Understanding Stochastic Systems

Stochastic Modeling

Estimated read time: 1:20

Summary

In this MIT OpenCourseWare lecture, the focus is on understanding and simulating stochastic systems. The lecture begins by continuing the discussion on the master equation, particularly on formulating it for complex systems with multiple chemical species. The exact GPE method for simulating stochastic systems is introduced, emphasizing the difference between probability distribution evolution and individual stochastic trajectories. The lecture further explores the Fokker-Planck approximation for modeling systems, highlighting its intuition for diffusion on potential landscapes. Questions about protein bursts, master equation applications, and the Gillespie algorithm's efficiency are explored to help understand stochastic processes better.

Highlights

The lecture demystifies various approaches to understanding stochastic systems, including the master equation and GPE method 🙌.
Explains how to formulate the master equation for complex systems with multiple species 🤓.
Introduces the Fokker-Planck approximation to bridge understanding between stochastic trajectories and diffusion intuition 🌐.
Discusses protein bursts in stochastic simulations and how they differ from deterministic expectations 🔍.
Highlights the Gillespie algorithm’s efficiency in simulating stochastic systems over naive time-stepping methods 🔄.

Key Takeaways

The master equation helps model the evolution of probability distributions in stochastic systems 🎲.
The GPE method offers an exact way to simulate individual stochastic system trajectories 📈.
Fokker-Planck approximation is useful for systems with large yet non-negligible fluctuations 🌊.
Stochastic simulations can manifest protein bursts not visible in deterministic models ⚡.
The Gillespie algorithm enables efficient and exact stochastic system simulations 🚀.

Overview

The lecture delves into modeling stochastic systems, focusing on mastering the distinction between probability distribution evolution and individual trajectory simulations. Using the master equation, students learn to model complex systems involving multiple chemical species.

Next, the Gillespie algorithm is introduced as a computationally efficient method for simulating stochastic systems, saving time over naive methods while remaining exact. Students explore the differences in interpretation when using deterministic versus stochastic approaches, particularly in phenomena like protein bursts.

Finally, the Fokker-Planck approximation is presented as a tool for gaining intuition about systems where fluctuations are significant but manageable. This approximation aids in bridging concepts of stochastic modeling with classical diffusion understanding, offering a holistic view of analyzing complex systems.

Chapters

00:00 - 02:30: Introduction and Overview The chapter introduces the purposes and goals of the lecture series, which is part of the MIT OpenCourseWare initiative. It emphasizes the importance of understanding stochastic systems and suggests that the subsequent lectures will focus on various approaches to modeling these systems. The material is provided under a Creative Commons license, and there is a call to support MIT OpenCourseWare for it to continue offering free educational resources. Additional materials can be accessed on the MIT OpenCourseWare website.
02:30 - 10:00: Master Equation Basics In this chapter, we delve into the basics of the master equation, furthering our understanding from previous discussions. The focus is on applying the master equation in various complex scenarios, such as systems with multiple chemical species. Additionally, we introduce the gespi method, a precise technique for simulating stochastic systems.
10:00 - 19:00: Gillespie Algorithm The chapter discusses the Gillespie Algorithm, emphasizing its computational tractability compared to naive methods. It highlights the qualitative differences between the Gillespie Algorithm and the master equation. While the master equation examines the evolution of probability distributions across a system, the Gillespie Algorithm focuses on generating individual stochastic trajectories. Starting with similar initial conditions in the Gillespie Algorithm can yield probabilities in specific contexts.
19:00 - 25:00: Fokker-Planck Approximation The chapter discusses the Fokker-Planck approximation in the context of the gaspie method and stochastic trajectories. It contrasts between thinking in terms of probabilities versus individual trajectory instantiations. The chapter aims to help understand when to use either approach effectively. Furthermore, it explains the Fokker-Planck approximation, which is useful for intermediate cases to make continuous approximations.
25:00 - 33:00: Production and Degradation Rates The chapter 'Production and Degradation Rates' begins with an introduction to concepts related to diffusion in effective potential landscapes. The speaker briefly pauses to check for any questions or administrative concerns, particularly reminding the audience about the upcoming midterm exam scheduled for next Thursday evening from 7:00 to 9:00 p.m. The speaker notes that students should email Sarab if there are any conflicts with the exam timing, emphasizing the importance of timely communication.

Stochastic Modeling Transcription

00:00 - 00:30 the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer highquality educational resources for free to make a donation or view additional materials from hundreds of MIT courses visit MIT opencourseware at ocw.mit.edu so today what we want to do is uh discuss various approaches that you might want to take towards trying to understand uh stochastic systems in particular how is it that we might model
00:30 - 01:00 or simulate a stochastic system now uh we will kind of continue our discussion of the master equation from last time hopefully now you've kind of thought about it a bit more in the context of the reading then we'll discuss kind of what what it means to be using the master equation and how to formulate the master equation for more complicated situations for example when you have more than one chemical species and then we'll uh we'll talk about the idea of the gespi method which is uh an exact uh way to simulate stochastic systems and uh it's uh it's both exact and
01:00 - 01:30 computationally um tractable as compared to what you might call various naive methods uh and the gespi method is is really qu sort of qualitatively different from from the master equation because uh in the master equation you're looking at the kind of the evolution of probability distributions across the system uh whereas the gpie method is really a way to generate individual stochastic trajectories okay so if you start with somehow similar initial conditions then you can actually get uh you can get for example the probability
01:30 - 02:00 distributions from the gaspie method by running many individual trajectories but but it's it's kind of conceptually rather different because of this notion of whether you're thinking about probabilities or you're thinking about uh individual instantiations of some stochastic trajectory so we'll try to make sense of uh when you might want to use one or the other and then uh and then finally we'll talk about this uh fuer plank approximation which uh as the reading indicated for intermediate uh ends it's useful to make this kind of continuous approximation and then uh
02:00 - 02:30 then you can get a lot of the intuition from uh your knowledge about diffusion on on effective potential Landscapes are there any any questions about this or administrative things before we get going uh I'll I just want to remind you that the uh midterm is indeed next Thursday evening 7: to 9:00 p.m. if you are if you have a problem with that time then you should have emailed sarab and uh if you haven't emailed him yet you
02:30 - 03:00 should have do it right now um and yes okay all right so let's let's uh let's think about the master equation a little bit more okay uh now before what we did is we thought about the simplest possible case of the master equation which is if you just have uh something being created at a constant rate and then being degraded uh at at a rate that's kind of proportional to the number of of that chemical species right and I'm I'm going to be using the uh nomenclature that's a little bit closer
03:00 - 03:30 to what what was in your reading just uh for hopefully Clarity and I think that some of my choices for from uh last lecture were maybe unfortunate uh so here this is for example uh M would be the number of mRNA for example in the cell this is the rate of creation of the MRNA and then uh the rate of degradation of the MRNA okay so m is the number of mRNA and uh if we want to understand gene
03:30 - 04:00 expression we might include an equation for the protein so we might have some P dot but it's some KP okay now oh sorry again I always do this all right so this we're going to have this be an N dot all right so now n is uh is going to be the number of the protein
04:00 - 04:30 okay now this really is kind of the simplest possible uh model that you might write down for gene expression that includes both the MRNA and the protein right so there's no uh no Auto regulation of any sort uh it's it's just that the MRNA is uh involved in uh in creating the protein but then we have we have degradation of the protein as well so what we want to do is kind of try to understand how to formulate the master equation here but then also we want to make sure that we understand what the master equation is actually
04:30 - 05:00 telling us and how how it might be used right so first of all in this model I want to know is there uh in principle protein bursts right so before we talked about uh the fact that in at least in in that in Sunny's paper that we read they could observe protein bursts at least in that in those experiments in eoli the question is
05:00 - 05:30 is there should should this model somehow exhibit protein bursts and you know why or why not all right I just want to uh see where we are on this and I think uh this is something that reasonable you know that depending on how you interpret the question you might decide the answer is yes or no but I'm I'm curious uh I I think it's it's worth discussing what the implications are here
05:30 - 06:00 and uh you know and the the relevant part of this is going to be the discussion afterwards so I'd say uh don't worry too much about what you uh what you think right now but I'm just curious um you know you know this model does it some Does it include somehow protein bursts all right ready uh three two one okay so we got um we got I'd say at least a majority of the people are saying no okay so the um can you know but then some people are saying yes so can we can somebody volun
06:00 - 06:30 ER yeah what's go you know why or why not yes I think the difference is if are we using this in a continu fashion or disc continu yeah okay all right all right so he's answering both both possible sides of the argument right and uh and and the point here is that if you just simulate this from the standpoint certainly for example you know there's continuous there right so if if you if
06:30 - 07:00 you just simulate this as a deterministic pair of differential equations then will there be bursts no okay all right because everything is well behaved here right on the other hand if we go and we do what like a full Gillespie simulation of this pair of uh pair of equations then in the proper parameter regime we actually will get protein bursts okay which is in some ways weird right that depending upon the framework
07:00 - 07:30 that you're going to be analyzing this in you're going to you can get qualitatively different behaviors for things right um but there's a sense here that the uh the deterministic uh continuous evolution of these quantities would be the average over many of these stochastic trajectories and the stochastic ones do have burst but if you average over many many of them then you end up getting some well behaved uh pair of equations so we'll kind of try to make sense of this more later on but I think that this just highlights that the uh that you you can get really
07:30 - 08:00 qualitatively different behaviors for the same set of equations depending upon what you're what you're looking at right and this is not just you know it's U and you know these protein births can be dramatic events right I mean that where the protein number pops up by a lot right you know so this this really then if you look at the individual trajectories here they would look very different right whether you were doing kind of a stochastic treatment or the deterministic one okay can somebody remind us the situation in which we kind of get protein
08:00 - 08:30 bursts uh in in the stochastic model particular will we always get will we always get these these discrete protein bursts or what determines the size of a protein burst yes okay right so okay so possibly there's a lag time between the time that an mRNA is created and and then the next
08:30 - 09:00 thing would be when right and then when the protein is totally right so that okay so there are multiple time scales right so one is U right so after an mRNA is created and that's through this process here okay so now out pops an mRNA right now there now there are multiple time scales right there's the time scale for mRNA degradation that goes as one over gamma M right there's a time scale for protein degradation you know after a protein is
09:00 - 09:30 made that goes as one over gamma P but then there's also a time scale associated with um kind of the rate of protein production from each of those mrnas and right and that's determined by KP right so we get these we get big protein bursts if what what what is it that what determines you know this the size of these protein bursts yes
09:30 - 10:00 right that right it's always confusing we talking about times or right but in particular we have protein bursts in the in the stochastic situation right if we do a stochastic simulation and and that's in the regime if KP right the rate of protein synthesis uh from the MRNA is somehow much larger than uh than this gamma
10:00 - 10:30 M right have I have I screwed up okay yes so just this is also like in the sense of being different from the deterministic equations we probably also want the total number right um yeah I think that it and this comes the question of of what mRNA number you need I mean it depends on what you mean by protein bursts I would say that if um if so long as this is true okay that what that means is that
10:30 - 11:00 each mRNA will indeed kind of lead to you know a burst of proteins being made where where the burst is getting geometrically distributed with some right now there's another question which is are those protein bursts kind of large compared to the steady state protein concentration right and that's going to depend upon U km and and Gamma p as well is is that yeah I guess so guess I guess dep all right well and you're saying
11:00 - 11:30 time resolution in terms of just measuring yeah okay yeah well okay but right now we're kind of imagining that we live in this perfect world where we know at every moment of time exactly how many of everything there is right so in some ways we haven't we haven't said anything yet about time resolution we're assuming that that our time resolution and and our number resolution is actually is perfect right but still depending upon the regime that you're in the protein all right numbers could look something like uh right so if you look at the protein number which is is
11:30 - 12:00 defined as this n right as a function of time then you know in one regime you're going to see it where okay so it's kind of low you get a big burst and then you kind of It kind of comes down and then and then a big burst and then it kind of comes down and burst and kind of comes down right so this this is in the regime where you don't you have infrequent mrnas being produced and then large uh large uh size bursts from each mRNA right and then and then you kind of get this effective degradation or dilution
12:00 - 12:30 of of the protein numbers over time right and this distribution if you take a histogram of it is what right so I when I'm I'm imagining that we we look at this for a long period of time right and then we come over here and we histogram it right so now we we come over here we turn to the left we say number as a function of this
12:30 - 13:00 is number okay this is the the frequency that we observe some number of proteins this a frequency okay um and you know this is going to do something right so what you know what what if I it may not be a beautiful drawing but but you're supposed to know the answer um I'm trying to I'm trying to review things for you because I hear that you have a big exam coming up and I want to make sure that uh
13:00 - 13:30 gamma it's a gamma right so this is U this is what we learned earlier so this is a gamma distribution and you should know what this gamma distribution looks like in particular there there are these two parameters describe the gamma distribution as a function of underlying parameters in the model maybe all right I I don't want to get get too much to this because I I
13:30 - 14:00 because well on Thursday we spent a long time talking about it so you know once we get going we'll spend another long time talking about it again but uh but you should you should review uh review your notes from from Thursday before uh next before the exam okay um okay so this thing is gamma distributed and if we looked at the MRNA number as a function time we did a histogram of that the MRNA distribution would be what it's Plus on right so it's important remember that you
14:00 - 14:30 know just because I tell you that oh the protein number is gamma distributed that doesn't immediately tell you exactly what you should be expecting for the uh the distribution of say the number of proteins is a function of time right I mean there are many different things I could have ploted over here they would all kind of come down to a gamma distribution over here right so it's important to kind of keep in mind the different representations that you might want to think about the data in okay um
14:30 - 15:00 okay right so what we want to do now is we want to uh think a little bit more about this master equation in the context of um if we if we're going to divide it up into into these states now I would say that any time that you are asked to write down the the master equation for something right so now if how many how many equations will the master equation the how many yeah if I I say master equation but there's really more than one maybe so how many how many
15:00 - 15:30 Master how many equations will be involved in the master equation dist uh kind of description of of this model infinitely many but there were infinitely many already when we had just one when we just had the MRNA distribution right well you know infinite times infinite is still infinite right so long as it's you know still it's a countably infinite number right okay but um yeah but it's still okay whatever you know
15:30 - 16:00 still infinite always all right so what we want to do is uh divide up the states all right so you know when somebody asks you for you know the the equations describing how those probabilities are going to vary you know really what we're interested in is some derivative respect to time of some probabilities described by MN right we want know der respect of time right for all MNS right so that's why there are infinite number because M goes in One Direction n goes in another lots of them okay
16:00 - 16:30 now it's always tempting to just write down this derivative and then and then just write down the equation uh if you can do that that's fine but I would recommend that in general what you do is you try to write a little chart out to keep track of what direction these things can go okay so for example here we have the probability of being the MN state right now there's going to be ways of going here and this is going to be going probability of being an m+ One
16:30 - 17:00 n what I'm going to do is I'm going to give you uh just a couple minutes and in two minutes I want you to try to write down as many of the rates the fs and n's that correspond to all these transitions you may not be able to get through all of them but if you don't try to figure out some of them then you're going to have trouble doing it at later
17:00 - 17:30 date all right all right do you understand what I'm asking you to do okay so next to each one of these arrows you should write something all right so I'll give you two minutes to to kind of do your best of of writing these things down for
17:30 - 18:00 [Music]
18:00 - 18:30 that for
18:30 - 19:00 all right why don't we uh why don't we
19:00 - 19:30 reconvene and we'll uh we'll see see how
19:30 - 20:00 we are um all right so this is uh this is very similar to what we did on Thursday right we have to remember that M's are the mras and this is what we uh this is what we solved before right where it's just a long long a row okay now first of all the MRNA distributions and
20:00 - 20:30 the rates do they depend on the um on the protein numbers no so what does that mean about say this Arrow as compared to some the arrow that would be down well down here it's going to be the same right because n does not appear up in that equation describing mRNA okay if we had Auto regulation of some sort then it would okay but um that's helps all right so let's uh let's go through all right what we're going to do is we're going to do
20:30 - 21:00 verbal yelling out okay ready this Arrow okay this one here is 32 1 km all right all right ready three 2 1 gamma M time M right 3 two 1 gamma M * m + 1 okay now remember that there are more mRNA over here than there
21:00 - 21:30 are here which means that the rate of degradation will increase right okay now uh coming here now this is talking about the creation and destruction of of the proteins changes in N all right this Arrow here already 3 2 1 it's KP right times M right so this is the rate of creation right going from um right going from n minus one to n that's fine uh you know I I was looking at my notes from last year and I I got one of
21:30 - 22:00 these things in correct so um right and then okay ready this one here three two one KP * m right so here it's the same rate and should we be surprised by that right so the number of proteins are changing but here it's the number of mRNA that matters because we're talking about the rate of translation right okay now uh this one here three two one gamma P * n and here 3 2
22:00 - 22:30 1 gamma P * n + 1 all right perfect now this is um of course as you can imagine the simplest possible kind of set of equations that we could have written down uh if you have other crazy things you get different distributions right if you have Auto regulation or if you have uh interactions of something with something else um or the same thing so forth but um it's uh I I think it's it's really very useful to to kind of write this
22:30 - 23:00 thing down to to clarify to clarify your thinking in these in these problems you know and then you can and then you can kind of fill out for change of probability of MN you come here and you just go around and you count take all the arrows coming in and those are positive those are ways of increasing your probability right the ways going out are ways of decreasing your probability okay now in all those cases you have to multiply these raw rates by the probabilities of being in all these other states okay
23:00 - 23:30 so can you use the master equation to get these probabilities uh if you're out of equilibrium out of steady state all right all right so that's question right so you know so the master equation useful out of steady state
23:30 - 24:00 yes all right ready three two [Music] one all right so we got um a a fair number of so it um there there is some disagreement yeah so it actually the answer is yes okay and that's because you can start with any distribution of probabilities across all the states you'd like it could be that all of the probabilities at one state it could be that it's just however you like right and the master equation tells you about how that probability
24:00 - 24:30 distribution will change over time okay now if you let that run forever then you come to some equilibrium steady state uh and that that tells and that's a very interesting quantity is the steady state distribution of these probabilities but you can actually calculate from uh from any initial distribution of probabilities evolving to any later time T what the probability would be later okay this comes to another question here all right so let's imagine that at time
24:30 - 25:00 t equal to zero I tell you that there are M mRNA and P and not I I always do this this is I don't know somehow my brain does not like this okay and not right because the P's I want to be probabilities okay um all right we start with M mRNA n protein okay now we um you know in and maybe it's a complicated situation you know we
25:00 - 25:30 can't calculate this analytically so what we do is we go to our computer and we have it solve how this probability distribution will evolve so that time t equal at some time if we'd like we could say this is T1 um I I'll tell you oh the probability of having M and N mRNA and protein is going to be equal to something P1 okay now the question is let's say I then go and I do this simulation again okay now I calculate some other at time
25:30 - 26:00 T1 again the probability that you're in the MN State the question is will you again get P1 all right so this is a question mark and a is yes B is no right I'm going to give you 15 seconds uh I think that this is uh very
26:00 - 26:30 important that you understand kind of what the master equation is doing and what it is not doing I'm sorry what's that right okay so I mean you know this is just uh you know uh you know you you program in your computer to solve this master you know this you
26:30 - 27:00 know to use the master equation to solve how the probabilities are going to evolve right I'm tell I'm just telling you you start with some initial distribution and you if you do it once it says oh the probability that you're going to have M and you know this time you're going to have m m mRNA and proteins is going to be P1 says 10% okay great now I'm asking just you know if you go back and do it again will you again get 10% or will or is um or is this output stochastic
27:00 - 27:30 it's okay that if you're confused by this distinction um I think that it's easy to get confused by which is why I'm doing this but all right let's let's just see where we are ready three two one okay um all right so I'd say a majority again you know but we're kind of at the 8020 75 25 a majority here are saying that uh yes you will get the same prob ility okay uh and this is uh very
27:30 - 28:00 important that we understand um kind of where this you know where the stochasticity is somehow embedded in these different uh representations of this of these modelings right so this this the master equation right it's a set of differential equations telling you about how the probabilities change over time given some initial conditions right now we're using these things to calculate the evolution of some random process but the probabilities themselves evolve
28:00 - 28:30 deterministically okay so what that means is that although these things are probabilities if you do if you start somewhere and you use the master equation to solve you get the same thing every time you do it okay now this is not true for the Gillespie simulation because that you're looking at an individual trajectory right an individual trajectory then the then the the stochasticity is embedded in that trajectory trory itself right whereas in the master
28:30 - 29:00 equation the stochasticity arises because these are probabilities they calculating and so any individual instantiation will be probabilistic as you are sampling from uh those different probability distributions okay now this is I think a sufficiently important point that if there are questions about it we should talk about it yeah how do you make simulations would you essentially can
29:00 - 29:30 you take a over different okay so it's true that you can you can do a sum over different gaspie no but I we haven't yet told you about what the gaspie algorithm is so I'm you know so I can't use that but um but indeed uh you can just use a standard solver of differential equations right so whatever program you use is going to have some way of doing this right um and and once you've written down these equations the fact
29:30 - 30:00 that these are actually probabilities doesn't matter right so those could have been something else right so this this could be the number of eggs on what whatever right um so then uh so yeah once you once you've gotten the equations then the equations just tell you how the probabilities are going to change over time yeah this maybe this is a silly question but in practice do do you have to like assume there all the probabilties zero above some number oh no it's not at all
30:00 - 30:30 a silly question um right so um yeah because because you know Compu Sol exactly right um and uh yes no that's a very that's a very good question uh and right so I told you this an infinite set of differential equations and U but you know but at the same time I told you this master equation is supposed to be useful for something right and um kind of at the face of it these are incompatible ideas right uh and and the basic answer is that um you you have to you have to include all the states where there is a you know sort of
30:30 - 31:00 non-negligible probability okay now uh let's let's we can be concrete though so let's let's imagine that we have that I tell you we want to look at the MRNA number here okay and I tell you that okay km is equal to um well okay well let me make sure okay gam M all right what are typical lifetimes of mras in in bacteria again right order a minute so so that that
31:00 - 31:30 means that we let's say this is 0.5 minutes to the minus one right to give a lifetime of around two minutes okay um all right and then let's imagine that this is then all right 50 per minute all right so an mRNA is kind of made once a minute uh there's 50 of them that's a lot but all right whatever there are a few genes all right I I
31:30 - 32:00 wanted to number to be something you all right so this a fair rate of mRNA production okay now um how many equations do you think you might uh need to simulate all right so we'll think about this first of all does it depend upon the initial conditions or not yeah it does so okay so so be careful right um in particular okay but let's say that
32:00 - 32:30 I tell you that we start with 50 mRNA right the question is how many equations do you think you might have to write down and let's say we want to we want to understand this you know you know once it gets to say the equilibrium okay all right number of equations all right give me give me a moment to come up with some reasonable options uh
32:30 - 33:00 all right well these are um so if you were to actually I mean let's say that you know this could show up on your homework right so the question is how many equations are you
33:00 - 33:30 going to program into your um into your simulation um and it may be you know doesn't have to be exactly any of these numbers but you know order do you guys understand the question all right so we need a different equation for each of these um each of these probabilities right so in principle we have uh an the master equation gives us an infinite number of equations right so we have D the
33:30 - 34:00 probability of having zero mRNA with respect to time right that's going to be does anybody any idea what this is going to be right so we have a minus km times what time p 0 right so this is cuz you
34:00 - 34:30 know if we start out down here p 0 now we have km all right so I I was just about to violate my rule and just write down an equation without drawing this thing but um right so it's you know it's km time P0 that's a way you lose probability but you can also gain probability right at a rate that is goes as gamma M times P1
34:30 - 35:00 okay all right so that's that's that's how this probability is going to change over time but we have a different equation for for P1 for P2 for P3 for P4 all the way in principle to P you know 1,683 th000 blah blah blah right so that's problematic right because if we have to actually in our program code up you know a 100 million equations or or could be worse right you know uh then then then you know we're going to have trouble with with our with our computers right so so you always have to have some
35:00 - 35:30 notion of what you should be doing and this also highlights that uh it it's it's really important to have some intuitive notion of what's going on in your system uh before you go and you start programming right because uh other if you know in that case well you're likely to write down something that's wrong you won't know if you have the right answer and you might do you know do something that doesn't make any sense right so um yeah so you have to have some notion of what what the system should look like before you even start um start coding it
35:30 - 36:00 okay my question is how many how many of these equations should we simulate all right okay okay let's uh let let's let's just see where we are ready three two one okay so I'd say that we have um you know
36:00 - 36:30 you know it's basically between C and D yeah um you know I would say uh you know some people are maybe more careful than I am all right can can one of the D's uh maybe defend why why they're saying D number okay the mean is 100 and when you say I mean I think that whatever you're thinking is correct but I think that the
36:30 - 37:00 the words are a little dangerous why and why am I concerned about you said the okay I mean is is the mean 100 for all time steady St at steady state right I think that that that was the key you know it's just that yeah so um for long times the the mean number of mRNA will indeed be 100 right so the mean number of m in this case will be km / gamma M which is going
37:00 - 37:30 to be equal to 50 ID that that gives us 100 okay now will it be exactly 100 no it's going to be 100 plus or minus what plus or minus 10 right because this distribution at steady state is what it's Pon what's the variance of a Pon distribution it's equal to the mean okay so for Plus
37:30 - 38:00 on the variance is equal to the mean right variance is the square of the standard deviation right which means that this is going to be plus or minus 10 that's kind of the typical width of the distribution right so what means is that at equilibrium we're going to be at 100 and it's going to kind of look like this all right so this is you
38:00 - 38:30 know this might be two Sigma so this could be 20 but you know each of these is 10 all right so if if you want to capture this that you you might want to go out to a few Sigma right right so let's say you want to go out to three sigma then you might want to get out to 130 maybe right you know so then if you want to be more careful you got to 140 150 but this thing is going to Decay exponentially right so you don't need to go up to a th right cuz the probability is going to
38:30 - 39:00 be 0 0 0 certainly once you're at 200 I say don't have to worry about it okay but of course you have to remember the initial condition we started at 50 okay so we started at at this point which means we definitely have to include that equation otherwise we're in trouble right now how much do we have to go to the um kind of below 50 any
39:00 - 39:30 my guess would be would be not much more than a few times because because if it were already an equilibrium that would be the me but it's not and so the driving force is still going to push it actually to right that's right so there's going to be a it's going to be a biased random walk here where it's going to be sort of maybe twice as likely at each step to be moving right as to moving left that means it could very well go to 49 48 but you know it's not really going to go below 40 say you know of course you have
39:30 - 40:00 to quantify these things if you want to be careful but certainly I would say going from I don't know 35 to 135 would be fine with me you would get full credit on your problem set okay and so we'll say uh I'm going to make this up kind of from 35 to 135 134 just so it could be 100 equations so I'd say I'd be fine with 100 equations okay so you simulate the change in the probabilities of P35 to
40:00 - 40:30 p134 for example right so although in principle the master equation specifies how the probabilities for an infinite number of equations are going to change um you only need to simulate a finite number of them depending upon the Dynamics of your system okay yes that's a very thank you for the question because it's a very you know very important practical thing yeah so in practice do you like in practice you don't know
40:30 - 41:00 right so the question is yeah in this in this case it's a little bit cheating because we already kind of knew the answer right um we didn't know exactly how the time dependence was going to go right how is it that the mean is going to change over time on average no exponentially right so on average you will start at 50 you exponentially relax to 100 right yeah but but in many cases we don't know so much about the system and I'd say that what you can uh what in
41:00 - 41:30 general what you can do is you have to always specify a finite number of equations but then what you can do is you can put in kind of maybe like reflecting boundary conditions or so on the end so you don't allow them you don't allow probability to escape okay but then and then what you can do is you can run the simulation and if you have some reasonable probability at any of your boundaries then you know you're in trouble and you have to extend it from there right yeah so you can for you know you can look to say oh is it above 10 the minus three or four you know whatever and and then if it is then you have then you know you have to go further
41:30 - 42:00 right okay any other questions about how you're actually going to be doing simulations of this so you know these are relevant questions for you um all right um okay so that that's okay that's that's the master equation uh but I'd say the key key thing to remember is that it is it tells you how to calculate the deterministic evolution of the
42:00 - 42:30 probability of these states uh given some potentially complicated set of interactions okay now a rather orthogonal view to the master equation is to use the Gillespie algorithm or in general to do direct stochastic simulations of individual trajectories Yeah question before we go we just set to zero right um okay so it's qu right so the question is whether we're somehow losing probability so what what I was
42:30 - 43:00 proposing before is that all right you you always want probabilities to Su to one otherwise you know it's not a probability and the mathematicians get upset um and and you know and the key the key thing there is that you know you you want to start with i you you you have to include all the states that have probability at the beginning right so okay so in that sense you you're given an initial distribution and you have to include all those States otherwise you're you're definitely like going to do something funny right so you start out with a normalized probability
43:00 - 43:30 distribution and then I guess what I was proposing is that you have a finite number of equations but you don't let the probability uh leave or come in from those ends right and if you do that then you will always have a normalized probability distribution of course uh at the ends you've kind of violated the actual equations right and and that's why you have to make sure that you don't have significant probability at any of your boundaries does that answer not
43:30 - 44:00 quite Z excuse so I was not suggesting that you set the probabilities equal to zero I was suggesting that you do what's kind of like what what what the equations actually do here which is that you don't allow any probability to leave you know there's no probability flux on this Edge right so for example out at p134 I I would just say okay well here's the probability that you're you have 134 Mr and I would get you know in principle there are these two arrows right but I I
44:00 - 44:30 would just but you can just get rid of them okay so now um now any probability that enters here you know can only come back right now and i' I've somehow violated my equations but if p134 is essentially zero then it doesn't it doesn't matter right okay um right so instead of looking at an individual um sorry instead of instead of looking at these abilities evolve kind of as a as a whole you can instead
44:30 - 45:00 look at individual trajectories right so the idea here is that if we start with the situation actually we can take this this thing here okay we have um so we know that at stud State it's going to be 100 starts out at 50 and in this case right with the master equation you say okay well you start out with all the probability right here so you have kind of a Delta function at 50 right but then what happens is this thing kind of evolves and over time you end this thing kind of spreads until you have something that looks like
45:00 - 45:30 right looks like this right where you have a pan distribution centered around 100 and uh and this this pan distribution is going to be very close to a gaussian right because you have a significant number right so the the master equation tells you how this probability distribution evolves okay now if we this is this is the number M and this is the uh this is kind of the frequency That You observe it right so we can also kind of flip things so that we instead plot the number M
45:30 - 46:00 on the y- axis okay and we already said okay the the deterministic equations will look like this and the characteristic time scale for this is what one over gamma M right so this thing relaxes to the equilibrium time scale determined by the degradation time of the Mr right so this is these are things that should be really you you want to be kind
46:00 - 46:30 of drilled into your head I'm trying to drill uh so you you'll hear them again and again all right now the the master equation indeed since everything's linear here the expectation value over the probability distributions actually does behave like this right so the mean of the distributions as a function of time look like that um and in some ways if we were to plot this we'd say okay well first of all it's all here then it's kind of it kind of looks like this right so the this is somehow
46:30 - 47:00 how those probability distributions are kind of expanding over time okay now for in individual trajectories if we run a bunch of stochastic simulations we'll get something that on average looks like this but it might look like this a different one might look like this okay um and and so on whoa although they shouldn't converge there you know because that's not consistent right so you what you and if if you did a histogram at all those different times of the individual stochastic trajectories you
47:00 - 47:30 should recover the probability distribution that you got from the master equation okay all right so this is a powerful way just to make sure that for example your simulations are working right that you can check to make sure that everything behaves um in a consistent way okay now there's a major question though how is it that you should generate these stochastic trajectories okay and um the the the sort of most straightforward thing to do is to just um divide up time into a
47:30 - 48:00 bunch of little Delta T's and just ask whether anything happened okay so uh let me uh so what we want to do is we want to imagine we have uh maybe M chemical species okay so now the these are different M's and NS let's be careful M chemical species they could
48:00 - 48:30 be anything could be proteins they could be small Mo something right and there are n possible reactions and indeed in some cases people want to study the stochastic dynamics of large networks right so you could have 50 chemical species and 300 chemical different reactions right so this could be rather complicated okay um and these M chemical
48:30 - 49:00 species have um we'll say numbers uh or if you'd like in some cases it could be concentrations uh X XI so then the whole thing can be described by some some Vector X and and the question is you know how should we simulate this okay uh the so-called what we often call the naive protocol and this is indeed what I did in graduate school because nobody told me that I wasn't supposed to do it um is that you uh you divide time into little
49:00 - 49:30 time segments delta T okay and small delta T and you just do this over and over and for each Delta you ask okay did anything happen okay if it did then you update if not you keep on going right now the problem with this approach well yeah I know what what is the problem with this approach
49:30 - 50:00 yeah time is continuous okay so this is um all right so one problem is that uh well you know you don't like discret time that's that's understandable uh but you know I'm going to say well you know the details you know delta T maybe is small so you know you won't notice I'm saying if I said delta T being small then I'm going to claim that you're not going to notice that that I've that i' slow right but then the simulation is slow right so there's a fundamental trade-off here and in
50:00 - 50:30 particular the problem with this protocol is that for it to behave reasonably delta T has to be very small right and and if um and and what do I mean by very small though that's right and right in for this to work delta T and this Delta B
50:30 - 51:00 has to be um such that um such that unlikely for anything to happen but this is already a problem because that means that we're doing a lot of simulations and then just and nothing's happening right you know uh okay how do we um how do we figure out what that probability is
51:00 - 51:30 right so in particular we can ask about well given um and okay possible reactions we'll say with rates um R sub I right so the probability that the I reaction occurs uh is equal to uh r i * delta T for for small delta T okay uh because each of these reactions
51:30 - 52:00 will occur kind of at a at a rate you know they're going to be uh they're going to be exponential distributions of the times for them to occur you know this is a Pon process right because it's random now uh what we want to know is the probability um that nothing is going to happen right because that's what that's how we're going to have to set delta T right well what we can imagine then is that you know then we can say all right well what's the prob ability that is say you know not not
52:00 - 52:30 reaction one and uh not two and okay dot dot dot okay well and this is this is in sometime delta T right well actually we know that if if the fundamental process just looks like this then we're going to get exponential distributions for each of those
52:30 - 53:00 right so we get end up with e to the R1 and indeed once we write an exponential we don't even have to write Delta Delta T this is just some time T right for this to be true it requires that delta T is very small right but if we want to just ask what's the probability that reaction one has not happened in sometime T this actually is indeed precisely equal to e to the r1t
53:00 - 53:30 yeah details all right this is e the minus r2t dot dot dot minus and there this we go up to n r to the n t Okay because each of those chemical reactions are going to be exponentially distributed in terms of how long you have to wait for them to happen Okay um and what's what's neat about this is that this means that if you just ask about the probability
53:30 - 54:00 distribution for um for you know for all of them combined I saying that none of them have happened this is actually just equal to you know the exponent exponent of minus now we might put we might pull the T out and we just sum over ri right so this is actually somehow a little bit surprising Rising which is that each of those chemical reactions
54:00 - 54:30 occur they're occurring at different rate some of them might be fast some of them might be slow the RIS could be different by you know orders of magnitude okay but still over these hundreds of chemical reactions if if the only thing you want to know is oh what's the probability that none of them have happened that is also going to end up that's going to Decay exponentially right and this actually tells us something very interesting which is that if we want to know the distribution of times for the first thing to happen that's also going to be
54:30 - 55:00 exponentially distributed okay and it's just exponentially distributed with a rate that is given by the sum sum of the these rates okay now that uh that's the basic Insight behind the galespia algorithm where instead of dividing things up into a bunch of time little times delta T but instead what you do is you ask um how long am I going to have to wait before the first thing happens okay and you just sample from an exponential with this rate r that is the
55:00 - 55:30 sum of the rates okay um oh and maybe it's even worth saying that okay so there's the the naive algorithm where you just divide a bunch of Delta T's you just take the little steps you say okay nothing nothing nothing nothing and then eventually something happens and then you update you keep on going there's the somewhat less naive algorithm which would be which is exact you know so it's not the same concerns that J had which is that you could just sample from n
55:30 - 56:00 different exponentials you know each with their own rates and then just take the minimum of them and say okay that's the guy that to that's the guy to happened first and then update from that that's an exact algorithm right the problem is that you have to sample from many possibly many different exponentials right and then and that's not a disaster but again it's computationally slow so the gaspie algorithm removes the required sample from those n exponentials because instead what what you do is you just say
56:00 - 56:30 um this you know so the the numbers of the concentrations um give all of the uh you know give you know the ri give you all the rates okay right and then what you do is you uh sample from U from from an exp exponential um with rate
56:30 - 57:00 um R which is the the sum over all the ri and that tells you okay when is the first reaction going to occur and then what you do is you ask well which reaction did occur because you actually don't know know that yet right and and there it's just the probabilities of each of them so the probabilities Pi is just going to be the rid the sum over the R so this Big
57:00 - 57:30 R okay so it may be that you had 300 possible chemical reactions but you only have to do two things here and they're both kind of simple right you sample from one exponential gives you how long you had to wait for something to happen and then you just sample from another Pro simple probability thing here that just tells you which which of the you know the N possible chemical reactions was it that actually occurred right and of course the chemical reactions that were occurring at a faster rate have a
57:30 - 58:00 higher probability of being chosen okay so this actually is an exact procedure in the sense that there's no uh no no uh kind of digitization of time or anything of the sort right so this actually is um it's comput computationally efficient and um and is exact you know assuming that you live assuming that your description of the chemical reactions was accurate to begin with right um right so then what we do is we update time and I this is in some ways you know
58:00 - 58:30 when you do computations when you when you actually do simulations this is maybe the annoying part about the gespi algorithm which is that now your times are not equally spaced and so then you just have to make sure you keep track you remember that you don't plot something that's incorrect right but because because your times are going to hop right at different time intervals um but you know that's doable up to you have to update your time and you have to update your your abundances okay and then uh and then what you do is
58:30 - 59:00 repeat okay I think the notes kind of allude to this glpi algorithm but don't qu are not quite explicit about you know what you actually do in all to go to go through this process right for the simulations that you're going to do in this class I would say that you you the you don't get the full benefits of the glaspie in the sense that you're not going to be simulating hundreds of differential equations with hundreds of different you know things um
59:00 - 59:30 but it's in those complicated uh models that you really have to do this kind of gpie approach as compared to even this somewhat more um you know better model which is a sample from the different exponentials right are there any questions about why this might work why you might want to do it yes right um what I mean is that you um you go to mat lab and you say you know
59:30 - 60:00 random um yeah so I'm sort of serious but uh sorry I'm trying to get a new um okay right so you have a the exponential right so it's a it's a probability distribution right so this is uh the probability as a function of time uh as a you know and then t right and it's going to look something like this right this thing is going to be
60:00 - 60:30 some well given that given that right in general it's going to be the probability T is going to be e to the minus RT and then do I put R here or do I put one over R is it one over R okay well what what should be the units of a distribution one over time in this case
60:30 - 61:00 right it's one over whatever is on this x axxis because if you want to get the actual honest to goodness probability right so this is the if you want the probability that um T is say between some T1 and T1 plus delta T right if you want an actual probability then this thing is equal to the probability density Z at T1 in this
61:00 - 61:30 case time delta T okay so that means this thing has to have a one over time and that gives us R here okay all right so now okay this is prob density right and and what I'm what I'm saying is that you want when I say sample from this probability distribution what it means is that it's like rolling a die okay but that it's a biased die because it's a continuous thing over the time right but just like when I you know when you have when you
61:30 - 62:00 have a six-sided die and I say okay you know sample from the die you're playing Monopoly you throw the die and you get you know one two three four five six and you do that over and over again right same thing here you kind of you know roll the die and and see what happens and indeed in uh you're going to get some practice with probability distributions on the homework that I think you're doing right now because you you're asked to demonstrate that you can take you can sample from a uniform distribution right which is something it's just equally probable across the unit um unit line and and do a
62:00 - 62:30 transformation and get an exponential distribution um now and it used to be that everybody knew all these tricks because you had to kind of know them in order to do computation but now I me mat lab or whatever program you use they know all the tricks so you just ask it to sample from an exponential with this uh property and it does it for you right but you still need to know what it's doing right so just to be clear what is the most likely time that you're going to get out from the
62:30 - 63:00 exponential zero right okay has a peak here but the mean is over here okay right all right any other questions about about what the you know how the gesp algorithm works and um all right can somebody tell me me how a protein burst arises right so we we had this original question about whether
63:00 - 63:30 there were protein bursts in that model that I wrote down right where we just had m dot is equal to okay now I what we had what we what what we said was that the master equation would not I mean the protein burst would somehow be there but you would never see them or you know somehow the protein burst would
63:30 - 64:00 influence how the how the mean and everything have evolved but you wouldn't actually see any big jumps right but then we said oh but if you did a stochastic simulation you would right so the claim here is that the that the gespi algorithm what I've just told you here will lead to protein bursts but what do I mean by when I when I when I make that statement what is it that I actually mean if we do a glas of this will the okay let's just hold on let me let me do a quick vote will we have cases where
64:00 - 64:30 Delta n is greater than one you know if I go through this process if I'm using the glaspie and I'm tracking how mRNA and protein number changing over time will I get you know these things protein burst where where Delta n is larger than one in one of these time Cycles okay okay ready three two
64:30 - 65:00 one okay all right so we got so most of the group is saying that's it's going to be no but uh but it's not you know again there it's it's mixed so so can somebody say why why we don't get yeah it seems like the structure of the simulation is to make that's right yeah so the simulation as written and you could imagine some sort
65:00 - 65:30 of phenomenological version of this where you allowed actually for protein but and as kind of specified is that we ask you know what's the time for one thing to happen right now uh why you but then but the claim somehow is okay well we can still get protein bursts from this and and how how does that happen yeah have the rate for something happening inre suddenly and that
65:30 - 66:00 would yeah for example right if we didn't have an MR before and then we got an mRNA what it means is that if you look at n as a function of time during one of these protein bursts but you know before I was drawing it just hopping up but really in the context of the gesp it would be that it would hop hop you know so there would be little time jumps right so this is a protein burst but it's that oh it's it's really before this mRNA is degraded you get one one one one right
66:00 - 66:30 so each of these is is Delta n of one right okay so this is uh you know whatever 67 right and then and then what can happen is that we get the MRNA degraded right and so then we're going to get a slower thing where it look you know it looks like that right so the gaspie everything is uh in uh everything's being created and destroyed in units of one but it could be that the time interval over this
66:30 - 67:00 burst is just very short so then it goes up very quickly but then it's slower to go away okay so what I want to do in just the last 15 minutes is talk a bit about the fuer plun approximation um I would say that you know all of these different approaches are useful to varying degrees in terms of actually doing simulations doing analytic calculations getting intuition and um and the plon approach it give you know I'd say it's
67:00 - 67:30 more or less useful for different people depending on um what you're doing okay so uh the basic ideas as kind of you you answered in the pre-class reading is that um in cases where n is is large enough that you don't feel like you need to take into account the discrete nature of the molecules um yet at the same time it's not so large you can totally ignore the fluctuations then the funer plun approach is nice because it allows you to get some sense of what's going on without all of the crazy details of um
67:30 - 68:00 for example the master equation um but uh and and then it also because of this idea of an effective potential it allows you to bring all the intuition from that uh into your study of these Gene circuits um now I'm not going to go through the whole derivation but uh but if you have questions about that please uh come up after class and I'm happy to go through it with you because it's um you know it's it's sort of fun um but um but the the notes you know do do go over it uh I think that what's
68:00 - 68:30 perhaps useful to just remind ourselves of is how it maybe leads to uh you know this g a gan with some width depending upon the shapes of the production degradation curse all right so the basic notion here is that depending on these F the fs and G's the production production degradation terms we we get different shaped uh effective potentials
68:30 - 69:00 okay all right now right so in general we have something that looks like okay we have some n dot there's some FN and then there's a minus GN okay so for example for something that has just simple expression uh in the case of let's just imagine now now that there was if you want we could say It's A protein that is just where it's just some K minus gamma n or if you'd
69:00 - 69:30 like we could say this is mRNA number you know but something it's just simple production and then uh first order degradation all right the question is how how do we how do we go about uh understanding this in the context of the fuer plun approximation and and it turns out that you can uh write it in what is a essentially some diffusion equation where you have some probability flux that is um that's moving around and and
69:30 - 70:00 within that realm you can you know write that the the probability distribution of the number is going to be something to describe so there's going to be some constant there's the f plus G and they're both these are both functions of N and then you have an e to the minus n right so the idea here is that this behaves as some effective potential
70:00 - 70:30 of course it's not quite true because f and g also are functions of N and they're not in here but this is the dominant term because it's in the exponential and here f is defined as the following so it's minus this integral Over N of the F minus G and f+ G DN and we my integrate Over N Prime
70:30 - 71:00 okay right and we we're going to uh kind of go through what some of these different fs and G's might look like to try to get a sense of why this happened okay uh it is worth mentioning that you can do this for any f and g that um where when it's just in one dimension right so you just have n okay once you have uh it in two Dimensions right so once you actually have mRNA and protein for example um you're not guaranteed to be able to write it as an effective potential uh right uh although I guess if you're if you're willing to invoke a
71:00 - 71:30 vector potential then U then uh then maybe uh then maybe you can all right but uh in terms of just a simple potential then uh you can do it in one dimension but not necessarily in more okay um and I think that in general our intuition is not as useful when you have the equivalent of magnetic fields and so forth here anyways all right what I to do is just try to understand why uh why this thing looks the way it does for um for this simple
71:30 - 72:00 regulation case and then we're going to ask okay if we change one thing or another how does it affect uh the resulting variance okay okay okay all right so for unregulated expression such as
72:00 - 72:30 here the if we look at the production and degradation is a function of n right so FN is G is just some constant K whereas GN is a line that goes up as gamma n okay okay now in this situation we if you do this integral and really what you can imagine is what this integral looks like right around that
72:30 - 73:00 steady um that steady state because that's kind of maybe what we want to know if we want to know something about for example the width of the distribution okay well there's going to be two terms in the numerator there's an F minus G in the denominator there's an f plus G okay now F minus G is actually equal to zero right at that steady state and that's why it's a steady state because production and degradation are equal okay now as you go away from that
73:00 - 73:30 location you're what you're doing is you're integrating the difference between the F and the g okay and you can see that around here these things are separating kind of um well everything's a line here and indeed even if f and g were not linear uh close to the um to that steady state they would be linear right what we can see is that as you're integrating you're integrating across something that is growing linear L sry that's what gives you a quadratic okay and that's why this
73:30 - 74:00 effective potential ends up behaving as uh as if you're in a quadratic trap okay now I encourage you to go ahead and like do that integral at some point I was planning on doing it for you today but uh we are running out of time once again I'm happy to do it just you know just after class uh and indeed what you can see is that because you're integrating across here you end up getting a quadratic increase in the effective potential right and if you look at what the variance of that thing is you indeed find that the variance is
74:00 - 74:30 equal to the mean here okay so what I want to ask in terms of trying to get intuition is what happens if we pull these curves down okay so in particular let's imagine that we have a situation where all right I'm going to reparameterize things so again we're kind of keeping the number the number at the equilibrium constant but now what I'm going to do is
74:30 - 75:00 I'm going to have an FN that looks like this and GN looks like so now GN is going to be some 1 half of Lambda and this FN is equal to K minus 12 of gamma n all right now the question is in this situation what will be the variance over the mean
75:00 - 75:30 well first of all the variance over the mean here was equal to what although should I should we do a vote okay here are going to be some options
75:30 - 76:00 question is variance over the mean in this in this situation all right I'm worried that this is not going to work but all right let's just see where we are uh ready uh three two one okay all right so I'd say that at least uh broadly everyone okay people are
76:00 - 76:30 agreeing that yeah the variance over the mean here is equal to one and again this is the situation that we've analyzed many times which is that in this situation we get a Pon where the pon only has one free parameter and that parameter specifies both the mean and the variance okay so for a Pon the variance over the mean is indeed equal to one right so the fuer flan approximation actually uh sort of accurately recapitulates that okay now the question is what will the variance over the mean be in the
76:30 - 77:00 situation that I've just drawn here right so I'm going to give you a minute to try to think about what this means and there are multiple ways of of figuring it out you can uh look at maybe the integral you can think about uh the biological intuition to make at least a guess of of where of what it should do right the question is if um if the production rate and the degradation rate look like this what does that um what does that mean for the the variance over the mean so I'll give you a minute to
77:00 - 77:30 kind of to play with it for
77:30 - 78:00 all right why don't we go ahead and vote just so I can and get a sense of where we are and also uh it's okay if you
78:00 - 78:30 can't actually figure this out or you're confused but I but go ahead and and make your best guess anyways because it's also useful to um if you can guess kind of the direction it'll go even if you can't figure out its magnitude all right so uh let's let's vote ready three two one okay so um all right so it's it's it's a mixture now I would say of you know a b c's D's um okay um yeah no I think this is this is I think hard and
78:30 - 79:00 confusing I maybe won't have um yeah all right I'll maybe say something because it may be that talking won't to each other won't help that much either um okay so in this case what's relevant is both the F minus G and the f plus G okay and and turns out that F minus G actually behaves the same way because at the fixed point or at the equilibrium it
79:00 - 79:30 starts at zero and then it actually grows in the same way as you go away from it the difference is that the f plus G where U that's very much not equal to zero and f plus g at the equilibrium this f plus G here is around 2K whereas f plus G over here is around uh 1K right what that means is that the in both cases you have a quadratic potential but here the quadratic
79:30 - 80:00 potential actually ends up being steeper so if this were unregulated then over here we still get a quadratic but it's a it's a with steeper walls okay so actually here um this the variance over the mean uh ends up being a half and it's useful to go ahead and just play with this play with these equations to see why that happens right and I think that a nice way to think about this is is in this limit where we pull this Crossing Point all the way down to
80:00 - 80:30 zero right now we have something that looks kind of like this right so very very uh low rate of degradation but then also the production rate essentially goes to zero when we're at this point that we could still parameterize as K Over gamma if we want you know with some you know but you know this could we could just think about this as as being at a 100 of these mRNA say but then
80:30 - 81:00 we're changing the production degradation rate right and the variance over the mean here does anybody have a guess of where that goes okay in this case it actually goes to zero and this is an interesting situation because really in the limit where there's no degradation and it's all at the production side what it's saying is that you produce you produce you produce until you get to this number which might
81:00 - 81:30 be 100 and then you simply stop doing anything right you're not degrading you're not producing right but in that case that all the cells will have exactly 100 maybe mRNA and what the the fuer plun kind of formalism tells you is that just because production and degradation rates are equal right Fus G is equal to zero doesn't mean that um that tells you that that's the equilibrium but it doesn't tell you how much spread there's going to be around the equilibrium okay if f and g are each larger that leads to a
81:30 - 82:00 larger spread because there's more Randomness okay whereas here f and g are both essentially zero at that point what that means is that that you kind of just pile up right at that precise value right we are out of time so I think we should quit but I am available for the next half hour if anybody has any questions all right thanks