BACKGROUND
cells have developed an extensive network of signalling pathways in order to mediate appropriate responses to varying nutrient availability and concentrations in the surrounding environment.
glucose, a rapidly fermentable carbon source, is the preferred carbon source of saccharomyces cerevisiae cells and regulates numerous nutrient signalling pathways  <cit> .
glucose is taken up by the yeast cell through multiple hexose transporters, which have a broad range of different affinities and transport capacities.
this enables the yeast cell to respond to a wide range of glucose concentrations  <cit> .
glucose sensing pathways that employ membrane-localized receptors, such as in the snf3-rgt <dig> pathway, are relatively well understood.
however, the sensing mechanism of intracellular glucose or metabolites of glycolysis is poorly explained  <cit> .
to study how metabolism is connected to these signalling pathways has proven to be a major challenge since it is difficult to uncouple signalling from metabolism.
the ampk/snf <dig> system controls energy homeostasis and is best known for its function in glucose signalling.
the snf <dig> protein kinase complex, which consists of three subunits, is activated by glucose depletion through phosphorylation  <cit> .
it is not well known how metabolism is connected with the activity of the snf <dig> complex.
when rapidly-fermentable sugars are available snf <dig> becomes dephosphorylated.
for the establishment of glucose repression only uptake and phosphorylation of glucose is required, but no further glucose metabolism  <cit> .
glucose repression is regulated, at least, at two different steps, i.e.
control of snf <dig> activation and its function on downstream targets such as mig <dig>  <cit> .
it has been suggested that the snf1-mig <dig> pathway works in a continuous on-off manner  <cit> .
however, evidence has emerged that mig <dig> shuttles in and out of the nucleus and shows transient behaviour at a single cell level  <cit> .
this indicates that the dynamics of the snf1-mig <dig> at single cell level is less simple than previously assumed.
to study the influence of glucose metabolism on the complex single cell dynamics of the snf <dig> pathway we decided to control the uptake of glucose into the cell while employing a microfluidic system to control the extracellular glucose concentration.
we show that glucose repression is regulated by glucose flux rather than the absolute glucose concentrations and that the snf1-mig <dig> system is closely regulated by glycolytic flux.
in our experiments we observed cell-to-cell variability.
to explain this variability, we developed a dynamical and nonlinear mixed effect model.
dynamical models of signalling pathways in yeast have previously been employed to describe the behaviour of populations of cells .
nonlinear mixed effects  modelling is a theoretical approach that provides a framework to account for cell-to-cell variability  <cit> .
nmle modelling is traditionally used in pharmacokinetic and pharmacodynamics studies since it allows for the analysis of sparse and unbalanced datasets  <cit> .
nmle has been proposed and used to model dynamic single cell data  <cit> .
recently, a simple phenomenological model describing the snf1-mig <dig> pathway using nlme approach has been constructed, capturing dynamics of mig <dig> localization, without taking into account parameter variabilities  <cit> .
our integrative approach reveals that the main source of variability is linked to transport of mig <dig> in and out of the nucleus.
our experimental data indicate that rapid degradation and cell size cause no or little contribution to the cell-to-cell variability, while variation in expression and translation of the hexose transporters is a possible source of cell-to-cell variability.
RESULTS
single cell time-scale fluorescence microscopy enables dynamic studies of the snf1-mig <dig> pathway
a high control of the cell environment is needed in order to study nutrient responses on single cells.
a microfluidics device allows for a fast and precise switch between different media and enables the nutrients composition in the media to be kept constant.
here we used a three inlet-channel microfluidics setup  <cit> , to achieve a high control of the cell environment and to study the influence of glucose concentration on saccharomyces cerevisiae.
upon deactivation of snf <dig>  mig <dig> is dephosphorylated and subsequently moves into the nucleus and is therefore a suitable marker for real-time snf <dig> activity  <cit> .
the nucleo-cytosolic shuttling of the transcription factor mig <dig> fused to a green fluorescent protein  served as single cell readout .
in the wild type  mig <dig> localized to the nucleus after being exposed to a glucose concentration of  <dig>  mm .
it has been shown that mig <dig> is phosphorylated in this range of glucose concentration  <cit> .
following shifts to higher glucose concentrations more cells respond to the upshift and the mig <dig> fluorescence intensity in the nucleus is stronger indicating that higher glucose concentrations result in a higher proportion of mig <dig> molecules in the nucleus .
the snf1-mig <dig> system responds to exposure to glucose and the degree of response is sensitive for the absolute glucose level which the cell is exposed to.
these results are consistent with those of a similar study in saccharomyces cerevisiae with a different genetic background  <cit> .
the response to increased glucose concentrations occurs rapidly after upshift, pointing to the fast adaption of cells to nutrients in the environment.fig.
<dig> localization of mig <dig> over time expressed as nuclear fluorescence intensity divided by the cytosolic fluorescence intensity.
each graph depicts the results of one experiment.
the y axis is distributed logarithmically.
each grey line represents the trace of one single cell and the average of all cells is represented with a thicker blue line.
between  <dig> to  <dig> cells were analysed in each experiment.
the different strains are displayed vertically  , hxt <dig> , hxt <dig> ) and the different concentrations are displayed horizontally
glucose uptake through only low affinity transporters results in a strong response in the snf1-mig <dig> pathway
the data obtained from the wild-type strain raised the question whether glycolytic flux correlates directly with snf1-mig <dig> pathway activity.
this would imply that the snf1-mig <dig> pathway is controlled by glucose metabolism by a quantitative sensor system.
to address this question we chose to control the flux through glycolysis via the glucose uptake into the cell.
a large set of isogenic strains expressing only a single hexose transporter is available  <cit> .
we employed strains that express either a low affinity or high affinity glucose transporter, respectively, under the control of the promotor of a high affinity transporter.
hxt <dig> is a low affinity transporter that is normally highly expressed under high glucose conditions; yeast cells expressing only hxt <dig> displays a high glucose transport capacity and a higher v
max then the wild type strain  <cit> .
the hxt <dig> cells already display mig <dig> nuclear accumulation when they were exposed to an upshift from  <dig> to  <dig>  mm glucose as in the wt however the fraction of the whole population displaying nuclear localization never exceeds 50%.
while for an upshift to 11 mm glucose a higher fraction of cells display nuclear accumulation but this nuclear accumulation never exceeds 80% .
while in the wt for both the upshift to  <dig>  mm and 11 mm more than 80% of the population displays nuclear accumulation.
for the other upshift to higher glucose concentration nearly all cells display nuclear localization.
the hxt <dig> strain displayed a large cell-to-cell variability of cells either responding or not responding to an upshifts of  <dig>  mm, 11 mm, and  <dig>  mm in glucose concentration in comparison with the wt cells .
only after an upshift to  <dig>  mm glucose almost all the cells of the population show mig <dig> localization in the nucleus.
at higher glucose concentrations, 55 mm or 220 mm glucose, a higher proportion of mig <dig> is localized in the nucleus than at lower concentrations.
the upshifts to 55 mm and 220 mm result in a higher mig <dig> nuclear accumulation compared to the wt and displays a higher cell variability compared to the other strains .
the response times of mig <dig> nuclear accumulation appear to be remarkably similar for all responding cells under all glucose concentrations, after 1 min the max is reached for the fraction of cells displaying mig <dig> nuclear localization of all strains and all upshift conditions .
overall, the data shows that the response characteristics of the snf1-mig <dig> system correlates well with the kinetic characteristics of the hxt <dig> transporter as the snf1-mig <dig> strain displays low response at an upshift to low glucose concentration but a strong response to upshift to high glucose concentrations.
the high affinity transporter causes a weak mig <dig> response to all glucose concentration upshifts
the hxt <dig> high affinity transporter is highly expressed at very low glucose concentrations.
the hxt <dig> strain displays a lower glucose uptake capacity than the hxt <dig> strain.
therefore the glucose uptake capacity is saturated at low glucose concentrations  <cit> .
the majority of the population shows mig <dig> nuclear localization after the cells are exposed to growth media containing glucose, in contrast with the low fraction of responders in the hxt <dig> strain .
however, unlike in the wt and the hxt <dig> strain the response is very similar for all glucose concentrations, and the intensity of mig <dig> in the nucleus is the same for all upshifts.
already at an upshift from  <dig> to  <dig>  mm glucose mig <dig> has reached a maximum mig <dig> nuclear localization for the hxt <dig> strain.
hence, even in the hxt <dig> strain the snf1-mig <dig> response characteristic corresponds to the properties of the transport system with high affinity but low capacity.
neither regulated degradation nor cell size is a major contributor to cell heterogeneity
cell-to-cell variability in mig <dig> localization upon changing glucose concentration has been reported  <cit> .
these studies however do not examine the source of the observed cell-to-cell variability, therefore we set out to explore the source of the variation.
it is known that the high affinity transporters hxt <dig> and hxt <dig> are internalized and degraded when cells are exposed to high concentration of glucose  <cit> .
degradation of hxt <dig> requires inactivation of torc <dig>  <cit> .
also hxt <dig> is actively internalized and degraded if glucose is depleted, an effect possibly mediated by downregulation of pka  <cit> .
internalization of the hexose transporters for catabolic degradation could lead to a decrease of glucose import.
we observed in the upshift experiments to higher glucose concentrations a slight decrease of the median after the nuclear localization reached its maximum value for all strains .
therefore, we reasoned that rapid activity adjustments of the hexose transporters could impact the mig1-snf <dig> pathway.
since this drop in mig <dig> nuclear localization differed between cells this mechanism could be a contributing factor to the observed cell-to-cell variability.
we exposed yeast cells expressing hxt7-gfp under the native promotor grown on 3% ethanol to 220 mm glucose and followed the localization of hxt7-gfp for 15 min .
the data was quantified by measuring the fluorescence intensity along an intersection through the cell.
we observed no significant change in the localization of hxt7-gfp during the experiment .fig.
<dig> study of the cell-to-cell variability observed in the snf1/mig <dig> system.
hxt7-gfp before and following a switch from ethanol media to media containing 220 mm glucose.
time lapse microscopic images, upper images show hxt7-gfp, the lower images show brightfield.
fluorescence intensity along an intersection through the yeast cells.
the fluorescence intensity is higher at the points the intersection line crosses the cell membrane and does not change over time.
the result of only one cell is displayed but multiple cells were analyzed and none of the cells showed a decrease in membrane localization of the hxt <dig> transporter after 15 min following the shift to glucose media.
the ratio 15 min after glucose upshift plotted over the cell size for the hxt <dig> strain.
the cell size is plotted on the x-axes.
as a measurement for the mig1-snf <dig> pathway response we chose the mig <dig> nuclear/cytosolic ratio.
upshifts to higher glucose concentration;  <dig> to 220 mm ,  <dig> to 55 mm  and  <dig> to  <dig>  mm glucose  result in higher final ratio while upshifts to lower glucose concentration  <dig> to 11 mm glucose ,  <dig> to  <dig>  mm  and  <dig> to 0 mm glucose  display a lower final ratio.
hxt7-gfp pregrown overnight in 3% ethanol media.
upper image shows the bright field image, lower image shows the cellular distribution of hxt7-gfp
we next asked whether cell size could influence the cell-to-cell variability.
fluctuations in cellular states, such as cell size, can cause extrinsic noise which could lead to the observed cell-to-cell variability  <cit> .
we therefore decided to test the influence of cell size by plotting the response of the snf1-mig <dig> pathway over the cell size.
as measurement for the snf1-mig <dig> pathway we used the mig <dig> fluorescence intensity ratio 15 min after the upshift.
the final ratio for the hxt1-strain did not show any correlation between the cell size and the snf1-mig <dig> pathway activity .
instead, the final ratio showed an even distribution around the average cell size with the values for the upshifts towards higher glucose concentration position higher along the y-axes.
this result excludes cell size as a major determinant for the cell-to-cell variability on the relatively short time frame of the experiments.
stochastic expression is a plausible cause of cell-to-cell variability in the mig1-snf <dig> system
cell-to-cell variability in dynamic adaptation responses might be caused by, among others, stochastic transcription activity  <cit> .
therefore, we reasoned that the expression pattern of the hexose transporters could lead to the cell-to-cell variability.
we therefore grew a strain expressing hxt7-gfp on 3% ethanol and followed the population distribution of hxt7-gfp.
the fluorescence intensity of hxt7-gfp differed significantly between the cells .
the lowest observed hxt7-gfp fluorescence intensity was only 10% of the maximal observed fluorescence intensity of hxt7-gfp.
the amount of hxt <dig> transporter molecules within each single cell varies and can therefore be a major contributor to the observed cell-to-cell variability.
these results show that, under our experimental conditions it is likely that expression and translation of hexose transporters is a major contributor to the observed cell-to-cell variability.
a mixed effect model suggests mig <dig> dephosphorylation as a new regulatory step
to better understand the effect of glucose upshift on the snf1-mig <dig> pathway we developed a mathematical model of glucose flux which controls snf <dig> phosphorylation and consequently mig <dig> localization .
the aim was to investigate if glucose uptake was able to regulate mig <dig> localization by controlling only one step in the snf1-mig <dig> regulatory system.
we assumed this step to be dephosphorylation of snf <dig>  since several publications identified this step to be controlled by the adp/atp ratio .
the adp/atp ratio is indirectly determined by glucose uptake and glycolysis, therefore the binding of adp to the snf <dig> complex could be the connection between glycolysis and the snf1-mig <dig> system.
nlme modelling was implemented in order to simulate the dynamics of mig <dig> localization for different yeast strains in various experimental conditions .
the model captures the characteristics of our experimental data .
by simulating parameters for multiple cells we could produce a distribution of the parameters and we compared this distribution between the wild type, hxt <dig> and hxt <dig> strains .
the model predicts that the vmsi, the parameter for snf <dig> dephosphorylation, increases with upshifts to increasing glucose concentrations .
however, there is no significant difference between the different strains.
the model suggests snf <dig> dephosphorylation to be active immediately after glucose is imported into the cell, but this process is influenced neither by glucose concentration nor by the strain it was simulated in.
this suggests that snf <dig> dephosphorylation is regulated more in an on/off fashion rather than in a dynamic fashion.
however, the parameter for mig <dig> dephosphorylation, vmd, did display the characteristics of the different strain .
at low glucose upshift the simulated vmd parameters of the hxt <dig> strain were higher than the hxt <dig> parameters.
only at the higher upshift concentrations the vmd parameters simulated for hxt <dig> strain where much higher than the vmd parameters simulated for hxt <dig> strain.
from this, the model suggested mig <dig> dephosphorylation as a regulatory step which is controlled by glucose flux.fig.
<dig> dynamic and nlme modelling of the snf1/mig <dig> pathway.
a schematic representation of the model.
the model consists of three main parts, namely the activity of glucose, the activity of snf <dig> and the activity of mig <dig>  b simulation of the distribution of the random parameters for the hxt <dig> strain.
the columns indicate the extracellular glucose concentration, ranging from  <dig>   <dig>   <dig> to 220 mm which are illustrated from the left to the right in the figure.
each heat map is generated by drawing  <dig> mixed effect random terms, that is η ∼ n, corresponding to the parameter vectors from the generated parameter distributions for the various strains and glucose concentrations.
the heat map displays various parameters on the y-axis, the individuals on the x-axis and the magnitude of the random terms are indicated by the colour scale shown above the figure.
the colour scale ranges from  <dig> to  <dig> where a red colour corresponds to a high random term and a blue colour correspond to a low value of the random term.
the white fields correspond to the parameters connected to the hexose transporters that are not active in hxt <dig> strains
a mixed effect model identifies hypothetical sources of variability in the snf1-mig <dig> regulatory module
since our model takes cell-to-cell variability into account we could use it to identify which parameters display highest cell-to-cell variability and under which conditions.
the wild type strain displayed increasing variance with upshift to the higher glucose concentrations 55 mm and 220 mm .
the highest variability was observed in the hxt <dig> strain with the upshift to the higher glucose concentrations .
the hxt <dig> transporter strain displayed a large cell-to-cell variability following shift to high glucose but a small cell-to-cell variability after upshift to low glucose.
the simulated variance of the hxt <dig> strain was lower in comparison with the wild type and hxt <dig> strains and did not increase with upshift to higher glucose levels .
the simulated variance is in correlation with the observed variance seen in the experimental data .
the model also allows us to predict the most important parameters that are the major contributor to the cell-to-cell variability.
we compared the magnitude of perturbation of each parameter for the simulation of the wild type strain in the upshift from  <dig> to 220 mm glucose .
a parameter which displays a higher perturbation error has a higher variability in that parameter.
parameters involved in dephosphorylation events have been ranked as low significance , suggesting that the dephosphorylation of mig <dig> and snf <dig> after glucose upshift is a minor contributor to the observed cell-to-cell variability.
the parameters which display the highest variance are kex <dig> and kim <dig>  which account for transport of mig <dig> in and out of the nucleus.
the respective perturbation errors for these two parameters are in the order of 10− <dig> while the other parameters have perturbation errors in the order of 10− <dig> and smaller.
indeed, it has been shown that movement of mig <dig> in and out of the nucleus shows considerable variability between cells  <cit> .
those data suggest that variability in the nucleocytoplasmic transport of mig <dig> would be the major contributor to cell-to-cell variability and not the dephosphorylation events following glucose upshifts.
discussion
it is well established that the snf1-mig <dig> system and hence the nuclear accumulation of mig <dig> are controlled by the level of glucose in the growth medium.
we have previously reported that glucose derepression senses glucose concentration in a highly dynamic fashion  <cit> .
but it remained unclear whether the observed dynamics were correlated with adaptations of sugar transport and glycolytic metabolism.
previous modelling approaches have suggested the importance of the kinetics of the glycolytic flux in signalling of glucose  <cit> .
to elucidate the influence of sugar transport on the snf1-mig <dig> pathway we studied the response of three different s. cerevisiae strains with different glucose uptake capacity when exposed to an upshift in glucose concentration.
our data show that yeast cells response rapidly to changes in glucose concentration and that there is little to no cell-to-cell variability in response time.
even between the strains and between glucose concentrations we observed no significant difference in response time.
this indicates that yeast cells are programmed and determined to rapidly respond to a change in glucose concentration.
this behaviour allows yeast cells to rapidly adapt to new environmental conditions and thereby to potentially outcompete other species.
although single yeast cells induced a response at almost the same time, there was difference between the cells in the magnitude of their response.
our data showed that the mig <dig> localization pattern after glucose upshift correlates well with the glucose uptake characteristics of the respective yeast strains.
it has already been shown that establishment of glucose repression is driven by sensing of an intracellular metabolite rather than extracellular glucose  <cit> .
our single cell data confirms that glucose repression is sensed through an intracellular metabolite rather than extracellular glucose.
it has been shown that in the strains we tested the reduced glucose uptake capacity results in reduced glycolytic rate  <cit> .
the data indicate a closer link between the glucose uptake rate, which determines the glycolytic rate, and the activity of the snf1/mig <dig> system than previously anticipated.
this indicates that the signal which originates from the glycolytic flux is very dynamic in response to the changing glycolytic rate.
a source for this signal might be the adp/atp ratio  <cit> .
since, it has been reported that adp binds to the regulatory subunit snf <dig> and this binding leads to protection of the catalytic subunit snf <dig> from dephosphorylation, which leads to increased snf <dig> activity  <cit> .
the turn-over of adp to atp or vice versa could be a sensor for the glycolytic rate.
also hexokinase  <dig>  an enzyme part of the glycolytic pathway, has been suggested to serve as a sensor for internal glucose by serving as a threshold for the interaction between mig <dig> and snf <dig>  <cit> .
our data showed a considerable cell-to-cell variability in glucose sensing.
this variability could have considerable impact on the snf1/mig <dig> pathway.
we investigated the causes of the cell-to-cell variability that was observed in our initial experiments.
our results indicate that this behaviour is not caused by the size of the cell or rapid activity adjustments of the hexose transporters.
a large variation is observed between the concentrations of hxt <dig> in the cells and hence the high cell-to-cell variability can be caused by the expression and/or translation of hexose transporter hxt <dig>  in a yeast single cell study of the shift between sulphur sources it was observed that the transcriptional adaptation displayed a large cell-to-cell variability  <cit> .
the variability in expression and translation of hexose transporters may cause a different uptake capacity within a population and consequently variability in further glucose metabolism.
such variability in glycolysis might lead to cells responding differently to nutritional changes and different subpopulations.
systems to restore unbalanced dynamics in glycolysis have already been reported  <cit> .
data obtained by single cell techniques coupled with mathematical modelling offer an opportunity to understand the variability within a population of cells.
this work employs nlme modelling in larger dynamical models providing a framework to deeper investigate and identify source of cell-to-cell variability in snf1-mig <dig> signalling pathway.
the variability in our model corresponded to that observed in our experimental data.
therefore, we can conclude that the implemented model can in fact account for the cell-to-cell variability of the nuclear/cytoplasmic mig <dig> ratio.
our approach classifies certain parameters as low significant, having very small perturbation errors indicating that they could potentially be neglected without losing the predictive capability of the model .
this partly explains that the simplified model proposed by almqvist and colleagues  <cit>  can capture the dynamics of the mig <dig> localization, but fails to provide more information about the relationship between the parameters.
to our knowledge, this is the first mixed effect model which is complex enough to allow identification of sources of cell-to-cell variability in signalling pathways.
our model suggested a new regulatory step at the level of mig <dig> dephosphorylation and this step would be controlled by glycolytic flux.
it is known that mig <dig> is dephosphorylated upon glucose addition leading to glucose derepression  <cit> .
however, the glucose activated process which dephosphorylates mig <dig> has not yet been clearly identified.
for instance, it is unclear if mig <dig> dephosphorylation is regulated by glucose or by a constitutive phosphatase counteracting snf <dig> activity.
our computational approach suggests that mig <dig> dephosphorylation is regulated and therefore probably an active step.
the model suggest that dephosphorylation of snf <dig> is regulated by absolute glucose concentrations while mig <dig> dephosphorylation is regulated in a more dynamic way related to glucose flux.
we were able to identify import and export of mig <dig> in and out of the nucleus as a possible source for cell-to-cell variability.
both mig <dig> import and export are regulated by glucose through phosphorylation and dephosphorylation of mig <dig>  <cit> .
mig <dig> shuttles in and out of the nucleus regardless of the glucose concentration and flip/frap experiments have shown that there is a considerable cell-to-cell variability in mig <dig> nuclear translocation  <cit> .
therefore, it is not entirely unexpected that this step is predicted to encompass a high cell-to-cell variability.
CONCLUSIONS
this work links the glucose flux to snf1-mig <dig> signalling.
although the control of the snf1-mig <dig> regulatory module is complicated by crosstalk with other glucose sensing signalling systems, we suggest that glycolytic metabolic reactions are playing a major role in the regulation of mig <dig> localization.
we show that the initial response time of snf1-mig <dig> pathway displays no cell-to-cell variability.
we further developed and presented a modelling approach which can model the cell variability observed in the data.
most importantly, we demonstrate the close correlation between glycolytic metabolism and glucose signalling metabolism.

 cells have developed an extensive network of signalling pathways in order to mediate appropriate responses to varying nutrient availability and concentrations in the surrounding environment. glucose, a rapidly fermentable carbon source, is the preferred carbon source of saccharomyces cerevisiae cells and regulates numerous nutrient signalling pathways. glucose is taken up by the yeast cell through multiple hexose transporters, which have a broad range of different affinities and transport capacities. this enables the yeast cell to respond to a wide range of glucose concentrations. glucose sensing pathways that employ membrane-localized receptors, such as in the snf3-rgt pathway, are relatively well understood. however, the sensing mechanism of intracellular glucose or metabolites of glycolysis is poorly explained. to study how metabolism is connected to these signalling pathways has proven to be a major challenge since it is difficult to uncouple signalling from metabolism. the ampk/snf system controls energy homeostasis and is best known for its function in glucose signalling. the snf protein kinase complex, which consists of three subunits, is activated by glucose depletion through phosphorylation. it is not well known how metabolism is connected with the activity of the snf complex. when rapidly-fermentable sugars are available snf becomes dephosphorylated. for the establishment of glucose repression only uptake and phosphorylation of glucose is required, but no further glucose metabolism. glucose repression is regulated, at least, at two different steps, i.e. control of snf activation and its function on downstream targets such as mig. it has been suggested that the snf1-mig pathway works in a continuous on-off manner. however, evidence has emerged that mig shuttles in and out of the nucleus and shows transient behaviour at a single cell level. this indicates that the dynamics of the snf1-mig at single cell level is less simple than previously assumed. to study the influence of glucose metabolism on the complex single cell dynamics of the snf pathway we decided to control the uptake of glucose into the cell while employing a microfluidic system to control the extracellular glucose concentration. we show that glucose repression is regulated by glucose flux rather than the absolute glucose concentrations and that the snf1-mig system is closely regulated by glycolytic flux. in our experiments we observed cell-to-cell variability. to explain this variability, we developed a dynamical and nonlinear mixed effect model. dynamical models of signalling pathways in yeast have previously been employed to describe the behaviour of populations of cells. nonlinear mixed effects modelling is a theoretical approach that provides a framework to account for cell-to-cell variability. nmle modelling is traditionally used in pharmacokinetic and pharmacodynamics studies since it allows for the analysis of sparse and unbalanced datasets. nmle has been proposed and used to model dynamic single cell data. recently, a simple phenomenological model describing the snf1-mig pathway using nlme approach has been constructed, capturing dynamics of mig localization, without taking into account parameter variabilities. our integrative approach reveals that the main source of variability is linked to transport of mig in and out of the nucleus. our experimental data indicate that rapid degradation and cell size cause no or little contribution to the cell-to-cell variability, while variation in expression and translation of the hexose transporters is a possible source of cell-to-cell variability.  single cell time-scale fluorescence microscopy enables dynamic studies of the snf1-mig pathway a high control of the cell environment is needed in order to study nutrient responses on single cells. a microfluidics device allows for a fast and precise switch between different media and enables the nutrients composition in the media to be kept constant. here we used a three inlet-channel microfluidics setup, to achieve a high control of the cell environment and to study the influence of glucose concentration on saccharomyces cerevisiae. upon deactivation of snf mig is dephosphorylated and subsequently moves into the nucleus and is therefore a suitable marker for real-time snf activity. the nucleo-cytosolic shuttling of the transcription factor mig fused to a green fluorescent protein served as single cell readout. in the wild type mig localized to the nucleus after being exposed to a glucose concentration of  mm. it has been shown that mig is phosphorylated in this range of glucose concentration. following shifts to higher glucose concentrations more cells respond to the upshift and the mig fluorescence intensity in the nucleus is stronger indicating that higher glucose concentrations result in a higher proportion of mig molecules in the nucleus. the snf1-mig system responds to exposure to glucose and the degree of response is sensitive for the absolute glucose level which the cell is exposed to. these results are consistent with those of a similar study in saccharomyces cerevisiae with a different genetic background. the response to increased glucose concentrations occurs rapidly after upshift, pointing to the fast adaption of cells to nutrients in the environment.fig. localization of mig over time expressed as nuclear fluorescence intensity divided by the cytosolic fluorescence intensity. each graph depicts the results of one experiment. the y axis is distributed logarithmically. each grey line represents the trace of one single cell and the average of all cells is represented with a thicker blue line. between to cells were analysed in each experiment. the different strains are displayed vertically, hxt, hxt ) and the different concentrations are displayed horizontally glucose uptake through only low affinity transporters results in a strong response in the snf1-mig pathway the data obtained from the wild-type strain raised the question whether glycolytic flux correlates directly with snf1-mig pathway activity. this would imply that the snf1-mig pathway is controlled by glucose metabolism by a quantitative sensor system. to address this question we chose to control the flux through glycolysis via the glucose uptake into the cell. a large set of isogenic strains expressing only a single hexose transporter is available. we employed strains that express either a low affinity or high affinity glucose transporter, respectively, under the control of the promotor of a high affinity transporter. hxt is a low affinity transporter that is normally highly expressed under high glucose conditions; yeast cells expressing only hxt displays a high glucose transport capacity and a higher v max then the wild type strain. the hxt cells already display mig nuclear accumulation when they were exposed to an upshift from to  mm glucose as in the wt however the fraction of the whole population displaying nuclear localization never exceeds 50%. while for an upshift to 11 mm glucose a higher fraction of cells display nuclear accumulation but this nuclear accumulation never exceeds 80%. while in the wt for both the upshift to  mm and 11 mm more than 80% of the population displays nuclear accumulation. for the other upshift to higher glucose concentration nearly all cells display nuclear localization. the hxt strain displayed a large cell-to-cell variability of cells either responding or not responding to an upshifts of  mm, 11 mm, and  mm in glucose concentration in comparison with the wt cells. only after an upshift to  mm glucose almost all the cells of the population show mig localization in the nucleus. at higher glucose concentrations, 55 mm or 220 mm glucose, a higher proportion of mig is localized in the nucleus than at lower concentrations. the upshifts to 55 mm and 220 mm result in a higher mig nuclear accumulation compared to the wt and displays a higher cell variability compared to the other strains. the response times of mig nuclear accumulation appear to be remarkably similar for all responding cells under all glucose concentrations, after 1 min the max is reached for the fraction of cells displaying mig nuclear localization of all strains and all upshift conditions. overall, the data shows that the response characteristics of the snf1-mig system correlates well with the kinetic characteristics of the hxt transporter as the snf1-mig strain displays low response at an upshift to low glucose concentration but a strong response to upshift to high glucose concentrations. the high affinity transporter causes a weak mig response to all glucose concentration upshifts the hxt high affinity transporter is highly expressed at very low glucose concentrations. the hxt strain displays a lower glucose uptake capacity than the hxt strain. therefore the glucose uptake capacity is saturated at low glucose concentrations. the majority of the population shows mig nuclear localization after the cells are exposed to growth media containing glucose, in contrast with the low fraction of responders in the hxt strain. however, unlike in the wt and the hxt strain the response is very similar for all glucose concentrations, and the intensity of mig in the nucleus is the same for all upshifts. already at an upshift from to  mm glucose mig has reached a maximum mig nuclear localization for the hxt strain. hence, even in the hxt strain the snf1-mig response characteristic corresponds to the properties of the transport system with high affinity but low capacity. neither regulated degradation nor cell size is a major contributor to cell heterogeneity cell-to-cell variability in mig localization upon changing glucose concentration has been reported. these studies however do not examine the source of the observed cell-to-cell variability, therefore we set out to explore the source of the variation. it is known that the high affinity transporters hxt and hxt are internalized and degraded when cells are exposed to high concentration of glucose. degradation of hxt requires inactivation of torc. also hxt is actively internalized and degraded if glucose is depleted, an effect possibly mediated by downregulation of pka. internalization of the hexose transporters for catabolic degradation could lead to a decrease of glucose import. we observed in the upshift experiments to higher glucose concentrations a slight decrease of the median after the nuclear localization reached its maximum value for all strains. therefore, we reasoned that rapid activity adjustments of the hexose transporters could impact the mig1-snf pathway. since this drop in mig nuclear localization differed between cells this mechanism could be a contributing factor to the observed cell-to-cell variability. we exposed yeast cells expressing hxt7-gfp under the native promotor grown on 3% ethanol to 220 mm glucose and followed the localization of hxt7-gfp for 15 min. the data was quantified by measuring the fluorescence intensity along an intersection through the cell. we observed no significant change in the localization of hxt7-gfp during the experiment.fig. study of the cell-to-cell variability observed in the snf1/mig system. hxt7-gfp before and following a switch from ethanol media to media containing 220 mm glucose. time lapse microscopic images, upper images show hxt7-gfp, the lower images show brightfield. fluorescence intensity along an intersection through the yeast cells. the fluorescence intensity is higher at the points the intersection line crosses the cell membrane and does not change over time. the result of only one cell is displayed but multiple cells were analyzed and none of the cells showed a decrease in membrane localization of the hxt transporter after 15 min following the shift to glucose media. the ratio 15 min after glucose upshift plotted over the cell size for the hxt strain. the cell size is plotted on the x-axes. as a measurement for the mig1-snf pathway response we chose the mig nuclear/cytosolic ratio. upshifts to higher glucose concentration; to 220 mm, to 55 mm and to  mm glucose result in higher final ratio while upshifts to lower glucose concentration to 11 mm glucose, to  mm and to 0 mm glucose display a lower final ratio. hxt7-gfp pregrown overnight in 3% ethanol media. upper image shows the bright field image, lower image shows the cellular distribution of hxt7-gfp we next asked whether cell size could influence the cell-to-cell variability. fluctuations in cellular states, such as cell size, can cause extrinsic noise which could lead to the observed cell-to-cell variability. we therefore decided to test the influence of cell size by plotting the response of the snf1-mig pathway over the cell size. as measurement for the snf1-mig pathway we used the mig fluorescence intensity ratio 15 min after the upshift. the final ratio for the hxt1-strain did not show any correlation between the cell size and the snf1-mig pathway activity. instead, the final ratio showed an even distribution around the average cell size with the values for the upshifts towards higher glucose concentration position higher along the y-axes. this result excludes cell size as a major determinant for the cell-to-cell variability on the relatively short time frame of the experiments. stochastic expression is a plausible cause of cell-to-cell variability in the mig1-snf system cell-to-cell variability in dynamic adaptation responses might be caused by, among others, stochastic transcription activity. therefore, we reasoned that the expression pattern of the hexose transporters could lead to the cell-to-cell variability. we therefore grew a strain expressing hxt7-gfp on 3% ethanol and followed the population distribution of hxt7-gfp. the fluorescence intensity of hxt7-gfp differed significantly between the cells. the lowest observed hxt7-gfp fluorescence intensity was only 10% of the maximal observed fluorescence intensity of hxt7-gfp. the amount of hxt transporter molecules within each single cell varies and can therefore be a major contributor to the observed cell-to-cell variability. these results show that, under our experimental conditions it is likely that expression and translation of hexose transporters is a major contributor to the observed cell-to-cell variability. a mixed effect model suggests mig dephosphorylation as a new regulatory step to better understand the effect of glucose upshift on the snf1-mig pathway we developed a mathematical model of glucose flux which controls snf phosphorylation and consequently mig localization. the aim was to investigate if glucose uptake was able to regulate mig localization by controlling only one step in the snf1-mig regulatory system. we assumed this step to be dephosphorylation of snf since several publications identified this step to be controlled by the adp/atp ratio. the adp/atp ratio is indirectly determined by glucose uptake and glycolysis, therefore the binding of adp to the snf complex could be the connection between glycolysis and the snf1-mig system. nlme modelling was implemented in order to simulate the dynamics of mig localization for different yeast strains in various experimental conditions. the model captures the characteristics of our experimental data. by simulating parameters for multiple cells we could produce a distribution of the parameters and we compared this distribution between the wild type, hxt and hxt strains. the model predicts that the vmsi, the parameter for snf dephosphorylation, increases with upshifts to increasing glucose concentrations. however, there is no significant difference between the different strains. the model suggests snf dephosphorylation to be active immediately after glucose is imported into the cell, but this process is influenced neither by glucose concentration nor by the strain it was simulated in. this suggests that snf dephosphorylation is regulated more in an on/off fashion rather than in a dynamic fashion. however, the parameter for mig dephosphorylation, vmd, did display the characteristics of the different strain. at low glucose upshift the simulated vmd parameters of the hxt strain were higher than the hxt parameters. only at the higher upshift concentrations the vmd parameters simulated for hxt strain where much higher than the vmd parameters simulated for hxt strain. from this, the model suggested mig dephosphorylation as a regulatory step which is controlled by glucose flux.fig. dynamic and nlme modelling of the snf1/mig pathway. a schematic representation of the model. the model consists of three main parts, namely the activity of glucose, the activity of snf and the activity of mig b simulation of the distribution of the random parameters for the hxt strain. the columns indicate the extracellular glucose concentration, ranging from to 220 mm which are illustrated from the left to the right in the figure. each heat map is generated by drawing mixed effect random terms, that is η ∼ n, corresponding to the parameter vectors from the generated parameter distributions for the various strains and glucose concentrations. the heat map displays various parameters on the y-axis, the individuals on the x-axis and the magnitude of the random terms are indicated by the colour scale shown above the figure. the colour scale ranges from to where a red colour corresponds to a high random term and a blue colour correspond to a low value of the random term. the white fields correspond to the parameters connected to the hexose transporters that are not active in hxt strains a mixed effect model identifies hypothetical sources of variability in the snf1-mig regulatory module since our model takes cell-to-cell variability into account we could use it to identify which parameters display highest cell-to-cell variability and under which conditions. the wild type strain displayed increasing variance with upshift to the higher glucose concentrations 55 mm and 220 mm. the highest variability was observed in the hxt strain with the upshift to the higher glucose concentrations. the hxt transporter strain displayed a large cell-to-cell variability following shift to high glucose but a small cell-to-cell variability after upshift to low glucose. the simulated variance of the hxt strain was lower in comparison with the wild type and hxt strains and did not increase with upshift to higher glucose levels. the simulated variance is in correlation with the observed variance seen in the experimental data. the model also allows us to predict the most important parameters that are the major contributor to the cell-to-cell variability. we compared the magnitude of perturbation of each parameter for the simulation of the wild type strain in the upshift from to 220 mm glucose. a parameter which displays a higher perturbation error has a higher variability in that parameter. parameters involved in dephosphorylation events have been ranked as low significance, suggesting that the dephosphorylation of mig and snf after glucose upshift is a minor contributor to the observed cell-to-cell variability. the parameters which display the highest variance are kex and kim which account for transport of mig in and out of the nucleus. the respective perturbation errors for these two parameters are in the order of 10− while the other parameters have perturbation errors in the order of 10− and smaller. indeed, it has been shown that movement of mig in and out of the nucleus shows considerable variability between cells. those data suggest that variability in the nucleocytoplasmic transport of mig would be the major contributor to cell-to-cell variability and not the dephosphorylation events following glucose upshifts. discussion it is well established that the snf1-mig system and hence the nuclear accumulation of mig are controlled by the level of glucose in the growth medium. we have previously reported that glucose derepression senses glucose concentration in a highly dynamic fashion. but it remained unclear whether the observed dynamics were correlated with adaptations of sugar transport and glycolytic metabolism. previous modelling approaches have suggested the importance of the kinetics of the glycolytic flux in signalling of glucose. to elucidate the influence of sugar transport on the snf1-mig pathway we studied the response of three different s. cerevisiae strains with different glucose uptake capacity when exposed to an upshift in glucose concentration. our data show that yeast cells response rapidly to changes in glucose concentration and that there is little to no cell-to-cell variability in response time. even between the strains and between glucose concentrations we observed no significant difference in response time. this indicates that yeast cells are programmed and determined to rapidly respond to a change in glucose concentration. this behaviour allows yeast cells to rapidly adapt to new environmental conditions and thereby to potentially outcompete other species. although single yeast cells induced a response at almost the same time, there was difference between the cells in the magnitude of their response. our data showed that the mig localization pattern after glucose upshift correlates well with the glucose uptake characteristics of the respective yeast strains. it has already been shown that establishment of glucose repression is driven by sensing of an intracellular metabolite rather than extracellular glucose. our single cell data confirms that glucose repression is sensed through an intracellular metabolite rather than extracellular glucose. it has been shown that in the strains we tested the reduced glucose uptake capacity results in reduced glycolytic rate. the data indicate a closer link between the glucose uptake rate, which determines the glycolytic rate, and the activity of the snf1/mig system than previously anticipated. this indicates that the signal which originates from the glycolytic flux is very dynamic in response to the changing glycolytic rate. a source for this signal might be the adp/atp ratio. since, it has been reported that adp binds to the regulatory subunit snf and this binding leads to protection of the catalytic subunit snf from dephosphorylation, which leads to increased snf activity. the turn-over of adp to atp or vice versa could be a sensor for the glycolytic rate. also hexokinase an enzyme part of the glycolytic pathway, has been suggested to serve as a sensor for internal glucose by serving as a threshold for the interaction between mig and snf. our data showed a considerable cell-to-cell variability in glucose sensing. this variability could have considerable impact on the snf1/mig pathway. we investigated the causes of the cell-to-cell variability that was observed in our initial experiments. our results indicate that this behaviour is not caused by the size of the cell or rapid activity adjustments of the hexose transporters. a large variation is observed between the concentrations of hxt in the cells and hence the high cell-to-cell variability can be caused by the expression and/or translation of hexose transporter hxt in a yeast single cell study of the shift between sulphur sources it was observed that the transcriptional adaptation displayed a large cell-to-cell variability. the variability in expression and translation of hexose transporters may cause a different uptake capacity within a population and consequently variability in further glucose metabolism. such variability in glycolysis might lead to cells responding differently to nutritional changes and different subpopulations. systems to restore unbalanced dynamics in glycolysis have already been reported. data obtained by single cell techniques coupled with mathematical modelling offer an opportunity to understand the variability within a population of cells. this work employs nlme modelling in larger dynamical models providing a framework to deeper investigate and identify source of cell-to-cell variability in snf1-mig signalling pathway. the variability in our model corresponded to that observed in our experimental data. therefore, we can conclude that the implemented model can in fact account for the cell-to-cell variability of the nuclear/cytoplasmic mig ratio. our approach classifies certain parameters as low significant, having very small perturbation errors indicating that they could potentially be neglected without losing the predictive capability of the model. this partly explains that the simplified model proposed by almqvist and colleagues can capture the dynamics of the mig localization, but fails to provide more information about the relationship between the parameters. to our knowledge, this is the first mixed effect model which is complex enough to allow identification of sources of cell-to-cell variability in signalling pathways. our model suggested a new regulatory step at the level of mig dephosphorylation and this step would be controlled by glycolytic flux. it is known that mig is dephosphorylated upon glucose addition leading to glucose derepression. however, the glucose activated process which dephosphorylates mig has not yet been clearly identified. for instance, it is unclear if mig dephosphorylation is regulated by glucose or by a constitutive phosphatase counteracting snf activity. our computational approach suggests that mig dephosphorylation is regulated and therefore probably an active step. the model suggest that dephosphorylation of snf is regulated by absolute glucose concentrations while mig dephosphorylation is regulated in a more dynamic way related to glucose flux. we were able to identify import and export of mig in and out of the nucleus as a possible source for cell-to-cell variability. both mig import and export are regulated by glucose through phosphorylation and dephosphorylation of mig. mig shuttles in and out of the nucleus regardless of the glucose concentration and flip/frap experiments have shown that there is a considerable cell-to-cell variability in mig nuclear translocation. therefore, it is not entirely unexpected that this step is predicted to encompass a high cell-to-cell variability.  this work links the glucose flux to snf1-mig signalling. although the control of the snf1-mig regulatory module is complicated by crosstalk with other glucose sensing signalling systems, we suggest that glycolytic metabolic reactions are playing a major role in the regulation of mig localization. we show that the initial response time of snf1-mig pathway displays no cell-to-cell variability. we further developed and presented a modelling approach which can model the cell variability observed in the data. most importantly, we demonstrate the close correlation between glycolytic metabolism and glucose signalling metabolism. 
our data indicate a close link between the glucose uptake rate, which determines the glycolytic rate, and the activity of the snf1/mig system. strains expressing hxt display a constant response to all glucose concentration upshifts. we further apply mathematical modelling to our data, both general deterministic and a nonlinear mixed effect model. microfluidics systemsglucose uptakenon-linear mixed effect modellingdynamical modellinghttp://dx.doi.org/ /501100004963seventh framework programme289995hohmann stefan http://dx.doi.org/ /501100001729stiftelsen för strategisk forskning253140901cvijovic marija issue-copyright-statement© the author 2017 glucose uptake through the low affinity transporter hxt results in nuclear accumulation of mig in response to all glucose concentrations upshift, however with increasing glucose concentration the nuclear localization of mig is more intense. our model suggests a presently unrecognized regulatory step of the snf1-mig pathway at the level of mig dephosphorylation. we show that differences in amount of hexose transporter molecules in the cell could cause cell-to-cell variability in the mig1-snf system. model predictions point to parameters involved in the transport of mig in and out of the nucleus as a majorsource of cell to cell variability. with this modelling approach we have been able to suggest steps that contribute to the cell-to-cell variability.
 the yeast ampk/snf pathway is best known for its role in glucose de/repression. when glucose becomes limited, the snf kinase is activated and phosphorylates the transcriptional repressor mig which is then exported from the nucleus. the exact mechanism how the snf1-mig pathway is regulated is not entirely elucidated.  glucose uptake through the low affinity transporter hxt results in nuclear accumulation of mig in response to all glucose concentrations upshift, however with increasing glucose concentration the nuclear localization of mig is more intense. strains expressing hxt display a constant response to all glucose concentration upshifts. we show that differences in amount of hexose transporter molecules in the cell could cause cell-to-cell variability in the mig1-snf system. we further apply mathematical modelling to our data, both general deterministic and a nonlinear mixed effect model. our model suggests a presently unrecognized regulatory step of the snf1-mig pathway at the level of mig dephosphorylation. model predictions point to parameters involved in the transport of mig in and out of the nucleus as a majorsource of cell to cell variability.  with this modelling approach we have been able to suggest steps that contribute to the cell-to-cell variability. our data indicate a close link between the glucose uptake rate, which determines the glycolytic rate, and the activity of the snf1/mig system. this study hence establishes a close relation between metabolism and signalling. electronic supplementary material the online version of this article contains supplementary material, which is available to authorized users. keywords microfluidics systemsglucose uptakenon-linear mixed effect modellingdynamical modellinghttp://dx.doi.org/ /501100004963seventh framework programme289995hohmann stefan http://dx.doi.org/ /501100001729stiftelsen för strategisk forskning253140901cvijovic marija issue-copyright-statement© the author 2017 