 The speaker is Andrea Waise and she's going to talk about mechanistic models of bacterial growth. Thanks very much. Do you hear me well? Alright, so I will talk about yet another approach of modeling bacterial growth. Perhaps most similar to what Sanjay presented yesterday. Right, so just before I start, I wanted to mention that in my group, what we do is we work to use our skills and do our humble bid to try and tackle the AMR crisis, the resistance crisis. So we do this not only at the level of bacterial physiology, we also work at the patient level where we work with clinicians to understand how transmission of the worst forms of antibiotic resistance works in hospitals. So this is pretty much data driven whereas most of what we do at the cellular level is a mechanistic approach. So you've seen that in Holly's talk on Tuesday on the RNA repair and trying to understand how that influences tolerance. There's also a poster outside by Elena where we're trying to better understand antibiotic responses and the dose responses. So another project that we're working on which is not mechanistic but rather data driven is one by Emily where we're looking at the early adaptive response of MRSA, so methicillin resistant staphylococcus aureus and the role of different RNAs in dynamically adapting to a host environment. So mechanistic projects are mainly based around this approach that I will present to you now which is about before I start with that. So if you find that things I present here or things that you hear from Elena and Holly are interesting to you, please talk to us. We do have PhD openings and also spread the word if you find that there might be candidates who might be interested. So basically the models that we work with, we see them sort of as a framework to predict growth rate in response to different kinds of inputs or perturbations. So those inputs could be different kinds of nutrient environments or drugs that they are exposed to as well as genetic modifications. The framework that we want to look at should ideally be dynamic and what I will talk about now will be split in two parts. In the first part I will talk about a framework that we developed of modeling bacterial growth responses deterministically, sort of the average responses of cells in a population. And the second part then will focus on single cell responses and how stochasticity drives heterogeneous responses in growth. Before I run out of time I would also like to thank my coworkers on those projects. So those are mainly spread between Edinburgh and Imperial College which were the two places I have been flip-flopping between with the exception of Vincent who deserted us to France. So we've seen a lot of different approaches and obviously we know that there's a lot of activity going on in the cell and it's highly complex and we're trying to coarse-grain this in some way. And there are various approaches that we've already seen here at the workshop. And for us now the choice was which kind of avenue to go for with the goal that we had of predicting emergent responses. So as I said already with the inclusion of different kinds of perturbations or inputs, actually a mechanistic approach lends itself very well to this. And so although we've actually taken a lot of input from the kinds of approaches that Terry has presented and has developed, so we decided to go for a mechanistic approach which kind of on this slide, the only truly mechanistic model that you see here is the whole cell model where there's an attempt of modeling everything in the cell which was also not the avenue we wanted to go for. The last approach down in the bottom is one that in terms of the coarse-grainness was actually most appealing to us, but it was still based on optimization which was something that we wanted to shy away from. So the approach that we took was one that had a very similar level of coarse-grainness as the third approach I've shown you there. But instead of relying on optimization we wanted to fully remain mechanistic. So the model that we developed here is one that largely builds on the central dogma, the central dogma based around a coarse-grain genome much inspired by the works of Matt Scott and Terrewah. So this basically describing the central dogma on this coarse-grain genome dynamically and then combining it with some sort of nutrient and energy influx. So the import of a nutrient is conversion to energy that can then be used to fuel the biosynthesis of proteins. And we focused our modeling approach on three fundamental constraints that all living cells will face. So one is that they will only have a finite amount of energetic resources that we coarse-grain with this one species A here. They all rely on a finite pool of ribosomes for their protein biosynthesis. And then their proteome is finite and it needs to include all the proteins that the cell needs to perform the different tasks. Now I will show you now, I will go through how we implemented those different constraints. So in the first one about the finite pool of energy, so basically we build this model as an ordinary differential equation model where we describe the temporal evolution of the different species. So here it's this energy species and this basically comes in at a rate that depends on the enzyme concentration of this green guy here as well as the concentration of the internal nutrient. And then to model different kinds of nutrient environments, we have this parameter that basically tells us how much energy do we get out from that type of nutrient. And then we have all the processes that are feeding from this energy resource and here we're making a huge simplification. We do assume that it is only a translation that dominates energy consumption. And this is based on some estimates that stated that energy consumption by translation is around 70% of the energy budget. And then finally all our cellular species are being diluted through growth of the cell. We have a mechanistic derivation as well of the translation rate of the different species that we're considering. So for that derivation we're considering a mechanism where basically a ribosome bound to an mRNA can reversibly bind to the energetic resource and then once it's bound one elongation step can happen. So obviously this has to happen in times within the number of amino acids in that protein. And then at the end the peptide is released and the complex falls apart. And so we do not model all of this explicitly, we only use that to derive the net rate of translation, which then turns out to be this kind of Michaelis Menten type kinetic where we have a maximal translational elongation rate and sort of an energy threshold at which elongation is half maximal. And further it depends on the levels of bound ribosomes of that type of mRNA as well as the length of that protein. Right, so in the next trade-off we look at the finite number of ribosomes and for that first we need to model how mRNAs are being transcribed and then how they competitively bind free ribosomes. So here we have the equation for the free mRNAs being produced through transcription and through the same logic as I showed you in the previous slide about translation we can argue that each elongation step in transcription consumes a finite amount of energy even though we neglect it we can assume this kind of same shape of the transcription function. And then I want to emphasize that we have those parameters theta here that principally can apply to all different genes but in our framework we infer them for ribosomal and non-ribosomal genes separately. And only for the kind of housekeeping proteins that are supposed to have a more or less stable concentration throughout different growth conditions we assume this kind of negative auto-regulation that keeps them more or less stable. And then the way we implement the constraint in ribosomes is in the equation of the free ribosomes where basically apart from the production and dilution obviously we have to sum over all the processes that either take away free ribosomes or feedback into the pool of free ribosomes. And that is basically summing over all possible genes, all possible mRNA types that ribosomes can interact with and we have the binding of mRNAs and free ribosomes as well as their unbinding and then the free up at the end of translation. Right and finally we have the constraint on the finite mass of the cell. Again here we're assuming that this mass is only made up of protein and we can sum up basically our total mass of the cell. So this is given basically by looking at all the different proteins that we have and multiplying them by their length. So we're considering the mass as a total of amino acids incorporated into the proteome. And then obviously we also have to consider the ribosomes that are being found by mRNAs. So when we write this we can then also write the ODE for the mass and that obviously is determined by how much protein is being produced and then how much of the mass is being diluted through growth. Now so here we still have this kind of undetermined growth rate and it is quite obvious that when we set M to zero so we are at kind of steady state growth. Then this mass is not the total mass. It's being diluted so it's like a concentration. It's different from... I mean we're not considering volume, we're kind of considering that... That mass is proportional to volume but this M I should think of it as a fraction or a concentration. There's a mass fraction or a concentration. Yeah. Thanks. So obviously when we set this to zero then we get that lambda should be proportional should be equal to the total translation rate divided by the current mass. And to define this rate dynamically we set the dynamic growth rate to the total translation rate divided by a mass that we take as the typical mass of an exponentially growing cell. And then what we guarantee is that at steady state so at balanced growth we will reach that typical mass. Right so this is just to show you how the full model then looks. We have a set of coupled oddities here. We have except for this auto inhibition term in the housekeeping transcription. We have no sort of switching functions or anything. And overall we then have 14 oddities so basically 3 oddities per gene that we're considering in terms of the free mRNA, the bound mRNA to the ribosome and then the protein plus the internal nutrient as well as the energy. Now obviously this model has a lot of parameters but the advantage of this mechanistic approach is that we could actually harvest quite a lot of those parameters from the literature. And the parameters that we were left to infer were actually those associated with the expression of those coarse grained genome sectors. Right so we used the data from Terry's lab to fit our model and what you see here is those lines are simulations based on samples from our posterior distribution of the parameters and overall we see that a lot of different parameter values seem to produce this behavior very robustly. We could also look at the actual values of those parameters and we saw that there was a huge range of variation in the actual parameters. So this was a very robust behavior which then led us to actually derive those growth laws analytically from the model and we could which I guess is not so surprising anymore. We've also seen that from Sanjay's model yesterday which has kind of the same structure although we fleshed it out a bit more with mechanisms. And then what this allows us is to kind of derive the relationships where we can determine the parameters in those growth relationships. We can link them back to parameters in our model. So for example here the slope related to the time it takes to translate the efficiency of the ribosome. This is something that Terry also mentioned already. And then the second growth law when inhibiting translation we could relate it back to the efficiency of the enzymes. The amount of housekeeping load that we need to produce. And then finally also related to the previous talk now the mono growth where we found that the maximal growth rate depends heavily on the housekeeping load as well as the enzymatic and the ribosomal efficiency. Right, so we also use this model now to study different systems. So as an example of one system where we kind of use the model to contextualize, provide a cellular context to systems that are often studied in isolation and that may behave very differently when we actually consider them within a dynamic environment of a cell. So one example that we saw on Tuesday was Holly's RNA repair system which we're trying to currently integrate into this growth framework to kind of see how her system dynamically interacts with the growth apparatus and kind of invokes those different kinds of tolerance states. Obviously as I told you in the very beginning we're very interested in antibiotic responses so this framework really allows us to model how the different antibiotics tackle different sports targets inside the cell and the framework has also been taken up quite a lot by the synthetic biology community or I have to say now engineering biology community it's what we have to say in the UK now. So where basically people use it to quantify the interaction of the synthetic circuits that they build with the host cell and kind of quantify the burden that they impose on the cell as well as the limits how the host physiology constrains the performance of the synthetic circuits. And I'm aware this, I'll mention it I know this is kind of makes me a target here but we've also used it to model competition experiments so I believe this is really a very flexible framework that allows us to modularly put different aspects that one might be interested in together and investigate them. So if there are no questions for this part I'll move on to the second part. We know that the growth of single cells can vary quite a lot so what you see here is the growth trajectories of individual cells and you can see that those differ quite a lot like how fast they grow, how big they grow, et cetera but also if we look at the growth rate of individual cells then those can vary considerably over time and this variation of growth rates can be influenced as well by the environments that the cells are exposed to. Now why is this interesting? This is interesting to us because it can have quite an impact in real environments when cells are exposed to antibiotics and when they might display tolerance to antibiotics which then leaves the treatment ineffective. So the framework, the reason traditionally people believe is how heterogeneity among isogenic cells arises is through gene expression because it is the process where we have mRNAs at very low copy numbers and this then typically results in a variation of levels at the protein level and then this can lead to different phenotypic responses. This has been studied extensively but typically at the level of a single gene or maybe a combination of a few genes. Now we're looking at this messy system and to study how variation in some of these components lead to variation in growth rate we again go back to our framework but now we consider it in a stochastic setting. So I apologize it's a different figure but it's exactly the same reaction, it's exactly the same species and now we consider every reaction as a stochastic reaction. So this really allows us to not impose stochasticity phenomenologically but kind of predict its emergent effects from first principles. Now I think at the single cell level it's not enough to only consider stochastic reaction kinetics but a major stochastic event in the cell is also cell division so what we did for that is we coupled the stochastic growth model with an established cell cycle model which is the Donicky model so basically we coupled cell growth to DNA replication where we used the Donicky model which basically says that a new round of replication is initiated whenever the initiation mass is passed the way we interpreted it was that the concentration of DNA origins goes below a certain threshold. One comment, so there's a lot of single cell data that challenges this initiation model probably it's irrelevant for what your purpose is but just to mention. So after initiation then there's a fixed period of time that takes for the cell to replicate. The chromosome, this is called the C period and after termination of replication so basically when this replication reaches this yellow point here there's the D phase, the division phase where the cell prepares for division most likely and those we consider fixed. There's also some contention about that. Sorry? Are your models into models that resemble what you actually measure in single cells which are sort of equivalently complex? So this now allows us to simulate the growth and division of a cell lineage over time. Just to clarify how we do the division so there are different ways of doing that and basically we assume that we have the cell as a bag of different types of molecules and what we do is then that we say that the cell decides on a septum position where the cell will divide and that septum position will on average be one-half but there is some variation around that and we found that this little bit of variation was actually important to reproduce some of the data that we looked at and then once this proportion of the daughter cell is determined then we do a binomial partitioning of all the different chemical species separately with that proportion. Right, so now we can stochastically grow our cell. We can divide it and we can simulate it over time. So here you see one such simulation where we can see how fluctuations and some mRNA species might have like a slow fluctuation over time and we can see how the slow fluctuation kind of propagates to a slow fluctuation in growth rate so we see that molecular fluctuations move on to more physiological fluctuations and we can look at different kinds of macro variables in our cell. Right, now this is hugely inefficient and we have to, in each of those cell cycles we have to simulate the production of millions of molecules so this is computationally infeasible and that's why we also looked at some approximation method for that. Again, I'll just quickly go over this. So here we have our variables which are the molecule numbers of different types each of those molecules is associated with a certain mass and I'll remind you that in the current setting we're only considering proteins that have mass and then we can sum up the masses of all the proteins to arrive to the total mass of the cell. And then assuming that mass density is fixed and mass is proportional to volume we can look at the concentrations of different species and also write down the cellular growth rate as the rate at which mass is being produced. Now the first approximation that we do is that we assume that those concentrations although they are discrete of course they are continuous and if we ignore division and growth for a moment then this is just a standard set of chemical launch event equations but of course we have division and growth and so we have some extra terms in our equation which account for the dilution of species through growth the biomass production as well as the partitioning of the cellular contents at growth. We also do have an equation for the mass and that overall gives us a couple of set of launch event equations which we still cannot solve. So we did another approximation here which is basically our small noise approximation where we say that fluctuations are small as compared to our system size so basically our M is typically very large. That's what we assume and then we can derive some closed form estimates that allow us really to use this model effectively. So by being able to solve this model quickly we can integrate different kinds of data and estimate parameters we can analyze the model and we can test different kinds of hypothesis as I will show you now. Right, so the data that we integrated is a mix of population, both data so this is the data that we've seen over and over again so the data from Matt Scott's paper on mean growth rates and ribosomal contents as well as single cell growth data and the fluctuations in terms of the coefficient of variation of the growth rate in different growth conditions. So we were able to fit the data that we used and then we used the inferred model to do some sort of sanity check on how we're able to reproduce independent data so one thing that we looked at is basically the unit size of the cell that's the size of a cell per number of origins DNA origins, replication origins and this has been observed to be fairly constant over different conditions which is what we reproduced this is not a surprise because it's a consequence of the Donnecke model that we assumed for the cell cervical and DNA replication but when we looked at our estimates of the unit size which were in the units of amino acids incorporated into the proteome compared to volume then actually we could see that we're reproducing very nicely the protein density that's been estimated in the literature and we also looked at cell mass which we're producing very nicely so we typically assumed here the black ones oh I forgot to say sorry so the different things that you see here is the squares are always the data that we used the dots are stochastic simulations and the line is the small noise approximation that we use which agrees very well with the stochastic simulations just as a sanity check so for the cell mass in most of our simulations we used a C plus D period of 60 minutes and there was another set of data where they had observed a longer C plus D period so when we inserted that we could reproduce also that set of data quite well and then I was very glad to see Ludovico's talk the other day which made me understand why our predictions of total mRNA in different growth rates have this linear type of growth across different conditions so I haven't updated here the data to the newer estimates so this is from the Bremer and Dennis data which were indirect measurements right so this makes us confident that the model does something right and so we went on to actually analyze where the noise comes from that we're observing so maybe as a first observation one prediction of the model is that the noise for maximal growth rate goes to zero so a question that I always get is so does that mean that there are no fluctuations anymore? I have a question that's not this question so your model should give a prediction for the effective noise of the growth rate along the lineage of single cells so is it additive or mixed or additive, multiplicative? Did you look at that? No we didn't so the other question I get is that noise goes down to zero which doesn't mean there are no fluctuations in the cell anymore it basically means that we're now in saturated levels of the growth rate where fluctuations don't lead to changes in the growth rate anymore so we do not observe noise in the growth rate anymore and then our noise, the decomposition so basically the noise that is predicted by the model we can decompose into the contributions of different reactions that cause that noise and basically one prediction that was made by the model is that throughout basically all of the growth conditions we have half of the noise coming from basically the decay of mRNAs and that is either through mRNA degradation or through their partitioning at division and the other half is coming through their production now this is just any type of mRNA but we can also further pin that down and one thing that we found when asking which mRNAs causing the growth noise was that this is quiet condition dependent so basically when we assumed that we have one rate limiting step in the import of the nutrient so this is the slowest reaction here then basically the model predicted that all of the noise pretty much comes from the mRNA that produces this transporter here and then conversely if the metabolic enzyme that converts the internal nutrient to energy is the rate limiting step then all of the noise came from the mRNA for this enzyme now since ribosomes are so essential to growth we also asked if ribosomes were maybe able to produce the observed levels of noise and in principle that was sort of possible but it meant that the mRNAs coding for ribosomal proteins had to be at extremely low levels and that the blue ones here didn't agree with values that had been reported in the literature whereas values that we estimated in those two conditions actually had a fairly good agreement so we could kind of discard this hypothesis here right so we can look at the sources of noise here but we can also look at how initial fluctuations those sources kind of trickle through the growth apparatus of the cell I actually didn't want to talk about this because it always takes a long time to explain so basically I will just sketch it and if you're interested in how we did that precisely then talk to me or check out our paper so basically this is looking at the stochastic simulations and then computing the cross correlation between species and then based on the lag time between different species we build those kind of minimal delay graphs so basically species that are next to each other in this graph have a minimal delay between in their correlation and we interpret that as it's the species that first feels fluctuations in the other species and now the reason I wanted to mention this was because Dan gave this very beautiful talk yesterday about how growth rate should influence the switching rate between different phenotypes and perhaps like at a mechanistic level one thing that we found was that growth rate was never at the bottom of the propagation so it was not a pure receiver of noise in this propagation rather it was usually quite high up and it would transmit noise to a lot of downstream processes which also means that if there are some switching gene systems that I don't know switch between persistence and susceptibility those could be affected by the noise that is transmitted through growth rate which would be higher at lower growth rates and finally I wanted to talk about antibiotic responses so the stochastic model can reproduce as well the inhibition data through chloramphenicol this is no surprise because the deterministic model did that already and here we are looking at the average responses of the stochastic model we also looked at this in more detail so how different like the whole space of different nutrient qualities and chloramphenicol doses how that affected average growth rate we didn't see any surprises here so basically with higher nutrient quality growth rate goes up and with increasing antibiotic dose growth rate goes down what was surprising to us was how complex the noise landscape was so basically the heterogeneity in growth rate which might indicate that cells that exhibit different growth rates might also exhibit tolerance when they are exposed to antibiotics so here we saw that this was a very non-monotonic landscape and basically if we're looking at any fixed nutrient condition in this range here and we're kind of in this context thinking as a nutrient and condition as a location in the body where we have an infection or so so those might be very different in terms of nutrients if it's in the bladder so if we look at any different any fixed nutrient condition then increasing the antibiotic dose doesn't have the expected effect of monotonically increasing noise because this is what we expect from what we've seen before that at lower growth rates we see higher noise but rather we see that actually we might be initially going up a noise and then the fluctuations take a dip so meaning that actually for many conditions there's this non-zero antibiotic dose that minimizes the growth heterogeneity which might be something desirable when we're trying to devise a treatment strategy that minimizes the possibility of invoking tolerance so with that I'm coming to the end and basically the key messages that I'd like you to take home is that by looking at those cellular mechanisms key cellular mechanisms and the kinds of constraints that we considered we were able to capture the fundamental growth relations that we've been talking all week about and this mechanistic approach gives us a lot of flexibility to modulally study different systems of interest and their emerging growth responses and also that this kind of stochastic framework that we developed I think is very powerful to pinpoint sources of noise integrate different kinds of data and then analyze how this noise goes to the cell and what kind of potential consequences this might have and with that I'd like to finish and thank you for your attention We have time for questions Thank you for your talk and thank you for putting in the slide on the noise transmitter which is the growth rate I was wondering you have this coarse grained model and that also per gene I guess quite high compared to what it would normally be per gene do you think this would really change the model if you have mRNAs that are most of the time zero can you still do the small noise approximation for example Yes we can I mean so I would assume that if we inserted now some sort of circuit of interest then and we would re-estimate the different like expression parameters of the different sectors I would assume that by fitting the actual system that we're investigating then we will get much higher expression levels on the other sectors What about the noise you have when you add antibiotics the very sharp transition increasing noise at a very high concentration of antibiotics is that increased due to that you have a growth rate equals to zero or is there another reason Well we think actually in that regime we're getting into what we investigated before this ribosome limiting regime when we get noise and ribosome availability as well as getting very close to zero Yes so in your model you can let's say the deterministic one you can measure the fraction of ribosomes that are translating a particular gene or set of genes and ask whether it's the ratio of messengers of that particular class of genes or genes to the total is it and do you have regimes where it's different things I'm not sure I understand What is the fraction of ribosomes that are translating a certain gene or class of genes in your model and how it's related to messenger concentrations Did you look into that? Well I mean we just have mass action binding and unbinding so it is completely related to the relative abundance of the mRNA But because you have the complete level of mass action you could have different regimes probably I was wondering if you get to different regimes or not In terms of what Ludovico for example I was very glad to hear his explanation because it made me understand much better why we're seeing this and that actually what he presented made a lot of sense to me because we did actually because we're modeling the binding explicitly we do account for this complex formation regime and then as far as I understood the linearity was only explained at the same time the elongation rate increases and this is also something that we had included already in our energy dependence from your distribution it sounds like you're probably not aware of our recent work on the total mRNA measurements but then from that study we were able