So thank you for the introduction. I'm really happy to be here. I'm really gratified to see my colleagues at Lunt University and affiliates with Lynx and really thankful and appreciative to be included in this symposium series. So today I'm going to talk with you about my group's research centering on what sometimes we call biofidelic colloidal interactions. So we do modeling of how biomolecules interact but at the colloidal scale. And we'll talk about advanced computational representation. And I'm going to sort of talk about the impacts on cellular biology but I'm hoping that the connections that are of interest to this group will be obvious and I'll point those out as we go along. And the talk today features the work of Jennifer Hoffman who's now at Prescient Genentech. Dr. Akshay Maheshwari who's now a Stanford medical student. Dr. Alpsunal who's now on a postdoc at Harvard. Dr. Emma Gonzalez who's now a senior research scientist at Dao. Theo Yang who's still finishing his PhD at Stanford. Co-advised by my colleague Dan Uros.  Okay so my group's work focuses broadly on the study of non-living and living complex fluids and most recently we're devoting a lot of attention to biological fluids which are ubiquitous in everyday life. Some common examples include bodily fluids like mucus, saliva, and sweat. Protein therapeutics like the monoclonal antibodies shown here and systems of living cells such as bacteria. And I'm going to focus today on our modeling of biological cells to some degree but much of what I show pertains to modeling interactions between biological macromolecules in vitro. So let's take a closer look inside one of these E. coli cells. For example in this classic illustration by David Good Sell you can see that even this relatively simple bacterium is organized into functional regions. A confining cell wall in green, the crowded cytoplasm in blue and purple, and the DNA containing nucleoid in yellow. Experimental and modeling techniques for interrogating intracellular behaviors are very well established but pretty much in two disparate regimes that have structural biology on one end which represents molecules and interactions with full atomistic resolution but often are limited to probing very small systems over very short time scales. On the other end of the spectrum systems biology focuses on obtaining bulk metrics and monitoring whole cell processes over minutes which is very helpful and important but often at the cost of abstracting space away entirely reducing it to a system of ODE's and sort of a subway map how i view it. However much of the biological complexity that we're interested in in this consortium and cell biologists in general are interested in emerges at the intermediate length and time scales which are inaccessible by these traditional approaches. So the key to bridging this gap is to recognize that these biological macromolecules are part of the broader class of colloids where small particles are suspended in a fluid and at this so-called colloidal length scale we can interrogate physical phenomena such as how crowded the environment is the mobility of the macromolecules and higher order behaviors such as phase separation. And this is work that I began at Stanford University with Drew Endi and bioengineering and together we mentored several PhD students including Akshay Maheshwari who drove much of our early ideas and Alp Sunol, Jennifer Hoffman, Emma Gonzalez and then most recently Theo Yang who's co-advised by Dan Yaroche. So by building models at this colloidal scale we can connect physical and chemical details about how individual molecules interact including if they attract or repel one another their size and if they chemically react to whole cell behaviors that we want to predict in engineers such as how fast the cell grows and divides or bulk properties of especially interest to this group such as the viscosity. And this modeling is actually quite difficult this cartoon that I was just showing in this movie is just that it's a cartoon there are no physics actually driving the interactions and those and so it's really quite challenging to put together these types of models and rather focus on the details of how hard we work to put together this model I just want to show you what we can do with our physics-based models. I'll give a little background just not knowing how sort of the breadth of the audience here so colloids broadly are particles that are about two nanometers to a micron in size meaning that they're small enough to diffuse due to thermal collisions with the surrounding solvent shown in the schematic as individual water molecules. However colloids are also large enough to interact with the surrounding solvent as a continuum of fluid if we like which allows us to model the solvent implicitly in coarse-grain particle shape and interaction features. This of course is is greatly enhanced by interacting with atomistic or molecular scale models that give us more information about how we can coarse-grain our models. The continuum approximation also allows us to use the framework of fluid mechanics to describe fluid motion the continuum mechanics and this is governed by the Navier-Stokes equations shown here in its viscously failed scaled form. The Reynolds number which describes the relative importance of inertial to viscous forces is of course vanishingly small in the case of small colloidal particles and in this limit the inertial term vanishes and the equations of motion simplified to the Stokes equations shown here and so I'll emphasize here that we're interested in timescales that that inertia doesn't matter but we could of course go down to if we wanted to go down to timescales that are involved in say protein folding then that would require different types of modeling and that's part of sort of the the multi-scale modeling of interest in this group. So we can then represent the coupling of motion between individual particles and our approach shown in this above schematic using the Langevon equation which solves a momentum balance on each particle on the suspension. In the Stokes flow limit inertia is negligible and so the left hand side is set to zero and particles can experience various forces including the hydrodynamic drag exerted on the fluid by the fluid on the particle and Brownian force which describes stochastic fluctuations due to solvent collisions and deterministic interparticle forces which depend on the gradient of the potential between two particles. And so here I'm showing an example of one such potential between two oppositely charged colloids which experience a hard sphere repulsion meaning they just can't overlap and also an electrostatic attraction due to their opposite charges so just a simple model we have and I if anybody has questions about it we've also developed a framework to iteratively interact between experiments and simulation using my collaborator Matt Helgeson a UCSB and I have developed a framework so that we can use his experiments with Rio sands and neutron scattering to get information about structure and then get detailed information about say creating a two-ucala potential or other very detailed potentials of particle interactions. But here I'm just going to focus on some simple potentials to illustrate the point. So this equation can then be integrated forward in time and dynamic simulations. Our group uses two computational approaches one is accelerated stochastic dynamics which I'll talk about a bit more at the end and the other is lamps which is a massively paralyzed molecular dynamic simulator which is capable of evolving systems of hundreds of thousands of colloids over milliseconds and we use the implementation of lamps where the solvent is implicit and then we've done a lot of modification, deep modification to lamps to make it to customize it for our use. So this part of our work which was led by my former graduate student Jen Hoffman focuses primarily on developing models of deterministic interactions to better represent biological systems in simulation. And it's important that we first understand the details of how we want to model these things before we bring in the heavy machinery of hydrodynamics because it's on the one hand this is straightforward for our group because we understand these interactions quite well but they also happen to be the most computationally expensive so we tend to bring those in toward the end. Okay so beyond developing new colloidal scale algorithms we've also applied these models to understand how colloidal scale physics and molecular interactions regulate whole cell function. And so I'm going to talk about four projects that we've worked on in this area shown here and today I'll focus primarily on the first two touch briefly on the third. So Jennifer's first project examined how ultra-weak interactions between proteins can alter emergent properties of bacterial proteomes. So interactions between proteins or PPI's organize the cellular interior and form complex networks for carrying out biological functions. These PPI's are typically quantified via an equilibrium dissociation concept KD which describes how easily a complex dissociates. So as such a stronger binding affinity corresponds to a smaller value of KD. The study of PPI's has traditionally focused on interactions that correlate with a strong observable phenotype and thus often have high affinities which are easily quantified by established biophysical methods. But improved experimental resolution has enabled increasing detection of lower affinity PPI's highlighted by sort of an explosion of interest in weak physical interactions underlying in vivo phase separation. Everybody's heard of liquid-liquid phase separation and this really was sort of a launching point for us becoming more interested in weaker interactions. These observations suggest that even weaker PPI's might also contribute to cell function but limited in vivo resolution has largely prevented study of PPI's with millimolar binding affinities. So we wanted to start interrogating this gray zone with a question mark. In this ultra-weak regime clearly defining bound states as is needed to calculate KD can be challenging. Instead interaction strength can be better understood in terms of its mechanistic origin. A competition between deterministic attractions that tend to pull proteins together and the underlying thermal fluctuations that drive Brownian motion that tend to separate particle pairs. And in the case of ultra-weak PPI's this deterministic attraction strength at contact which will characterize with a characteristic value v0 is pretty close in strength to the thermal energy KT. And one thing that I'll get into in some detail here is it depends on how close a pair of particles get as to what is really causing an interaction. There's some distance with which a pair of particles can approach each other and these physical chemical interactions dominate what's going to happen. And then if it allows a pair to get close enough then they might get into a very close radius of interaction where they can do chemistry. So here we've applied colloidal modeling to explore how these ultra-weak PPI's alter emergent properties in the dense cytoplasm. We modeled interactions across a range of attraction strengths v0 of our KT on the x-axis which corresponds to the depth of the interaction potential at particle contact. And of course contact is kind of a loose thing and that requires some exploration in any given system especially if particles are squishy or we're not sure what a hydrodynamic radius is and we have methods to determine where we want to set contact to be. The form of the potential captures both entropic exclusion meaning the particles cannot overlap and an electrostatic attraction which we choose as a model protein-protein interaction PPI. The Kappa parameter is the inverse of the device screening link that controls the range of the interaction which is set by the ion concentration in prokaryotic cytoplasm. So in our simulations we then calculated the apparent KD to identify the region of ultra-weak interactions with strengths less than about 2.5 KT. And as expected a stronger attraction strength produces a smaller that is stronger KD. We can then look at visualizations of our simulations where we've colored molecules based on the number of other particles with which they're coordinated from red to blue and here I'm not saying bonded I'm saying coordinated because one has to be careful whether you're saying some particles sort of form a group or whether they're actually in a durable aggregate. On the left we show that even without deterministic attractions entropic exclusion still drives measurable encounters between proteins and I think in the college community this is no surprise but in the biological community this was sort of I think mind-blowing that entropy is responsible for local sampling of positions of particles near each other. We have a way that we describe that we that we understand that but it can be challenging to put that in terms that are of interest and relevant to the biological community but what it's basically saying is that we need to learn to see Brownian motion and entropic forces as a sort of in some cases associative part of how particles hang out together. In the opposite limit moderate affinity protein-protein interactions are strong enough to form an arrested gel network where proteins are bound like long-term to many napers and obviously that's not a biologically relevant scenario so this gave us some framing for you know how we wanted to characterize our interactions. In between these two limits PPIs at the ultra-weak threshold drive measurable complexation but they form small clusters rather than an arrested network and so this is sort of the sweet spot that we're looking for that is biologically relevant and so the question then is if these interactions last long enough to impact biological functions in vivo and more broadly I think for this audience how does this affect how a solution of say monoclonal antibodies will are these bonds strong enough to create a viscosity that's going to be problematic in the flow of a material. So to answer this we quantified particle dynamics where we track the state of individual particles as either bound or unbound and over time quantify the duration spent by each protein in encounters and freely diffusing so are they in an encounter or are they just wandering around in the fluid. We then do this for all proteins in the suspension and calculate the probability distribution of the duration of these binding and search events. In the heart sphere case we found that proteins tend to spend most of their time freely searching but they still experience many brief encounters with other proteins. In the moderate affinity limit proteins spend a lot more time bound together than they do diffusively searching. Ultra-weak protein-protein interactions however operate at a dynamical sweet spot between these two limits sometimes you call it the Goldilocks zone not too long not too short but just right and these drive transient binding and they also speed up search times for particles to sample interactions with other proteins. So this could be of use if we're trying to enable biological molecules to continue to sample the the cytoplasm or the in vitro environment that they're existing in but we want to limit their attachment to other molecules. So to answer how this all works we quantified particle dynamics where we track the state of individual proteins as either I'm sorry we've got let me move on to the faster this faster sampling emerges from a transient cluster formation which can keep particles close together over much longer time scales. So to track cluster lifetime so a group of particles gets together and that same particles with the same ID hangs out together for some time. So to track the lifetime of these clusters we first identify clusters of particles at an initial time and calculate how long it takes for all particles to dissociate from those clusters. In the case without attractions small clusters exist but dissociate rapidly whereas moderate affinity protein-protein interactions form ultra-durable networks that can last on the order of days. So clusters formed in between these two regimes by ultra-weak protein-protein interactions are transient meaning proteins are still mobile but they can co-localize for an order of magnitude longer than the case without attractions and this is probably a very practical limit to consider. So zooming back out what we really want to know is if this co-localization lasts long enough to facilitate meaningful biological functions in VEVO or with things like monoclonal antibodies if we need chemistry to happen over time can we facilitate that without causing arrest and formation of durable networks that harm the ultimate goals for viscosity. So compared to the time scales of intracellular processes in phenomena because we ask ourselves well what's fast and what's slow it's all relatives not you know we have to compare it to something to see whether it's biologically relevant and in the cell we decided to identify okay what's the time scale of chemistry what's the time scale of an enzymatic reaction what's the time scale of some biological function this tends to be much longer and so we compared these to show that ultra-weak protein-protein interactions can drive meaningful co-localization meaning biological stuff can happen with clusters that last as long as the fastest enzyme catalyzed reactions and so these transient associations operate below the time scales of phase separation, gelation, and whole cell processes such as growth and division that means it's really relevant to pretty dynamical processes. On a broader level we also want to understand how ultra-weak PPIs can shift the bulk behavior of proteins inside a plasm so to do so we calculated the overall likelihood of finding a protein in a binding event for all attraction strengths. When a protein spends more than half of its time in encounters with other proteins whether it's the same partner or it's changing partners we say that it is in a binding dominant versus search dominant regime and in our simulations we found that a subset of protein-protein interactions are both ultra-weak and therefore largely unexplored in experiments yet also strong enough to shift behavior into the binding dominant regime so these results demonstrate the impacts of ultra-weak protein-protein interactions in a model system but then a real question became how common are they inside living cells like would biologists or those working with biological proteins care about these results so to investigate we turn to the model organism mycoplasma genitalium which is naturally occurring near minimal cell meaning that any interactions or phenomena observed there are likely conserved among a limited set of life essential genes and processes so I'll emphasize that what we learn here can be potentially quite broadly applied to all cells so we focus on cytoplasmic proteins and protein complexes and we used established charge data to calculate a set of electrostatic interaction strengths which we classify as b0 over kt which was the x-axis in our previous plots for all possible protein pairs in the cytoplasm of this organism so from these distributions we can interrogate the prevalence of binding dominant ultra-weak interactions and identify specific protein protein interactions in this regime so first we focus on cytoplasmic proteins alone where the upper plot shows the overall probability distribution of attraction strengths across all pairs and the lower plot shows the probability distribution experienced by individual proteins shown in each row each color corresponds with that protein's broad functional role so color is function such as translation metabolism or protein folding among others we find that almost all such electrostatic interactions are ultra-weak and that a small fraction are strong enough to be binding dominant so we were pushing these through a sieve to look for these things that have evaded detection before but still matter so specific protein protein interactions in this regime include pairs of chaperone proteins and enzymes involved in consecutive metabolic processes where we might imagine transient co-localization and shortened search times could possibly facilitate more rapid substrate exchange between these proteins so cytoplasmic proteins also interact with durable protein complexes including the ribosome so if we include the ribosome and its subunits into our calculations these distributions we found that interactions are shifted to higher attraction strikes including doubling the likelihood of finding binding dominant ultra-weak protein-protein interactions this special regime and this is represented in a number of pairs involved with translation and because translation limits prokaryotic growth rate we might expect that transient binding and shortened search times between these molecules could support more efficient protein synthesis and overall cell growth finally we considered another set of important complexes dimers between amino acid tRNA synthetases and tRNAs interestingly we found that these complexes shift the interactome primarily into the binding dominant ultra-weak regime up to 14 percent rather than the long tail induced by ribosomes this is showing that there is this previously unstudied regime which we can now interrogate and find by using special tracking techniques and by understanding dissociation constants in terms of physical chemical interactions things the strength of the interaction relative to kt so a deterministic force that wants things to associate and a brownian force that in general wants things to dissociate so things can sample the whole space so in this summary in this project we highlight how colloidal simulations can be applied to interrogate transient interactions that are currently inaccessible by current in vivo assays we found that such ultra-weak ppis can increase the duration of associations up to biologically relevant time scales while at the same time speeding the diffusive search process we then demonstrated the ubiquity of these interactions in the cytoplasmic proteome of engenitalium where transient binding and faster transport could speed life essential processes such as translation okay now i'm going to shift a little bit and i'll talk about our original work in physics-based modeling of biological cells so this is sort of where we got started so we have been broadly interested in the physics that limit how fast a cell can grow and divide and i'm going to sort of preface this by saying that one of the things that we've always been interested in is how colloidal scale physics instantiates life itself not just an innocent bystander that creates diffusion and lets things sort of blind they wander around and let the important genetic DNA centric view take place but we've wondered for quite some time whether the central dogma is missing a physics piece and so this is an overarching theme of research in my group that's currently being led by a research scientist that works with me Vishal Sivisankar i won't be presenting his forward-looking work today but that's sort of the backdrop of where we're going with this but here i'm going to show you how we have focused on E. coli which can grow rapidly doubling in as little as 20 minutes and this means that it has to replicate all of its contents in a very short time so it's got to synthesize the majority of its dry mass every doubling so that means translation elongation is really going to be cooking so the majority of these contents are proteins so it's unsurprising that protein synthesis limits how fast a cell can grow and this process which i'm sure all of you are familiar with is called translation and it happens in the crowded cytoplasm at ribosomes which then use information stored in mRNA to assemble amino acids into protein products so to do so a molecule called the ternary complex must quickly drop off the requested amino acid in the correct order and often here i show a movie that was made by the Nobel prize winners who defined what happens inside a ribosome and what we see is a sort of black background around the ribosome where all of these amino acids according to previous models get dropped off in exactly the right order in exactly the right time like this combinatoric miracle somehow happens and that's the view that one gets when you completely discount physics and what we've done is build up a model to show how this actually happens and so here's that movie the detailed chemistry that happens within the ribosome during this process has been extremely well studied but is illustrated in this cartoon video the transport process that happens outside the ribosome was pretty much not not even not understood it was completely discounted and like i fond of saying i you see that these ternary complexes are dropping off these amino acids in exactly the right order in exactly the right time a combinatoric miracle and the background is black like deep space because nothing was known about how these actually get here so this is a fantasy and so our group has devoted a lot of time to understanding how if you count how long translation takes inside the ribosome versus how long the extension of that polypeptide chain you see growing takes there's 30 percent of this process that is mechanistically unexplained by previous models so we've published a couple of papers in this area and showed that it is absolutely essential to model diffusive and combinatoric physics in a model we call stoichiometric crowding to explain the speed up of of E. coli at least at the at past or growth rates so experiments have shown that the rate of this protein synthesis process per ribosome speeds up by about two fold as cells grow faster but precisely how the speed up occurs was unknown each simulation contained and our so we decided to build the model of this nobody had even really drawn a sketch of this process because the physics supposedly didn't matter and so we built a model of this process and you're looking at a voxel here this work was started by my co-advised student Akshay Maheshwary who was co-advised with my colleague Dr. Professor Drew Endi at Stanford so to understand this this collaboration between my lab and Drew Endi's lab built a physically resolved model of this process in E. coli cytoplasm each simulation contained hard spheres representing the ternary complexes ribosomes and other native proteins and these were at physiological abundances and densities extracted from experiment so i'm sort of speeding through you know months and months of scraping the literature to pull together all of the abundances and physical measurements and relative abundances of translation molecules in this very well-studied model organism so from these concentrated mutation data we found that the speed up in translation with growth rates shown in green is accompanied by an increase in the cytoplasm volume fraction in black from 13% to 41% and this quantity describes how much of the volume is occupied by macromolecules and on the right i'm showing the simulation snapshots of a solution of single size colloids at 10 and 50 volume fraction to give some intuition on how crowded these suspensions are so we would expect this increase in crowding to slow transport at faster growth rates and in fact that's what the literature has said previously when i tried to address this idea however the concentration of ribosomes and ternary complexes outpaces that of native native crouter proteins which are shown in red meaning the relative stoichiometry of translation molecules improves at faster growth rates and i want to emphasize here again that the idea is that E. coli synthesizes protein faster as it grows faster but if you count the number of ribosomes all over all physiological representations there's not enough ribosomes to explain this speed up and there's not enough speed up in chemistry inside the ribosome to explain this so we hypothesize that somehow the productivity of individual ribosomes must have gone up we wanted to know how that happened and we surmised that something going on outside in the cytoplasm of colloidal scale was speeding this process up so we might expect this whole sort of densification to speed up transport since it would be easier for translation molecules to encounter each other and indeed we found that ternary complexes are more than four times closer to the nearest ribosome at faster growth rates due to this more advantageous stoichiometry and there's a lot more that i could unpack here for the audience but that's sort of a separate talk but the question is then which phenomenon wins out is it that things are closer together and they can find each other faster or that things are more crowded and everything slows down because both of these things are happening and so to test this and we use dynamic simulations to explicitly track the time each ternary complex spends freely diffusing and in chemical reactions with matching or mismatching ribosomes and here a match means that a ternary complex is carrying the correct amino acid to match the mRNA codon at the A site so in this schematic we colored green so i'm sort of oversimplifying a lot of what we did but we built thousands and thousands and thousands of translation voxels to represent the full physiological reality of the cytoplasm and E. coli at a range of growth rates and then we use reactions that follow well established kinetics of codon recognition and incorporation within the ribosome so we have chemistry and physics and overall sort of the making a longer story very short these transport and reaction times all contribute to the overall translation time which is what is measured in experiment so we found that the transport contribution indeed speeds up with growth rate suggesting that improved stoichiometry wins over increased viscosity so the reaction time does not change significantly with growth rate and is up to five times faster than transport meaning the physics are the rate limiting process for translation elongation and E. coli and it is both an increase in proximity and change in stoichiometry and the relative abundance of translation molecules it causes this to happen so summing these transport and reaction times produces the overall elongation time and our bottom up unfitted simulations in symbols can predict the qualitative speed up observed in experiment which is a green and demonstrate the mechanistic origin of the speed up which is faster transport rather than just chemistry inside the ribosome however the absolute predictions of translation time from this initial model are about three times slower than in vivo suggesting there's additional mechanisms left unexplored and we were curious if the gap could be closed by other physical mechanisms that speed up the transport process so specifically I asked Jennifer to focus on attractions between ribosomes and ternary complexes shown here in green which are hypothesized to preload up to four ternary complexes onto ribosomes at one time and these attractions can provide a local pool of ternary complexes to draw from thus shortening transport time and further speeding up elongation closing that rest of that quantitative gap and my other graduate student Theo Yang also helped Jennifer with this project so my project is focused this this project is focused on developing a colloidal scale model of these attractions in the dynamic simulations to test this hypothesis and it turns out that it's fairly difficult to characterize this interaction and experiment due to the movement of these ribosomal arms which with ternary complexes they wiggle around but these this is where the ternary complex is bind however one crystallography study used a proxy molecule to infer the location and character of the binding site shown here and based on this information we calculated the microscopic attraction strength to be about 16 kT or about 39 kilojoules per mole we then implemented this attraction strength as the well depth of the interparticle potential epsilon between ternary complexes and ribosomes in translation of oxal simulations as an initial model we first assumed that the attractions are exerted isotropically over the entire molecule surface meaning they don't depend on orientation so we want to test the hypothesis that these attractions can drive co-localization of translation molecules and thereby speed up the transport process looking at the case with 16 kT attractions we do see for sure that the ternary complexes in light red are co-localized near ribosomes which are gray however the attractions are also strong enough to cause the system to gel which obviously would kill a cell in fact we found these isotropic attractions increase the effective viscosity felt by ternary complexes to orders of magnitude which is in red relative to the case without attractions in blue these kind of viscosities would be untenable serve for cell survival and are also more than an order of magnitude higher than estimated in vivo so that model obviously is not correct so what was the issue with that initial representation we can look back into real biological interaction which has limited valency meaning patchy surfaces each ribosome can only bind up to four ternary complexes and each ternary complex can only bind to one ribosome so to build a more biofidelic model we first estimated the accessible surface area of the mobile l12 arms and the size of each binding site to match structural data in our new model the estimated 16 kt attractions are then only felt between these protruding binding sites which are shown in green and the little nubbies there so first we want to know if limiting the valency prevents gelation which we saw on the isotropic model and we tested this by calculating the radial distribution function g of r which quantifies the likelihood of finding a ternary complex ribosome pair at a given separation for systems without attractions there's a non-zero probability of finding a ternary complex nearby because they have finite size and their system is crowded and let's take a look in the isotropically attractive case where there's a pronounced peak at contact corresponding with pairwise binding and a second peak that emerges from longer range structure along the gel network so neither of these cases is the landing spot our limited valency model the dashed red line can drive short range binding without a second peak that would be for arrest so the wide first peak is due to a set of equally likely center to center distances with varied angles between the protruding binding sites we also show that the limited the limiting the attraction valency prevents unphysically high viscosities like we saw in the isotropic case and recovers the estimated in vivo viscosity quite well as another validation of this model we compare against in vivo binding metrics including the fraction of ternary complexes that are bound to ribosomes and this value has been measured to be about 60 at one doubling per hour shown as a black symbol we can also estimate how this bound fraction might change with growth rate using whole cell crowding and stoichiometry data and our limited valency model agrees quantitatively with the measured and extrapolated values so we think we got that right we can also measure against in vivo binding of the affinity measured to be about 20 micromolar at one doubling per hour and once again our limited valency model quantitatively recovers this value so taking a step back we've demonstrated that our model can quantitatively predict in vivo measurements of co-localization affinity and rheology the question now is if these interactions close the gap in translation rates between our prior model without attractions and that value compared to experiments so to test this we implemented a chemical reaction framework developed previously that we had developed in our last model in lamps and since the previous simulator could not incorporate deterministic potential so we sort of really got under the hood in lamps and and installed this new framework we again ran dynamic simulations where we explicitly track the transport and reaction times of translation molecules as they search for their match when we include limited valency attractions between ternary complexes and ribosomes we find that elongation is fed up two to three fold relative to the baseline non-attractive simulations which are shown in black so the absolute latency is now in much better agreement with experiments and our predicted speed up with growth rate actually quantitatively recovers the in vivo trend so we were pretty pleased with that these attraction speed elongation by facilitating colloidal scale transport in the case without attractions a ribosome that finishes testing and rejecting a non-matching tRNA must wait for the next tRNA to arrive to continue the process however when ternary complexes are preloaded onto the l12 subunit a pool of new tRNA are nearby and can almost immediately protested thereby shortening wait times and speeding elongation so now we're getting a little closer to this sort of combinatoric miracle with this mechanism in mind we can ask other cell biology questions for example why does E. coli have only four of these l7 l12 subunits and i think i'm probably running a little short on time so i'm going to probably skip over this but this is a really compelling biological question about that shows there's been some evolutionary pressure to get to this state but i'm going to skip over that and talk about optimize let me get past this get to something that's more essentially of interest so the summary of that is that the we've really enhanced the biofidelic interactions between translation molecules i think this is sort of the message i would like everybody to take home is that we've learned a lot about how to extract biologically relevant features topographical features and chemistry interpret these translate these into very efficient course-grade models and then relate these to dynamics and rheology okay so i'm gonna do a quick check-in on time i've got probably another five minutes is that gonna work for the timing of the seminar yeah yeah that's okay that's completely fine okay great okay so i'm going to touch just briefly on a third project which demonstrates how short-range molecular interactions can impact cellular organization and dynamics over much longer length scales so this work was conducted in close collaboration with the chevitz lab at princeton they're led by the first authors alp sunal and diana velverde menbez so alp will be telling the full story of is has told the first full story of this project and some of his talks but i'm going to focus here specifically on i think the part that would be most interesting to this group which is the electrostatic interactions so the chevitz group has developed methods to track the motion of individual particles inside e coli and 3d which is a really beautiful achievement the way that they've designed these experiments and the particles they use called gems are genetically encoded multi-american nanoparticles are fluorescent and are tunable by both size and charge so everything that you see here experimentally is coming from the chevitz group and has worked done by a doctor velverde menbez recently graduated from josh chevitz group so in tandem with their experiments we built a coarse grained whole cell model of e coli which includes a spherically confining membrane a porous nucleoid which is the genetic material in a basically a network and freely mobile cytoplasmic complexes and this also includes gems these tracking particles that are being used by the chevitz group so molecular characteristics including size abundance and charge were set in our simulations according with literature values and in this particular what i'm going to highlight here is where jennifer and alp developed the electrostatic interaction model where the particles either attract or repel based on their relative sign and magnitude of their charges and i think what's i mean there's a lot of biologically interesting things here and how we show the the ability of charge and size to regulate sort of the distribution of particles throughout the cell but i think it would be also nice just to see the ability for us to interact with experiments and translate that into a model so using our model we can interrogate particle localization and dynamics in direct comparison with the chevitz group experiments for example here we're plotting the likelihood of finding a gem inside the nucleoid which is the red region on the left as a function of its charge our simulations agree quite well with those from the 3d single particle tracking experiments with more positively charged gems spending less of their time inside the nucleoid our simulations can also suggest the mechanistic origin of the observed experimental behaviors for example we find that positive gems bind strongly to ribosomes which are abundant in themselves preferentially excluded from the nucleoid negative gems on the other hand interact only transiently with less abundant positive set of plasma complexes which permits them to more easily enter the nucleoid and these findings support previous experiments that suggested ribosome surface charge might limit the proteome composition to be coli preventing unnecessary binding of positive proteins to ribosomes overall these results emphasize that we're a molecule located where a molecule localizes and how it moves inside a cell depends on the broader context of the cellular interior for example the relative size charge and abundance of other macromolecules and so if we sort of extrapolate that to the sorts of things being thought about in this group we we have to think about not only just interactions between pairs but the surrounding environment the stoichiometry the relative size and how mobile in general the cytoplasm is cell scale features such as nucleon morphology and confinement are also crucial physical regulators of intercellular organization and i can talk to you about that in a in a separate talk and work that is being published by alp sunal and diana and our collaborators at princeton so zooming back out these three projects so far illustrate the broad role of colloidal scale physics in orchestrating life essential processes inside cells and more broadly how they orchestrate the behavior of biomolecules that have complex physical chemical interactions between them in the future this work can be used as a foundation for building increasingly biofidelic representations of living suspensions biological suspensions of living cells so for example incorporation of many body hydrodynamic interactions has helped interrogate additional collective and non-equilibrium behaviors and other work in my group and more finely resolved models are also useful for proteins with poorly defined structures or for suggesting specific sequence targets for mutagenesis and so theo yang in my research group in our collaboration with my colleague dan euros has developed a framework for studying the interactions between biomolecules that have both disordered and ordered structured regions and we've developed a pipeline for how to use both alpha fold and protein database to develop very nice coarse-grained models that still highly accurately represent the residues expressed on these proteins and RNA binding domains etc and this has led to some really nice interrogation that we've been able to do in my group for formation of clusters and things like prion-like proteins and things that are relevant to monoclonal antibodies so concurrently experimental testing could help validate and demonstrate the broad relevance of our results and it would be really interesting to sort of inspire experimentalists to do more work in that area and ultimately these tools can be applied to predict and engineer whole cell behaviors by tuning molecular and mesoscale thick features but what i'm excited about to talk about with this group and that will i think be in a follow-up meeting is how our work understanding how to get these detailed interactions between these structured and unstructured domains on protein and other biomolecules has led to some really interesting aggregation behavior and so going forward one of the things i'm most excited about is revealing how colloidal scale physics instantiates life and biological cells and to do that we're going to expand our existing we are expanding our existing computational models for colloidal physics and biological processes and we're building a physically and biochemically resolved and complete model of the JCVI CIN3A synthetic minimal cell in collaboration with Dr. John Glass at the J. Craig Benner Institute and we're going to use that to probe the matter life nexus how physics and life meet and this is work funded by the Sloan Foundation and i think that brings me to the end of my talk i want to thank my collaborators Professor Drew Endy, Josh Shevitz and the students involved with this project Jen Hoffman, Akshay Maheshwary, Alpsunal, Emma Gonzalez and Theo Yang and then at probably a follow-up would love to tell you about the really great success we've had with expanding all of this to include many-body hydrodynamic interactions and sort of the the the the spoiler there is that it's a lot more work and a lot more computational complexity we've gotten it to be very fast and we know exactly where it matters and how it contributes whether it's lubrication interactions or many-body interactions that matter with that i'll be happy to take your questions


Welcome to today's lecture on design analysis of orbit motor 2, which is geometric volume displacement. In earlier lecture, the in part 1, I have discussed about the geometric design of the star and envelope that is ring of an orbit motor. In this lecture, I shall show how to estimate the geometric volume of a chamber and then swept volume and how to find out the volume displacement rate with the shaft rotation. So, as I discussed earlier, the epitaphoid generated ropey mass that is rotary piston machines orbit and 0 to units, the envelope ring forms the chambers whereas, the epitaphoid star acts as a piston. If you look into this, this is the chamber. So, one particular chamber is shown here which is formed by you can say that the outer member that is envelope that is ring is the cylinder and this is acting as a piston. Unlike the cylindrical piston machines, in this case the area varies and so variation of volume with the shaft rotation is equal to the variation of area multiplied by the thickness of the star and ring which is constant. It is also discussed the number of chambers is equal to the number of chronodes of the envelope that is equal to z. So, the number of chamber will be equal to the number of roller in this case or it might be sometimes integral part of the envelope whereas, we know the lobe of this star will be 1 less than this. So, here there will be 6 lobes if we take z is equal to 7. Again in earlier discussion what we have seen that the ripple is less in case of odd number of chambers. So, while we shall discuss all the formulations particularly we have to be careful formulations are like that z is always an odd number that means number of chambers are always odd. A complete cycle of piston action comprises of two phases namely volume expansion and volume compressions. Let us look into this. So, this is the star and this is the ring. This is modified epithocate what we have learned and this is basically envelope and this modification is nothing, but the constant difference inward shift of the original epithocate generated. So, for which the envelope actually goes through this centre point envelope and then if you modify this envelope what we find this contact portion becomes circular arc. If you notice this rotation you will find that this contact is not up to the full arc it is some angle the maximum leaning angle you can say in both directions from the centre. So, but the thing is that we can replace this we can either make it integral or we can replace this integral part by roller. Advantage of roller I have already discussed which is that we can replace this roller if this is worn out. Now other part of the envelope is that this circle on which this rollers are kept I mean on the outer body this is slightly less than the circle the which is called bit circle slightly less than that. So, this roller does not come out and it is selected on the basis of at the top most position of this roller. Now this we can call that at this position the piston at its top dead centre top dead centre. Now when it will rotate you will find gradually this space is in being increased the black space is being increased that is the change in area. So, if we can estimate the change in area then if we multiply with the thickness of this star and this envelope which is equal nominally which is equal we can estimate the volume displacement and that is that we will learn today. But one important thing here I would like to discuss that if we consider this is the bottom dead centre then at this position you would not find any other chambers in top dead centre. Top dead centre will be which one will be the top dead centre in next while it will start rotating you will find that this one will become the it will reach at top dead centre this one just a small rotation. Let us see you can see this it is already had bottom dead centre and then again say suppose this is the minimum at this position you will find after a slight rotation this is becoming at top dead centre this angle can easily be calculated from the phases of such or bit motor from the analysis of the phases of such or bit motor. It is important to examine for the flow distributor valve action if we analyze the flow distributor valve action the duration of those phases in terms of shaft rotation for a single cylinder as well as the multi cylinder action. So, what we have to do we need to analyze how long a chambers or what with respect to the shaft rotation at what angle this bottom dead centre and top dead centre is occurring and according to that we have to also look into this and the distributor valve they should act accordingly. So, this phase analysis is not shown it will not be shown in today's lecture, but we will accept that when this is in expansion mode then one the while in channels are connected and when it is in the compression mode the while out channels will be connected. Now, in the analysis the initial position of the output shaft that is the shaft rotational angle zeta is taken as 0 what we are considering with this axis when this crest is at top dead centre that angle is 0. The centre c and o lie on the x, x dot axis this is x dash and here is the x. So, it is lying on this axis both this axis of the star as well as the axis of x axis of the envelope they coincides whereas, y axis is away at the centre distance c o whereas, this one is the this one y axis through the centre of the envelope and that is one of its dead zone we have called it the top dead centre and this is in the threshold of expansion phase that means immediately after that if we rotate in the clockwise directions here the expansion mode will start ok. Now this means that the centres are interchanged from their original initial position during generation. So, in the previous lecture I have explained that how this generation is done in that generation technique what we took initially that the centre of the star was here or the centroid fixed for star was here and the centre of the star sorry the ring here or envelope was here whereas, when we have started operations we have just interchanged the centre. This means that while we are we shall consider the geometry for the envelope we have to take care of this transformations ok. Now the positive z axis is along this z do not confuse this z is not the number of loads, but this is the z axis x axis y axis and z axis is in the upward perpendicular direction of the plane of the paper that is in this it is from the here will be the z axis which is coinciding with the axis of the output shaft. The output shaft is also directed outward so z axis is coinciding with that which I have just told you. Now we have taken some axis on the star and ring what are those we have taken a C S 1 C S 2 C S z be the central axis of the cylinders this means on the envelope we are calling that envelope is acting as a cylinder. So, on the envelope from C the centre of the envelope to the centre of this portion that is you can say the centre of the chambers that we have designated as C S. Now according to the number of chambers we have number the chambers also we have given the S 1 S 2 S 3 etcetera. Now here I have mentioned as a jth chamber any jth chamber, but only with respect to the figure which I have drawn we should say this is chamber 1 this is chamber 1. In the phase analysis I have shown that this is the chamber 1 next chamber is chamber 2 whereas the chamber on the x axis along the x axis when the rotation is 0 in that case this chamber is chamber 7. So, this is you say if you put this number this is h 7 this is S 1 and C is the centre. Similarly we have taken O T 1 O T 2 capital T O T z minus 1 be the central axis through the crest of the convex portion of the epitrocordial lobes. So, this is the convex portion the if you take the most point the crest point then we number is O T 1 O T 2 etcetera, but if you look into this case we have taken as if the O T 1 is here it can be named otherwise this is the T into z minus 1 because there will be 6 lobes 1 lobe less than z. So, that is why number will up to the z minus 1. Similarly we have also taken O small t 1 O small t 2 O t 2 that is the that is through the lowest point of this profile that is the concave portions we have taken this. Now these lines while we are analyzing the phases then which one is coinciding with which one depending on that we are giving the number we are assigning this is easy to understand, but in the present analysis our main concern is to estimate this area at any instant. We will find this area and you may find that these are not much useful here, but while you are writing something presenting something you can say suppose O S 1 has coincided with O T 1 capital T 1 or O T 2 like this that is easy to understand. To estimate full active volume of a chamber the volumes at top dead center and bottom dead center are calculated. Suppose if we would like to find out what will be the volume expansion of a chamber we calculate the volume at top dead center and bottom dead center. Top dead center means when the piston is at the top that means this is the top dead center whatever the volume at that condition that is not the active volume that is not the varying that is a constant volume that will be always there. So, we have to calculate separately this volume as well as we have to calculate when is the maximum volume at bottom dead center and then if we subtract this we will get the volume of a chamber. When the shaft rotational angle is equal to 0 O T 1 coincides with C S z in that case C S 7 if we take this z is equal to 7 and chamber z is at its top dead center. Now, after an angle that is it a 0 it a 0 which is calculated as pi by z into z minus 1 we will arrived and another chamber which is given by z minus z plus 1 by 2 th chamber reaches at its bottom dead center. But I have doubt I think this will be minus this will be true if we consider this is chamber 1, but we have consider this is chamber 1. So, possibly this will be z minus 1. So, that we can examine let us see again say if we let us consider this is 1 no this is 7th. So, 1 2 3. So, third one is reaching there. So, this will be z minus 1 by 2 not the z plus 1 it will be z minus 1, but this angle say this was at its top dead center when the shaft rotational angle was 0. Whereas, this will be at its bottom dead center then the shaft rotation can be calculated as pi divided by number of lobes on the envelope into the number of lobes on the a bit ok that is star the ring number. So, suppose in this case we have taken 7 and this is 6. So, pi divided by 42 it is around 4.3 degree or 4.27 degree for after which this angle will be a this chamber will be at its bottom dead center that means the maximum volume. So, what we can do we put the shaft rotation angle 0 we consider this 7th chamber we calculate the volume and next moment after 4 degree of rotation we consider the chamber 3 and we calculate the volume then this volume minus this volume will give the volume of the one chamber one maximum volume displaced of a chambers. So, this we have to do, but we must calculate this area now. So, this what I have discussed this is told here. Now, next slides what we will say that referring to this figure the area A j A j means area of the jth chamber act an instant when this has rotated by any triangle it is bounded by this area bounded by g h b d g h b d. Now, it is necessary to express the chamber area with respect to the axis fixed on the envelope we will find out this area considering this axis the fixed axis the fixed to the envelope or the ring. So, the axis x c x dash and y c y dash fixed on the envelope are also axis of the fixed reference frame that means this is fixed. So, this you can say the axis of the fixed reference. So, we will transform all coordinates with respect to this to estimate this area. Assuming the simplest modified form of the envelope what does it mean we have assumed this envelope is nothing, but that there is a circular ring and this is on a ring with the inner circle of this one. So, this is the simplest form which we have considered and the instantaneous area A j of jth chamber is given by A j is equal to A 1 minus A 2 where A 1 is the area bounded by the curve joining the points c d g h b c. So, we first we shall calculate c d g h b and again c this area we shall calculate minus the area bounded by the lines and epitropodal curve joining b c d. So, if we subtract now this area from this area we will get this instantaneous area of this. Now, how it is done? So, geometrical this is pure geometric analysis referring to this figure what we find the this is the area indicating c d g h b c is equal to area c e g h then f then c first we consider this area. So, this is like that we are considering this area. Now, next we subtract c e g d c now we subtract this area and then we add c f b c this area and then finally, we subtract h f b h f b h thus that area we exclude. So, ultimately we are getting this area. So, this you can check yourself this equations and you will find that we are calculating ultimately this area. Now, area in right hand side can be expressed as follows this is purely geometric analysis first we have considered this one this is we have taken pi r s where r s is the radius of this inner circle pi r s square by z that is the whole area minus r s square this is we have taken one seventh of this area because this area will be one seventh and then we have taken r s square arc cos you can you can just understand this how it is this is the cos inverse arc cos mean cos inverse of this angle. So, this is this will give one pocket and then next we shall consider the another pocket we will get this one. So, this you just you have to take this figure and this you have to understand here a 0 square is this is the a 0 this is square then r s square we have taken this this square minus r m square is this one cos inverse of that into r s square will give you this area. Now, plus r m into a 0 r m into a 0 into 1 minus this will give this area. So, we are going to get this area out of that next we consider the area c e g d c c e d g and again c e d g and c that means we are getting this area this is again if you can find out that first this is a triangle we have consider a 0 into this is perhaps not r 0 it will be r m theta r m this will be r m. So, this is 7 this is r m r m that means we are getting this perpendicular a 0 is this one. So, we are getting the area of this triangle this one then plus half r m square into cos inverse of this angle that we are getting perhaps this area this area we are getting this area minus no minus this area we are getting this area and then minus this. So, we are ultimately getting this area here now next we will consider c f b c c f b c this is simply half into a 0 r m by sin phi j. These angles are called leading angle that can be calculated for rotation of this shaft which I have discussed in the last lecture and finally h f b h is nothing but this angle. So, this area this is r m square into cos of that plus half r m into phi g that means we have considered this area first and then we have considered this area. So, now we are we will substitute this value in equations 832 and in non-dimensional now here is the question of non-dimensionalization which I have shown in earlier case what we find that area any distance we have put a bar. The bar means it is the non-dimensional value non-dimensional parameters. So, in that case what is done each and every parameter say if it is a length that it divided by r 0 what is capital R 0 what is capital R 0 capital R 0 is the radius of the centroid the bigger centroid the centroid for the envelope. So, we have divided by R 0 to make non-dimensional of any length a 0 r m etcetera. So, this means that while we are considering area suppose we have calculated this one what will be the actual area then a 1 bar into R 0 square capital R 0 square will be the area that you should remember now in the non-dimensional form this can be expressed as that I am not reading it. So, this when this equation is available you can just sum up and non-dimensionally we will arrived into that. So, now we have calculated only one area a 1 in the next page what we do we find out how they a 2. Now, for this a 2 the coordinates of b and d with respect to x y where this angle this angle is with the reference to the original generation of profile. Epitocids you just look at this angle indicates the shape rotation whereas, this angle with respect to the original development of the profile. Now, what you can find say this angle for the generations we take for the j th chamber psi j is equal to beta, beta is in this case pi by z that means half of this angular spread between two rollers plus the angle of rotation. So, beta plus angle of rotation we will consider for this one similarly, for the other one this angle this angle we will consider beta plus 2 pi by z plus this angle then subrotations. So, in the original formula of the profile we will put this value and this value to get the coordinates of these two contact points. So, therefore, this coordinates can simply be written in this form if you use this one you will find this coordinates. So, this is being transformed from the original coordinates what we have calculated with respect to the reference frame. First of all we have to calculate these two and then we will transform in this coordinate system. Next here I have mentioned this beta may be pi by z or any other angle only we have to take care of the proper geometry. Now, next we will express the A 2 in the dimensional non-dimensional form. Now, this is from this epitropoidal geometry it can be shown that this integration will give us this area A 2. If you would like to find out this area then knowing this coordinates of these two we can differentiate like this. Now, here I have presented this formula this is from the earlier lecture. So, this is the formula for expressing the coordinates of the envelope at any this psi angle. Now, in that case we have calculated this angle separately and we are adding to that. Remember we have to calculate the leaning angle while we are trying to calculate this coordinates of any contact point that is you have to also take the formula in the earlier lecture to calculate this properly. Now, if we evaluate this integral we will arrived into this formula, but still one integration part will be there for which this can be expressed in this form. Now, this integration is not presented in the any closed form closed from solution is not given here. So, we have to go for numerical integration to evaluate this, but may be a mathematician can he can find out something, but otherwise we have to go for this numerical integration. This is that you can find out this also this is not difficult to find out this is the leaning angle that means this angle. So, finally, before going into calculation this chambers here one interesting note is given that now we have calculated in the as a to this area. Now, at any instant if we add all such areas together that must give the area of the rotor which is important to find out the envelope I mean total area under the profile of the star which can be simply if we add all the area calculated like this at any instant we can simply get this area of the star. Now, the second term in is nothing, but the area bounded by d o b this equation can be modified to the form this one reference you will find that one reference he developed the formula in other way it was to me it was more cumbersome, but I have simplified this formula and to one we arrived into in this form which we will see in the next slide. So, what I have done I have substituted all the findings all the formulation in non-dimensional form and ultimately we get this will be the area of a chamber at any instant that means when the shaft is rotated by this angle we will get this angle or so to say I would consider if we consider any j and whether it is rotation is 0, 1 degree whatever it might be if we substitute this values we will get this area of a particular chamber. If we consider the next chamber we have to take all such values accordingly. The rate of change of this area with respect to the output shaft rotation is expressed in general form for both inward and outward modifications as follows. Now, what we have done we have first calculated this area. Now, here we have derived this as a rate of change of this area with respect to shaft rotation which now becomes in this form. If we look into this this is basically you will find that this motion of this epitropodal area this has no connection of the other geometric relations it is obvious the fixed part is eliminated in this case it clearly indicates that whatever shape we take for this envelope keeping this active envelope portion intact that means instead of taking this portion and this fixed portion as a this circular even if we take a big hole or even if we make a curve like this this equation will remain same. So, because this is only the variation of the volume which is working volume. Now, if we get this one then as a here I would like to mention one thing that this modification may be inward and outward this profile can be modified outward also. See in case of wonkel engine this is modified in the outward directions like that is small modification of course. So, to accommodate this whatever may be the modifications in that case of obviously it is inward directions what we have done we have used a equation force phi 0 in fact this phi 0 we can put along this axis. So, when this angle is pi that means this becomes minus 1 this is for outer modification that means in this formula for the orbit motor what we have considered we will put minus 1. So, this will become plus whereas, if we modify in the outward directions we will put this is equal to 1 that means in that case this angle is 0 and in inward modification this phi 0 angle is pi. So, it is minus 1 that we have to remember. Now, again so this variation this is obviously will valid for whether in compression mode and whether it is expansion mode, but we have to identify very carefully as long as this is in say expansion mode we will add with the other chambers which is in expansion mode because this machines in orbit machines is also a DC machines not alternating machines all the flow is being mixed. Suppose at an instant we find the chamber 1 chamber 2 and chamber 3 this is in the positive value. So, we will put this DAP that is in expansion mode and whenever we will get say for the chambers we will get this is minus we will put that separately and we will that make that this is in the compression mode, but the interestingly we will find that summation of these positive and negative will always constant. That means flow in flow out we will remain constant and another interesting information I would like to give you here although it is not shown, but if you take 1 chamber to the corresponding chambers you may find that the volume in and volume out is not same or in other words when say 1 chamber is in the expansion mode. Let us consider in this case or if I consider the fixed axis in that case this angle will be nothing, but these 2 pi by z that is the angle for any phase is expansion on compressions. Now, this phase what we will find 1, 2, 3, 4, 5, 6, 7 degree say such we will make it the interestingly we will find that for a particular angle the compression and expansion of a chamber is not equal whereas, in case of ordinary piston machines this is equal. So, whatever may be the rotations whether it is suction or compressions you will find a single degree of rotations this volume is constant. In this case these volumes are neither constant nor it is matching with the expansion phase and compression phase whereas, if we consider the overall that is matching and it has to match otherwise this machine will become impossible it will not rotate smoothly. Anyway our purpose is to find out the swift volume. So, what we it is to be noted that this rate of change of area is independent of the shape of the inactive envelope. This is inactive envelope may be say suppose this is the contact point up to this. So, this portion is active and this circular arc and this circular arc all inactive. So, it really this formula is independent of this area whatever it might be sorry whatever the area in equation 9 which is different for other approximated curves is only useful to calculate the area and volume at an instant. Only thing suppose if you would like to find out the trapped volume unused trapped volume then you have to consider the geometry. The expression of for flow rate speed ripples etcetera for such a rotary piston machines with different kinematics that is zero tour orbit units with star output ring output etcetera can is can easily be derived with the help of equivalent system concept. What it is? Now orbit motor what we have seen the ring remains fixed and output is taken through the star. Star is having two rotations one rotation is the revolving actions around the central axis that is the axis of the outer ring and it has also rotation about its own axis which is the output in case of orbit motor and orbit this version is only used as a motor because other pump version is not efficient or beneficial from the transmission point of view volume displacement point of view it needs if we make a pump with orbit principle it will need large torque for slow speed which is not available from the engine. So, it is not beneficial on the other hand when we make it fixed axis then which is called zero tour also a special name is assigned g roller when we use this type of roller when it is integral usually call it is zero tour or we should call in general case it is zero tour unit that is fixed axis unit. What is fixed axis unit? In that case both rotates about their own axis, but even they rotates about their own axis you will find this compression expansion that variation of this area will occur. So, that can be used as a pump as well as that can be used as a motor also. Now, what we have done we are analyzing all such area and everything with respect to the orbit motor, but it is possible that if we know the kinematics. So, same formula also can be used for the other kinematics that means zero tour units out let us see. Now, what we do the general flow we can express that what we have derived that this is 2 pi nr nr is the output rotations the b is the width. So, d A p and d gamma this gamma is general angle of rotations we have given. So, this simply if you add them you will find out what is the rotations and in case of orbit motor this will put eta and this is the output rotation of the shaft. Now, this angle is nothing, but transmission ratio into this angle this transmission ratio we have derived separately which is not given here. So, directly if you would like to use this formula you had to know this transmission ratio in terms of the gearing in between this as star and ring which is tabulated form available one of the references. Now, in case of obviously in case of orbit motor we will put this as a 1 orbit motor this as a 1. So, we can easily calculate this one or simply this will be replaced by this one in case of orbit motor. Now, here I have written is in case of 1 for orbit unit. So, without following the table you can still calculate this if you follow this one for 0 to width unit width ring rotation when the ring is rotating simply you put 1 by 1 by z minus 1. So, we will get this value and then this flow rate will be for that 0 to unit width ring rotations for the star rotations this will be 1 by z and we will find this same expression. In case of orbit motor this is 1 however, the generalized dimensions less shift volume V s can be expressed for unit width width is given by B of star ring as follows. What we do in case of so this first we calculate this I t is 1 for say orbit motor and then we multiply with this. This is flow rate volume in and we integrate up to this angle and then we simply find out. So, this is I would say rather it is a complicated form to find out the swept volume, but automatically from the flow rate we should be able to get the swept volume. So, this is shown here, but I will show you the simplest way of calculating the swept volume. Now, this is again the same thing I have described here while you are using this formula. The average geometric volume displacement Q A average you see this if we estimate this one with respect to this there we will find the flow fluctuation. So, what we should do I mean once we calculate this we simply multiply with the width factor and because this is for the unit volume unit width. So, we have to multiply here with B and then the rotational speed. Now, the alternative method of calculating swept volume this is just to calculate the swept volume what we can do? We can calculate the BDC and TDC at area of the chambers at BDC and TDC and which I have earlier explained that if you for the geometry we have considered if you consider this angle is equal to 0 then you can first calculate the this area at chamber 7 which is the chamber at top dead center and then after a rotation of pi by z into z minus 1 angle you can take the other one that is z minus 1 by 2 that is in this case this will be 3. Third chamber after this after that rotation of pi by z into z minus 1 and then you will get the bottom dead center area. So, and then what you would do the working volume is nothing but the area difference of these two areas into the width. Now, the swept volume is calculated then as follows for orbit motor what we do simply this volume of one chamber into this action in one revolution that means how many such chamber actions will be there in case of orbit motor z into z minus 1 time. So, you simply you multiply this you will get the swept volume orbit motor interestingly in orbit motor this is one respect to the other that means either star rotating with respect to the ring or ring rotating with respect to the star. So, in that case there will be single formula to find out this swept volume whether the star is rotating or ring is rotating whether whereas in case of zero tour unit this will be either v c into z or v c into z minus 1 depending on ring and star rotation respectively because in that case suppose your star is output both are rotating star is output. In that case you will find that when the star one full rotation is done actually 6 chambers are displacement is there not the 7 chambers whereas if the ring is rotating then 7 I have mentioned 7 or 6 in that case z minus 1 and z. So, carefully if we use we can calculate the swept volume of the ring I mean zero tour depending on which one is the output. Let this star ring of an orbit motor have the following data we will take a numerical example. Now in that case we have taken z is equal to 7 e 0 bar is 1.625 r m bar is equal to 0.405 beta b is approximately 0.7 it is slightly not 0.7 some value is there and r s means this one is 1.55 whereas r 0 is 19.6805 approximately 20. So, you can imagine what is the actual dimensions then this is r 0 is the radius of the inner centroid centroid for the epitroquate which is 16.869, but this is not required because we have non-dimensional lies with respect to this one. Interestingly say we use a 0 r m bar, but we never use this C 0 bar because C 0 bar is nothing but 1 by z because this you will find this and relation r 0 by capital R 0 by small r 0 is nothing but z by z minus 1. So, as their difference will be 1 z. So, this means that C 0 bar is 1 by z. So, in the formula you will find that not C 0 bar is used only 1 by z is used. So, with this data if we calculate this area what we have done for the swift volume a z we have taken first a 7 we have calculated and then this is again I have made a mistake this will be minus 1 by 2 the third chambers and this is a 3 and we will get this now this area these two areas are a TDC and a BDC respectively. Therefore, we will now use equation 14 to calculate the working full volume of a chamber which becomes this and you see this I have not shown the calculation, but to calculate this area we have used this formula and ultimately we have found out that this full volume of a chamber working volume will be around 2 cc very close to that. So, therefore, the swift volume of an orbit motor of with this data it will be 84 cc per revolution the flow required is 84 cc. So, you can imagine that in case of the fixed axis it will be either 7 times less or 6 times less than that for one revelations this is one. Second thing look at the size, size is say a 0 is 1.625 if you multiply with this 20 it will become 20 plus 12 30 or may be 35 millimeter here and this might be another say 20. So, it is 55 to 60 millimeter at the most. So, diameter will be 120 millimeter which is less than 5 inch 120 millimeter will be something like this. So, this is the size of the motor whereas and thickness is only 14 millimeter if you multiply with this it is close to 14 millimeter, but you see this how much volume is required for one rotation. Obviously, this is due to the gearing action in inside. So, in this way you can calculate the swift volume also you can calculate you have to obviously you have to make the numerical integration to find out the volume displacement of such machines. So, these are the references I would say that this paper you will not be able to follow because this is in this is old as well as this is in German language I think not in not even German language it is something it is not exactly German language I think these two people we are from Austrian people. They this if you read this paper you will have some idea about this fixed axis machine not orbit motor also Colburn he derived such displacement formula for fixed axis and Thoma he is I have used this how to phase is changed another this you can read this book also, but if you would like to know that phase differences and volume change etcetera. So, you can read this paper. Thank you.