 in biosynthesis flexes so okay so it's a model on cell growth so I'm studying cell growth and for that I'm starting from the central dogma of biology so transcription and translation from DNA to mRNA to protein production yes is it better like this is it better okay okay so yeah so I'm studying the transcription and translation to fluxes to study the production of proteins and so I'm starting from the I want to get those fluxes to understand cell growth so the first thing as I'm not really looking at the enzyme and metabolic pathways so I'm looking at the distribution of the RNA polymerase over the different chains of interest so for now it's a really very simple model I've just got three proteins of three sectors of interest the ribosomes RNA polymerase and all the other proteins so there is four ribosomes and it's for RNA polymerase and O4 orders so from that distribution of RNA polymerase over the genes I can get to my mRNA pool and from that mRNA pool with the ribosomes and the different initiation rates to the mRNAs I can get to my distribution of ribosomes on the mRNAs on the different sectors of mRNAs and that then gives me my prediction of proteins for the end my distribution of the proteome so again very simple sectors so they don't need to be the same sectors but here for simplicity they are so this gives me my repetition of the proteins and this will then determine along with the proteins the number of proteins my fluxes of transcription and translation and so I want to to go from there to cell growth and I want to do it to a mechanistic approach so I'm looking at those fluxes but like with a traffic model so I'm looking at how fast the ribosomes and RNA polymerase go over the DNA and the mRNA and to get to that protein production so I will first talk about the so that transcription and translation model and then see a few applications that we can use with this model so first off the traffic model that we use as the base the totally asymmetric simple excision process so it's a traffic model and which where particles are going over lattice with three important parameters the alpha k and beta respectively the initiation rate the elongation rate and the termination rate and from those three parameters we can get to every property that we're interested on in the in the model so the density of the lattice and the flux that you get for for those for those rates and so so in the simplest case we have this this flux that depends on the density row and you get to a saturation if the density gets too high so just a fair warning before this is and also the older equation you will see for the case where the particle over the lattice is of size one and that is really the case actually that is not the case in the cell as the ribosomes it's more this size so we can still solve analytically all the all the equations but for simplicity I just show the case where the size is equal to one so we've got to the model for we've got the the traffic model and so we want to apply it to translation for example so here translation we just have every lattice every mRNA is a lattice and the ribosomes go over it and the initiation rate depends on the free ribosome pool so it's every active in our ribosomes and the initiation rate then depends on the free ribosome concentration so as the new number of ribosomes increase you have a higher ribosome pool there's a higher initiation rate and that's your amount of ribosomes on the mRNA will increase so if you see can you see well no in the top left you have the how the density will change with the with the amount of ribosomes so if you increase your ribosomes your density increases as long as you're in that in a certain low density phase until until you get to saturation and as you get to saturation you change phase you get to a maximum current phase so the density is optimal for the for the current and no matter how many ribosomes you add in there you will still keep the same density and so the same flux over the mRNAs so we call that either the maximum current or the saturation phase and it is only limited by the inongation rate so we got we get actually a third phase but where the termination rate is limiting but we don't really see that in practice so I don't really show it here so in in practice we have just two phases the low density and the saturation phase one where the initiator and initiation rate is limiting on the inongation rate and so we can the flux will then depend on the amount of ribosomes you have on your mRNAs that's your bound ribosomes and that will depend on the so initiation rate so that's alpha zero times the concentration of ribosomes times the inongation rate and that's for one side so you multiply that by the number of sites you have or columns you have in your in your mRNAs so the average length of the mRNA times the total mRNA and which this number then saturates in the saturation phase at a half of all those sites so we get the the model for the translation process and we do the exact same for the transcription process the lattices are DNA the particles are in a polymerase but nothing changes so we have the wall the wall model and we can just find the fluxes depending on your amount of ribosomes and the parameters of the initiation rate and the inongation rate of DNA and mRNA and so you get a very simple gene expression model where you will be able to see the evolution of mRNA depending on your gene copy number the transcription flux and the degradation and you also see the production of proteins depending on the mRNA and translation translation flux so this is so this is the model and we don't want to see how that relates to the growth and so as you saw there are two phases for each each flex or transcription and translation so if you look at the growth you will have also four phases for each possible phase in which the fluxes are one where the everything is in low density and as either ribosomes or RNA polymerase will be abundant so you will get to a saturation phase for either transcription or translation or for both of them where the growth rate will be maximal as we will see so now we want to see what happens when you look at the growths in efforts really easy but it is you're just looking at the bulk growth so if you look at the mean value of the cell so you can make two hypothesis protein concentration is a constant for given growth rates so in a given condition so p is constant and the gene copy number concentration is constant to not for every growth rate for just for a given growth rate that means your flux of transcription and translation will also be constant over time for that given growth condition and so for each condition we can find a unique value of the RNA polymerase concentration and the ribosome concentration so first we can see a bit more about the transcription flux how that will in relate to how the flux will change for different values of the concentration of the genes and RNA polymerase so you see that it will saturate so this is the flux normalized by the maximum transcription flux so it will saturate for certain value and otherwise be in that low density phase so as long as you are above the black bar you are in low density phase and as soon as you go below it that's the saturation phase and the same for the translation process so also low density saturation and when we apply this to to find the prediction of proteins you get the phase diagram of the growth rates for different values of the RNA polymerase and ribosome concentration and so you either have both of the processes in low density or again ribosome or RNA polymerase to be abundant they will lead to saturation and you have again this part where both ribosome where both transcription and transcription are saturated and that's where we expect our maximum growth rate and indeed if you look at the normalized growth rate so the growth is every growth rate divided by the growth rate you have in the saturation part you see that it's indeed saturated where we expected it to be so in the top right and then it it decreases for the ribosome concentration and RNA-PE concentration when you are in the low density phase so this gives us a value of the growth rate for every possible condition but that's not every possible con not every condition is possible in the cells so we're looking at what we expect to find and as it happens the ribosome density of mRNA is supposed to be constant over the mRNA over the different conditions so those are lines where the density of ribosomes is constant so you'll have a relation between the RNA polymerase and the ribosome for different growth rates for a given density of ribosomes and so we expect our actual values in the cell to be following one of those lines now a slight problem you might have noticed in this diagram is the values of the ribosome concentration and the RNA polymerase concentration which are way too high for biological values so if you look more at what we expect for values to be found so for E. coli in normal growth rates so we'll find something more like this which works between zero and three because it's still you still have a saturated bit saturating part on the top right and values of the density of ribosomes that are closer to what you will expect and what we actually what is actually measured for the density of ribosomes is this line so that's our prediction for the ribosome concentration and RNA polymerase concentration for a given growth rate so we want to check that with data so we use the is the black vision and data from 2022 and you see we see that it's a very good prediction so it's not a fit it's just so starting from the density of ribosomes of RNA what you get for the ribosome concentration RNA P concentration for a given growth rate so for those of you who know the bio question and paper know that the RNA polymerase is actually the available RNA polymerase in the in the paper which is set by enzymes so but as we ignore the metabolic pathway the available RNA polymerase that's just our total RNA polymerase and so it's it is the it is the same but to be certain of that we can also look at the mRNA synthesis flux so the actual effect of the RNA polymerase for the yes so for this prediction you're looking at the low density phase for both transcription and translation here as it happens they're both in low density phase okay it's not it's not necessary I mean it's no no but it happens to be there just that was a question so that's the prediction and so if you look at the mRNA synthesis flux it also if it predicts the data very well so is the prediction based on assuming a constant linear density of ribosomes over M&A yes okay that is the the starting yes otherwise you have way too many possible values of RNA polymerase concentration and ribosome concentration for the same growth rate so of course you could get to this repetition two different ways just fitness evaluation or something like this but here it's just just a question and how far are you from from the saturation phase so here it is actually quite far but we still have traffic so if you it's low density but there's still traffic because you still have interaction because between the ribosomes and RNA polymerase so I mean I don't show it here but it's you all have a difference in the prediction if you remove the traffic from the from the model yes so if you look back if you look at the current density relationship you're not in the linear regime yes so yes you are in so this regime where so in the red the parts over there okay so that's the a bit the the results for this model on the prediction on cell economy and so we wanted to go a bit further see what we can do with this model in different applications so we looked at single cell dynamics and the first thing with single cells is that the gene concentration to be assumption that the gene concentration is constant is no longer true in small times so if you look at so before DNA replication or before division the cell is growing but you don't have more genes so the gene number is constant and why this is a work in progress so be careful about that but we we can make a prediction of what will happen because of not saturation but because of those traffic the impact it will have on the growth so if you look at the so the growth during a gene number of constants so during this space you will have that the ratio between the RNA polymerase and the number of genes will change because the protein number increase but not the gene number so you have more RNA polymerase for the same amount of genes and at some point you will get more traffic until saturation if you get saturation for the RNA polymerase you will then also have a limited amount of memory production and if you have limited amount of RNA production you can also have saturation for the ribosomes and that leads to saturated protein production of proteins so over a long time you will see that the production of proteins will saturate and so you have two two parts in this plot one where the production is still kind of auto catalytic so the you produce more proteins but the effect of having more proteins increases the the production of proteins so it is still a process that is kind of exponential but once it is saturated you so this saturated phase you will expect the production to not never change with the amount of proteins and that will lead to linear growth so that's the protein number over time over long time and when the production of proteins is limited that's you get a linear regime and when you are still the auto catalytic auto catalytic process you will have a pseudo exponential growth pseudo because it's still the transition between exponential growth where gene concentrations constant to that linear growth in effect that means if you look at the growth rate of your protein at a given amount at a given time so just the normalized production of proteins it will decrease slowly over time and once you in linear growth it will just exponentially decrease to zero very long time where you don't have exponential growth anymore the but in effect this time is very long so for a given given growth rate the actual time yourself will be in that phase of constant gene numbers will be relatively short so if we normalize the time by the time it needs to to divide you will see that the amount of of genes of the decrease in growth rates is actually quite small so there's two things we can see here one is first while it is a small decrease the instant growth rate is still decreasing that means you're not exactly in exponential growth rate during the gene constant number part but the actual decrease is really small so less than 6% even for very small growth rates so the actual approximation of that the gene is concentrating this constant is actually seems to be quite reasonable now we want to first use this to then make prediction on how long it takes to to get to the linear phase we already have that but this is still a toy model and then to make experiment to see if we can see that the transition by inhibiting DNA replication but in conclusion it's still we still have a model that can predict the economy in different conditions and while the single cell dynamics for now don't really have an impact on single on normal growth conditions we expect to see more effects on doing different stresses so when we add antibiotics especially with effects on transcription and we hope to see to make experiments with DNA replication so that's thank you for your attention and if you have any questions I will be glad to answer thanks a lot we have time for questions okay since we had already question let's thank the speaker again