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Module 1: Constraint-based Methods

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Robustness Analysis and Flux Sampling

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Metabolic EngineeringProf. Amit GhoshSchool of Energy Science and EngineeringIndian Institute of Technology, KharagpurLecture - 21Robustness Analysis and Phenotypic Phase PlanesWelcome to metabolic engineering course, today we will discuss about the robustness analysis and phenotypic phase plans for the metabolic network.(Refer Slide Time: 00:36)So, the topics will be covered today is about the mathematical formulation of objective function and then shadow prices and reduced costs analysis, phenotype phase plane analysis from this network analysis the different optimization scheme will be used to actually understanding the overall network how you can actually design and how you can bring out new properties?(Refer Slide Time: 01:00)So, these are the constant based methods which are already available in the previous class we discussed about flux variability analysis. So, this one we discussed. So, we actually have a range of fluxes that is v 1 min and v 1 max, v 2 min and v 2 max. So, these ranges of fluxes we have to we can estimate because there are many multiple solutions which exist in the biochemical network.So, given an objective function we can have a range value of fluxes which can give raise to the objective function that is the biomass objective function value which is unique. So, the biomass objective function for a given network is unique just the all other internal fluxes may have a range which can goes from low minimum value to maximum value and any value within that range can give rise to the maximum biomass.And then we discuss about gene deletion algorithm is we have discussed in the previous class we learned about the FBA deletion (())(02:09). So, these are the 3 techniques we learned. So, many things you can do by constraint based method FBA that we learned the flux variability analysis and today we are going to learn about robustness, phase plane, flux coupling we already learned in the previous class. So, as we progress we learn more or more of these constraint based method using the metabolic network we can do various analyses constraint based analyses to understand the network.(Refer Slide Time: 02:38)So, this is the FBA formulation that we discussed previously the FBA optimization problem for every objective formulation actually starts with the objective function. So, a function that can be minimized or maximized to identify the optimal solution and the constraint you put on the allowable fluxes that is the lower bound and upper bound of different regulatory constraints in terms of fluxes you can put and then you have the steady state approximation that is there is no metabolite accumulation and then you maximize the objective function.(Refer Slide Time: 03:16)So, this formulation of maximization is basically is represented in terms of flow reaction. So, we have say the slide illustrates the formation of objective function using simple example. So, there are 4 metabolite fluxes, so there are more metabolite fluxes we have and the objective function you have to design is actually maximize ATP production this will be maximized, to maximize ATP production therefore, the c has 0 for all other the c you define acolumn vector c that have all the components 0 except where ATP is there, we have the V ATP.So, the only V ATP that position we have 1 and other values are 0. So, we multiply the c dot v any we actually get the 0 multiplied by they should be transposed. So, then you multiply v dot c or c dot v then this would be a column letter this is transferred this is cancel this. So, when you multiply c dot v, then what you get is 0 multiplied by v G6P and then 0 multiplied by flux vector F6P and 1 multiplied V ATP.So, all other terms get 0 because you are multiplying by zero with all other fluxes except V ATP. So, you maximize since you I want to define an objective function I want to maximize ATP production. So that is why I kept the c to be nonzero at V ATP position. So, the coefficient of ATP flux is positive since it is being maximize. So, this way you define the objective function and remaining constraint S dot v = 0 and a boundary condition you already know.This way you can formulate the objective function. So, any biomass you can optimize any flux also you can maximize or minimize. So, in flux variability I have seen that you are minimizing and maximizing fluxes provided and your objective function that is growth rate you are keeping fixed to that maximum value, so this way you can formulate an objective function and define cv that is v is basically is a column vector and c is a row vector.So, c is a row vector. So, when you multiply c and v then you get one value that is V ATP that you want to maximize this one you want to maximize or maximize V ATP provided you have c vector and v vector which is basically summation of c and v.(Refer Slide Time: 06:24)So, the growth function which you have already know the growth function that required this was the requirement for making 1 gram of E. coli. So, this is the growth functions which have been defined here, see the Z. So, Z we are maximizing here. So, Z you can define as well as a linear combination of different component in the biomass. So, we need we V ATP multiplied by the coefficient this coefficient you get it from experiment.So, this coefficient you get it from experimental value for 41.257 3.547 NADH. So, these are the component you have to see for each cell it varies from cell to cell. So, this is defined for E. coli do you suppose you use Saccharomyces cerevisiae this component will change the coefficient, the coefficient value will be changing. So that this means that for cell to grow all the components must be provided in this amount for getting some growth in the cell you have to this much flux that is the coefficient value you should be able to the cell should provide otherwise it will not grow.So, thus a balance set of metabolic demand makes up the growth objective function which is shown over here the biomass composition thus solve to define the weight vector c, the full growth function of E. coli is much more complicated than one which is shown over here. Since various maintenance functions need to be considered, for simplicity we consider a simple equation where the coefficients are shown over here which is experimentally determined and then it would add this other objective function for the growth.So, this the NADH, how much NADH? How much NADPH? G6P, F6P the amount it is required that comes in as a coefficient in the biomass equation.(Refer Slide Time: 08:32)So, now, we will come to another concept that is the shadow prices that is dual problem, what is shadow prices in designing metabolic engineering strategies an important question is that what extent can specific fluxes be altered because you are doing metabolic engineering you are basically changing the fluxes the inter cellular fluxes are changed when you do metabolic engineering what happened you either removing a gene or adding a gene or up regulating or down regulating gene.And then what happened the fluxes inside the are going to altered, the whatever genetic perturbation you do a ultimately the fluxes get altered this is a regular phenomena what and you have make sure the effect will be on this cellular process of interest including growth and product formation. So, whatever perturbation you do at the genetic level, it should be it should affect on the cellular process.If it is not affecting the cellular process then you have an effort like you have generally this metabolic engineering strategies are actually labour intensive and it takes a lot of time. Suppose, you want to remove a gene it takes 2 to 3 months or even more depending on the organism you are changing. So, this includes growth and product formation. So, your main aim is to actually product formation and also including the growth.So, if the fluxes are not affecting growth and product then it is the genetic perturbation is not use. So, you have to design in such a way that the specific fluxes with that you want to alter it has an effect on cellular processes like growth and product formation, these issues can beaddressed within the LP formulation the linear programming which we discussed in previous classes.That linear programming formulation can be used, where the sensitivity analysis can be performed for the optimal solution. So, the sensitivity analysis can be calculated in terms of shadow prices, this is another term which is used to actually calculate the sensitivity of the metabolic network, the shadow prices are the derivative of the objective function. So, what is shadow price is basically a derivative of the objective function at the boundary with respect to exchange flux.So, it is basically a derivative which is given by gamma i. So, Z is the objective function you already know is biomass of objective function and then you do a derivative of the objective function with respect to exchange flux. So, not internal fluxes is the exchange flux that you make a derivative and these shadow prices, so that it can be used to determine whether this cell is limited by a particular constraint or not.That is the exchange flux that the flux which is entering the cell, whether it has a limitation or a constant in the cell growth that you can check this feature has been prevent to be very useful in interpreting optimal solution and metabolic decision making then especially in metabolic engineering. So, whether the excess flux is actually important for the cell or not that you can check using this equation.Using this equation the shadow price precisely define the incremental change in the objective function, if a constraining flux is the incrementally change, shadow price may change discontinuously as the excess flux is varied, these shadow prices can be used to determine whether a optimal functional state of a network is limited by the availability of particular compound. So, whether the network is actually constrained by any compound that is up taking or producing that you can check.Sometime what happen the many products which are synthesizing the cell are actually constraining the network? So that using shadow prices, you can even check whether it is limited whether the availability of a particular compound is actually constraining the network or not. So, this is very important to the shadow price calculation is important to check thelimitation of the network or the robustness of the network you can check very easily on the sensitivity of the network.(Refer Slide Time: 12:42)So, another term we introduced is the reduced cost the sensitivity measure the reduced cost can be defined as the amount by which the objective function can be reduced if the corresponding enzyme is forced to carry a flux which was not carrying a flux before. So, you turned on it or express that gene, so that it carry a flux. So, when metabolic engineering also this is very crucial where you add gene now all of a sudden it carries a flux.But as soon as the reaction carrier flux because of the enzyme availability of the enzyme, then how much the objective function is going to reduce whether the objective function will reduce or not? The objective function is basically the growth rate whether there is a effect on the growth rate or not? Moment you add a gene. So that also you can check using the metabolic model in the analysis of metabolic systems several important questions arise that can be addressed with an analysis including the reduced costs.Using a reduced costs you can check you can do metabolic system analysis and you can address several costs and that may arise because of the metabolic engineering, the reduced costs can be used to analyse the presence of alternate equivalent flux distribution. So, whether it has a equivalent flux or not? If the right set of reduce costs are 0 or not that also you can check.Additionally, the reduced costs are important in examining the effect of gene deletion. As I told in metabolic engineering, we do a lot of gene deletion whether this the deletion of genes actually affecting the overall function of the metabolism that can also change from the reduced cost. So, reduced cost analysis is another parameter, the reduced costs can be defined as the amount by which the objective function will change with flux level through an internal reaction that is not a not in the basis solution.That is a flux that have a 0 net flux, several important question that can be addressed using reduced costs, the reduced costs can be used to analyse the presence of alternate equivalent flux distribution. If a reduced cost is 0, it means that the flux level through the corresponding reaction does not change the objective function, so the reduced cost is 0 suppose, I get a 0 for that reaction that means he does not actually correspond to does not change the objective function.So, your objective function that is the growth rate or the biomass equation is not changing, the growth rate is not changing just reduce costs can be useful in examining the effect of gene deletion. So that is why the gene deletion and other experiment you want to perform on the bench you can check before if the value is 0 by adding a gene then it is not affecting the growth, then it is beneficial for metabolic engineering similarly, for deletion also you can check. So, this reduce cost measure is actually sensitivity of the metabolic network that you are designing or the phenotype you are planning.(Refer Slide Time: 15:44)So, in summary, the shadow prices and reduced costs what we learned in shadow prices, we defined dZ / db for each constraint or metabolite that is importing or exporting. So, dZ / db the shadow prices is less than 0 means the metabolite is required by the cell and it will increase the objective function in shadow prices a shadow price is greater than 0 that means that metabolite is not required by the cell and you can remove the metabolite from the network.So, these are the 2 important conclusion from the shadow price, the shadow price is less than 0 that means that metabolite is required by the cell and if you provide more of the metabolite and then what will happen your growth will increase that is Z will increase. Similarly, the reverse is the case where the shadow price is greater than 0, when shadow price is greater than 0 that means that metabolite not required by for the cell growth.That means by removing the metabolite would increase the Z, if you remove that metabolite and it will increase. And the reduced cost what we see that it reduced costs is less than 0 that means increasing flux would reduce Z, so if the reduced cost is increasing means if it is less than 0 that means the flux if you increase the flux, so that reaction that will increase Z. So, these are the 2 important conclusions that you draw from the shadow prices and the reduced cost.(Refer Slide Time: 17:29)Now, we will discuss about the robustness analysis, what is robustness analysis, the sensitivity of the optimal property of a network can be assessed by changing parameter over a given range of value and repeatedly computing the optimal state. Both environment andgenetic parameters can be considered. So, here in that what will happen you change the sensor, check the sensitivity of the optimal properties of the network like the growth rate by changing some parameters over a range of values.So, you change one of the parameters, what a range of values like from 0 to 10 you change and that those parameter can be environmental or genetic parameter that can be considered because robustness you want to check how the network is changing with changing one parameter at a time. So, you can change one parameter at a time. And then you see how one parameter variation in one parameter how the network is changing.So, one parameter can be varied in a stepwise manner. And the LP problem solved for every incremental The isoclines can be either vertical or horizontal then what happened? The alpha value in these phases will be 0 and infinity and these phases arise when the shadow prices one of the subset goes to 0. And thus has no value to the cell. So, the shadow price that is gamma in the 2 the shadow price for A one of the uptake rate goes to 0. So, when shadow price goes to 0 then it has no value to the cell. And in the third case, what happened? The phases in the phenotype phase plane can have a positive alpha value.So, alpha in this case we consider alpha to be greater than 0. So, in these phases, one of the substrate is innovated it towards obtaining the objective function and this substrate will have a positive shadow price. So, this is when you consider alpha greater 0 what happened is one of the shadow price become positive. The positive side of the metabolic operation in this ways is a waseful, in that it consumes substrate that is not needed to improve the objective function.The post peaks, the phases with positive alpha values are expected to be physiologically unstable. For example, under selection presser cell would move their phenotype state out of the phase. So, in this case, what happened? They consumed substrate that is not needed to improve the objective function, so that you can remove that metabolite the alpha when alpha is positive, the phases with positive alpha are expected to be physiologically unstable. And then what we the fourth one is basically finally due to stoichiometric limitation.There are infeasible steady state phases in the phenotype phase plane, if the substrate are taken up at the rate represented by this point the metabolic network is not able to obey the mass energy and redox constant while generating the biomass, the metabolic network can only transiently operate in such a region. So, it is basically this is the region the infeasible region where it is the stoichiometric; this is due to limitation of the stoichiometric network that creates infeasible steady state phases in the phenotype phase plane. So, this is a region is in feasible region.(Refer Slide Time: 36:52)So, what we have is basically the one alpha is actually negative and the alpha one is positive and these are the region which are formed bigger when the alpha become positive or negative we have the futile region and we have the limitation region these single substrate limitation and the futile region and the dual substrate limitation is comes under these are these are the 4 region which is shown over here.So, these are the infeasible region which you are not important and then we have the futile region and the substrate limitation region and this single substrate limitation region, this you can calculate from the phenotype phase plane.(Refer Slide Time: 37:37)So, in summary what you get is the infeasible region where the flux do not balance, so the stoichiometric. This is the limitation of stoichiometric and the region of single substrate limitation where alpha = 0. So, this say if you go to previous one where the substrate limitation happened and this is the substrate limitation where alpha = 0 or alpha = 0 or infinity. In this case the substrate limitation happened and then we have a region where alpha less than 0 and this is the dual substrate limitation.(Refer Slide Time: 38:18)So, this is the region where you have the dual substrate alpha less than 0. So, alpha less than 0 that is the dual substrate. So, this is the alpha less than 0. So that is the region we have thedual substrate in limitation and then we have futile region where alpha is greater than 0, So, this is the case we have the alpha greater than 0 and the line of optimality the isoclines the constant height in topography maps and the line of optimality correspond to the maximal biomass yield.The line of optimality will lies in the region where we will get the maximum biomass line of optimality correspond to the maximal biomass yield that you are observing by fixing the carbon updated and optimize the biomass using a FBA, this will give you the one point of the optimality. So, once you fix the carbon uptake rate and the biomass and after you optimize the biomass you will get a single point on the line of optimality.And that is the region you have to operate to get the maximum biomass and also you will see that they go add that oxygen carbon oxygen carbon uptake rate, you will be able to get the maximum biomass while doing the metabolic engineering. So, this is the growth phenotype phase plane that you can get for a while doing this ceylon acetate equal ceylon acetate but you can see the oxygen uptake rate which is changing with time and then the acetate uptake rate will be change.And this is the line of optimality which is shown over here where the growth is maximum, this is the region where we want to operate you can see that this is the region if you can operate the cell in that region then you can get the maximum this you can see that is my biomass is actually greater than 0.27 and this is where you want to operate the cell.(Refer Slide Time: 40:16)Similarly, if you want to grow the cells which are on succinate, then you can see the line of optimality where the growth rate is maximum is shown over here and this is the region you want to offer the cell this is the end and then beyond that, it was dual substrate limitation. This way you can characterize the do the sensitivity analysis and reverse analysis in terms of to uptake rate that is oxygen uptake rate and the substrate uptake rate. So, varying these 2 parameters, we will be able to draw the phenotype explain and try to calculate the line of optimality where the growth rate are maximum.(Refer Slide Time: 40:53)So, in calculation the flux balance and the capacity constant from a closed polyhedral spaces. So, basically you unable to do the flux balance analyse you get a own kind of structure this is this polyhedral is basically you actually contain all the flux balance solution and the linear programming can be used to find the optimal solution in this space. So, using the linear programming, you will be able to get the optimal solution is a single point where you have the maximum growth rate.And also you can shadow prices and reduced costs are used to characterize the optimal solution. So, using the shadow prices and the reduced costs formula that we discussed today can be you characterize the optimal solution how the optimal solution is good or bad that you can characterize and all possible combination of the value of 2 parameters can lead to the definition of a phase plane.So, you can draw from the network you would be able to by considering 2 parameters like oxygen uptake rate and the substrate uptake rate you can draw a phase plane and theboundaries in the phase plane edges only the polyhedral cone. So, the boundaries we already get from and you can get it from the polyhedral cone and thus the boundaries represent the systematic pathway.So, this is what we learn today. And for references you can read the book written by Bernard Palsson is that assistance various properties of reconstructing network and also you can read about the nature review in microbiology. Also you can read the Journal of bacteriology. These are the references you can read for you for the study. Thank you for listening. Hopefully you enjoyed the class.Metabolic Engineering Prof. Amit Ghosh School of Energy Science and Engineering Indian Institute of Technology - Kharagpur Lecture - 22 Flux Sampling, Optknock and Optstrain Welcome to metabolic engineering course, today we will discuss about flux sampling optknock and optstrain. Flux sampling is very important concept where you can sample the solution space of the metabolic network that you are actually analyzing. It can be E. coli metabolic network or yeast metabolic network and optknock is a strain design algorithm and optstrain is also a strain design algorithm or you can design the strain in silico. And then once we have a better prediction you can go in the lab and perform the experiment. (Refer Slide Time: 01:02)So, this involve the sampling flux solution space that we are going to learn today and the strain design algorithm optknock and optstrain. These are very popular algorithm which is used by industries, academia and then we have a lot of success. (Refer Slide Time: 01:20)So, this is a constraint based method which we discuss every class for last 2, 3 days, where you can see that the we learned about flux variability analysis, which is shown over here and then when then will yesterday we learned about flux sampling and then today we are going to learn about sampling. And flux coupling we learned in previous class robustness analysis also we learned in previous class. So, dynamic simulation we learned in previous class, gene deletion also we learned in previous class. So, this today we are going to learn about the sampling and also optknock. So, these are the 2 things we are going to learn today. (Refer Slide Time: 02:09)So, I will just introduce how you can actually characterize the metabolic solution space. So, the solution space you can see is actually constraints is bounded by the constraint and how you can sample the solutions with a dot-dot you can see we can have to sample the solutionsspace that you can get and get your metabolic network very well defined. And that is the generally use a FVA flux variability analysis to get the range of fluxes but sampling is the best way where you can sample the solution space accurately and then characterize your metabolic network. So there are method to characterize metabolic solution space, such as extreme pathway analysis, this we will learn in subsequent classes and then the effect of imposing constraint can also be study using randomized sampling methods. Defining the size of the solutions space and how the space changes. So, you define the space of the solution space. And then how the solution space changes with the constraint or, with there is a function of time, you can see how the solution spaces are actually changing with time. And then we will randomize sampling can be used to characterize flux solution space concentration solution space also kinetic solution space and the randomized sampling can be used to understand the network capabilities enzyemopathy disease state that is their diabetes condition of the cell or ischemia condition. These are the disease condition which you can characterize using the flux sampling method. So it can be applied to spaces that are linear or nonlinear can be applied to convex or non convex spaces, and solution spaces. So mostly we will be applying in convex solution space where the sampling is needed to understand the type of network, the network capabilities in different disease conditions that also can be evaluated. (Refer Slide Time: 04:14)So let us see what is a sampling a space? What do you mean by sampling a space? So let us suppose what is the area of this object can you calculate the area of the object it is very complicated. To carry a can calculate this 2 dimensional object I am showing over here. So a 3 dimensional object can be more complicated. Make it simple, we considered a 2 dimensional object and the 2 dimensional objects you can actually calculate the area just by sampling. So if you do uniform sampling of this area, then you will be simulation these studies have led to several notable results 3 of which are briefly I will discuss briefly the histogram provide information about shape of the solutions space. So, the histogram you can see that each flux has the glycolysis and the PPV pathway, you can see that the histogram these into histogram actually gives you provide about the shape of the solution space and how likely the flux are to fall into certain numerical values. For instance, some of the histograms are flat implying that every numerical value for a flux with a particular reaction is equal likely. So, some reaction fluxes are actually flat in this case, in this case, you can see that it is flat. So, all these values are equally probable. So many of them are flat actually so, all these value on this range is actually equally probable and the cross correlation between any pair of flux can be computed. So, you can go also correlate 2 fluxes and correlation all values also can be computed. And you can see how the flux values are correlating and this is another way you can actually say how 2 reactions are correlated the measurement of poorly correlated flows is likely to be more informative than measuring highly correlated fluxes. So, some reactions are very highly.