probability densities: the density between internal state variables p, and the density for evaluation of observational noise p. the proposed model is further defined as follows: observational equation: yt = vxt + ηt transition equation: xt = wxt- + εt where v is a d × n observational matrix, w is an n × n internal state transition matrix, d-dimensional vector ηt is observational noise, n-dimensional vector εt is transition noise. the vectors x εt, ηt are generated according to the following equations: x ~ nn εt ~ nn ηt ~ nd the next step is to define the relevant probability densities. np is a probability density function when a p-dimensional probabilistic vector x obeys a gaussian distribution whose mean vector is m, and covariance matrix Σ. np≡−p/2|∑|−1/2exp⁡ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgobgtdawgaawcbagaemicaahabeaakiabcicaoiabhiha4naaeeaabagaecyba0maeiilawiaeyyeiuuaeiykakcacaglhwoacqghhjiucqggoaakcqaiyagmiigacqwfapaccqggpaqkdaahaawcbeqaaiabgkhitiabdchawjabc+caviabikdayaaakmaaemaabagaeyyeiuoacaglhwuaayjcsdwaawbaasqabeaacqghsislcqaixaqmcqggvawlcqaiyagmaagccygglbqzcqgg4baecqggwbaccqggbbwwcqghsisldawcaaqaaiabigdaxaqaaiabikdayaaacqggoaakcqwh4baecqghsislcqwhtbqbcqggpaqkcqggnawjcqqhjowucqggoaakcqwh4baecqghsislcqwhtbqbcqggpaqkcqggdbqxaaa@6259@ we assume that the observational and internal transition noises are both gaussian, and therefore the relationship is a first-order markov process. p = pp the model parameter of – is defined as the parameter set θ: θ = {μ σ w, σε, v, ση} note that the model corresponds to a kalman filter when θ is known. the initial state x is defined as: p = nn from equations and, the following function is obtained: p = nn from equations and, the following function is obtained: p = nd using these results, the following joint probability is obtained: p=p{∏t=2tp}{∏t=1tp} mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqwgwbaccqggoaakcqwgzbqwdawgaawcbagaegymaejaeiooaojaemivaqfabeaakiabcycasiabdifaynaabaaaleaacqaixaqmcqgg6agocqwgubavaeqaaowaaqqaaeaaiigacqwf4oqcaiaawea7aiabcmcapiabg2da9iabdchawjabcicaoiabhiha4naabaaaleaacqaixaqmaeqaaowaaqqaaeaacqwf4oqcaiaawea7aiabcmcapmaacmaabawaaybcaeqaleaacqwg0badcqgh9aqpcqaiyagmaeaacqwgubava0qaaiabg+givdaakiabdchawjabcicaoiabhiha4naabaaaleaacqwg0badaeqaaowaaqqaaeaacqwh4baedawgaawcbagaemidaqnaeyoei0iaegymaedabeaakiabcycasiab=h7axbgaay5bsdgaeiykakcacagl7bgaayzfaawaaiwaaeaadagfwbqabsqaaiabdsha0jabg2da9iabigdaxaqaaiabdsfaubqdbagaey4diunaaogaemicaanaeiikagiaecyeak3aasbaasqaaiabdsha0bqabagcdaabbaqaaiabhiha4naabaaaleaacqwg0badaeqaaogaeiilawiae8hudehacaglhwoacqggpaqkaiaawuhacagl9baaaaa@7622@ the parameter optimization follows a standard em algorithm. criterion for detecting internal state transitions using the resulting estimated parameter, the log-likelihood with respect to the present time point t when all time points are given is defined by equation: log lt = log p this is calculated using the e-step formula after parameter estimation using the kalman filter. when the log-likelihood value log lt becomes much lower than log lt- then yt cannot be explained by y1:t- i.e., the cellular internal state has changed at time t. in this study, the point at which the log-likelihood value becomes relatively low between whole time points is defined as the state transition point. if the log-likelihood value remains low over a certain period, then the cells are changing their states continuously during that period. analysis of the bacillus subtilis data we first analyzed the relationship of cell population to state transition time on transcriptome data of bacillus subtilis. here, the exponential growth phase and stationary phase are commonly used microbiology terms referring to the state of the cellular population, as measured by the optical density. the transition from exponential growth to the stationary phase was observed in culture media: lysogeny broth, minimum glucose medium, glucose starvation, phosphate starvation, competence medium, difco sporulation medium, competence sporulation medium and dsm plus glucose glutamine. we confirmed that the log-likelihood index produced by lds was smaller at the transition time between two phases. next, we fitted the index calculated by the model to the phase transition data. for cell populations growing under two culture conditions, namely lb and mgm, we found that bl-som yielded different classification results for gene expression. this result indicates that expression of the genes responsible for the transition varied between the different environmental conditions, although their transitions appeared similar. for cells grown in either csm or dsm, two transition points for sporulation were detected. the first was the well-known transition from exponential growth to the stationary phase. however, the second was a novel transition detected by this approach. at the first transition in csm at around time point log-likelihood values show a sustained drop. the analysis suggests that cells take a long time to adapt to the csm culture environment. the second transition point in the sporulation media was further investigated by analysis of feature maps generated by bl-som. the candidate genes for the second transition were those activated just before the transition point and repressed soon after the transition point. these genes are listed in table and include those related to lysis of the mother cell, such as cwlh. thus, the second transition corresponded to mother cell lysis, a type of apoptosis. using the analytical approach described here, we not only succeeded in detecting the well-known transition from exponential growth to the stationary phase, but also identified another, novel transition point. this result suggests the possibility that, even in periods that are assumed to be eventless, cells may be invoking their adaptive systems. analysis of the arabidopsis data as described in the methods section, we analysed changes in gene expression and metabolite accumulation in arabidopsis plants following their transfer to sulfur-deficient conditions. we detected a transition between and hours in both gene expression and metabolite accumulation profiles in both leaves and roots. in addition, we detected a second transition at the final time point in the metabolite accumulation profile in roots. at the transition point of 12– hr, glucosinolate biosynthesis was decreased in leaves and anthocyanin biosynthesis was initiated in roots. the predicted transitions obtained by this analysis are consistent with those identified previously, indicating that our method can reliably identify candidate genes and metabolites involved in transition points. the transition time point detected for root metabolites at the end of the experiment showed that even after this period of time roots of a. thaliana continued to change in response to sulfur deprivation, at least in terms of metabolite accumulation. on the basis of the estimated transition results, coupled with prior knowledge and the feature map subtraction obtained by bl-som, we identified metabolites whose accumulation profiles showed changes that coordinated with the predicted transition point. these metabolites were found to be involved in biochemical pathways that are critical for the response to sulfur deprivation stress, for example, glucosinolate biosynthesis in leaf and anthocyanin biosynthesis in root. our results also suggested the presence of lipid metabolic responses in arabidopsis to sulfur stress. the accumulation patterns of detected ion peaks whose mass-to-charge ratio values corresponded to molecular species with various acyl groups, such as phosphatidylglycerol, phosphatidylethanolamine, phosphatidylcholine, phosphatidic acid, and sulfoquinovosyl diacylglycerol, are shown in figure because the accumulation profiles of these compounds showed similar patterns, we predict that lipid biosynthesis was also co-ordinately repressed at the transition at hr. a profile of sulfate accumulation was generated using capillary electrophoresis. in comparison with the control condition, the accumulation of sulfate was strongly repressed immediately after the shift to sulfur deficiency. under sulfur deprivation, it was believed that sulfate levels would only decrease. during the transition period from to hr after the shift to sulfur deficiency, however, sulfate levels temporarily ceased declining and stayed relatively constant as compared to the control. from these results, we hypothesize that sulfate is in an active form and is distributed throughout the plant at the transition time. during this period, in order to maintain the intracellular environment, membrane lipids are temporarily degraded and re-synthesized after the transition. this suggestion is consistent with the reported decrease in lipids under conditions of sulfur starvation.  in summary, by using a linear dynamical system, we have identified transition times in the adaptation processes of bacillus subtilis and arabidopsis thaliana to environmental stresses. by focusing on transition information based on a well-defined probabilistic index, we obtained novel observations on apoptosis in bacillus subtilis and the regulation of lipid metabolism connected with sulfur-stress responses in arabidopsis thaliana. as this approach uses probabilistic values to detect the transitions, the results are objectively supported without the risk of misinterpretation due to experimental noise. the results of this approach will enable us to more effectively design experiments specifically tailored for functional identification of genes and metabolites. by obtaining time series data with higher temporal resolution around the transition time points, we can obtain more precise information on the details of the responses. the strategy described here was successful in identifying a small number of candidate genes and metabolites, from the vast number of genes and metabolites in comprehensive "omics" databases. 
modelling of time series data should not be an approximation of input data profiles, but rather be able to detect and evaluate dynamical changes in the time series data. this method was applied to time series datasets from bacillus subtilis and arabidopsis thaliana grown under stress conditions; in the former, only gene expression was studied, whereas in the latter, both gene expression and metabolite accumulation. here we demonstrate the effectiveness of a markov model, named the linear dynamical system, to simulate the dynamics of a transcript or metabolite time series, and propose a probabilistic index that enables detection of time-sensitive changes. objective criteria that can be used to evaluate dynamical changes in data are therefore important to filter experimental noise and to enable extraction of unexpected, biologically important information. our method not only identified well-known changes in gene expression and metabolite accumulation, but also detected novel changes that are likely to be responsible for each stress response condition.
 modelling of time series data should not be an approximation of input data profiles, but rather be able to detect and evaluate dynamical changes in the time series data. objective criteria that can be used to evaluate dynamical changes in data are therefore important to filter experimental noise and to enable extraction of unexpected, biologically important information.  here we demonstrate the effectiveness of a markov model, named the linear dynamical system, to simulate the dynamics of a transcript or metabolite time series, and propose a probabilistic index that enables detection of time-sensitive changes. this method was applied to time series datasets from bacillus subtilis and arabidopsis thaliana grown under stress conditions; in the former, only gene expression was studied, whereas in the latter, both gene expression and metabolite accumulation. our method not only identified well-known changes in gene expression and metabolite accumulation, but also detected novel changes that are likely to be responsible for each stress response condition.  this general approach can be applied to any time-series data profile from which one wishes to identify elements responsible for state transitions, such as rapid environmental adaptation by an organism. 