 So, students welcome to today's lecture. We are discussing protein dynamics proved by NMR spectroscopy and here we looked at how we can use the R1, R2, NOE to understand the basic protein dynamics. So, today I am going to understand like explain more about like how we can probably use these techniques for the interpretation of relaxation data. So, as you know relaxation equations are dependent on the calculation of a spectral density that we had discussed in the first class, which is j omega, so spectral density at any frequency omega. So, if you look at T1, T2, NOE or R encoded in this spectral density function. So, like for T1 or R1 it depends upon frequency of j omega H, spectral density function of the proton frequency, nitrogen frequency and so on and so forth. So, R1 depends upon j omega H, j omega N and so on and so forth. R2 again can be written in terms of the spectral density function like a frequency, spectral density at frequency 0, spectral density of frequency, hydrogen frequency, nitrogen frequency and so on and so forth. So, R2 also has a component REX again here, the NOE similarly can be written in terms of the just spectral density function. So, essentially what I mean to say that all the relaxation equations are dependent on the calculation of a spectral density function and or vice versa using these also we can calculate the spectral density function. Now, how to interpret this data? So, suppose we are dealing with a folded protein. So, typically for a folded protein these relaxation data are interpreted in terms of modal free approach of Lipari and Zabo. Now, this modal free approach talks about like a talks about the separatability of internal and global motion, this is typically for a folded protein. So, folded protein it is kind of consider it is here. So, what is motion here? This is a local motion and a global motion, internal motion and global motion they can be separated based on assumption that can be separated global and internal motion. And this dynamics is described in terms of the overall rotational correlation time like a M overall time, the internal time is E like internal correlation time tau E, T M or T E and then another term called like order parameter which describes the amplitude of this internal motion, how much it is, what is the amplitude of this internal motion. So, Lipari Zabo basically gives you three parameter, the dynamics can be described in three parameter, the overall tumbling time, the internal correlation time and the order parameter which describes the amplitude of internal motion. Now, so when overall tumbling time of protein can be described by a single correlation time and internal motion takes place much faster like so it is overall tumbling happening slowly and internal motion happening very faster. So, then correlation time the correlation for this two motion, the global motion and local motion can be separated and the total correlation function can be given by this function. So, C at any time T is C 0, T and C 1, T. So, the correlation function for overall motion assuming that isotropic diffusion is given and these are the consideration for like a the overall tumbling time and if you do Fourier transform of this, we can get this spectral density function like j omega C. So, now, T M being a rotational correlation time for overall tumbling of a protein. The only consideration that internal motion is happening too fast here. Now, the internal motion can be written like in terms of this order parameter C i of T internal motion at any time T can be written as S square plus 1 minus S square exponential T by tau E that is a internal correlation time. So, S is called generalized order parameter and T E is effective correlation time right. So, for a completely restricted motion if a protein is completely restricted or a peptide bond is completely restricted, the generalized order parameter S square is equal to 1 and if it is completely disorder or flexible this is 0. So, typically for any realistic consideration the S square varies between 0 to 1 right. So, basically here in Leparage above we are considering the Brownian rotational diffusion of any molecule in solution and this diffusion can happen in terms of ellipsoid. So, like a protein is like ellipsoid or a spheroid or a sphere what is how it is diffusing that basically comes from this Brownian rotation diffusion. So, these considerations are in Leparage above interpretation of the dynamics data and this is called modal free assumptions. So, in modal free assumptions essentially we are assuming that overall rotation of this protein molecule is isotropic there is no anisotropy here there no orientation dependent here. So, it is isotropic motion and this method for characterization the overall rotation is a priori implies that internal motion intramolecular motion for most of the protein is too fast. The internal motion is too fast compared to the global motion that is a primary consideration and also overall rotation is an isotropic right. So, usually that parameter that govern the relaxation of N 50 nuclei that we are measuring in this protein they are kept fixed at some predetermined value in this modal free analysis and it assumes that intramolecular motion are independent or overall rotational motion first assumption is too fast then it is independent. So, conventional modal free approach assumes that protein does not aggregate. So, protein stage in a folded state without self association happening at the concentration that we are measuring relaxation study. So, with this assumption this modal free analysis can be done right. So, what are assumption isotropic motion internal motion is too fast then the global motion and the actually intermolecular motion are independent of overall molecular rotation. So, just to remind you again spectral density function j omega j of omega can be written in terms of tau c and just to put things in prospective if you plot j omega versus frequency and tau c are denoted by this. So, that means if tau c is 100 nanosecond it is it is tumbling slowly it is a spectral density function rapidly decay if it is tau c is 1 that is fast it starts with some value and slowly decay. So, what it implies a spectral density function j omega which is a Fourier transform of correlation function just says that rapidly relaxing domain signal give a broad line. So, if something is rapidly, rapidly relating sorry relaxing that means a bigger protein it gives a broad line if slowly relaxing it gives a sharp line. So, this makes sense right a molecule that tumbles very rapidly can sample very rapidly can sample like here can sample a large range of frequency ranging from say few megahertz to few like 100 megahertz and the molecule that some tumbles very slowly like a bigger molecule that can sample only shorter like a narrow range of frequency right. So, very long correlation time like 100 nanosecond or so and samples fewer frequency that is omega spectral density function j of omega c. So, taking that consideration the original leperie-jabeau j omega c is a square tau r rotational correlation time. So, globular and internal can be written in there if there are two motion on two different time scale you can just separate those two and with a anisotropic overall rotation you can add some anisotropic consideration as well. So, essentially you have to solve these functions get the generalized order parameter overall rotational time and internal rotational time. So, essentially in the modal free analysis data the parameter that we get it from R2 by R1 ratio is tau e which is less than 100 picosecond and tau r overall rotational time is more than 100 nanosecond rate. So, that is a wide range and these are adjustable parameter that can be obtained by fixing the relaxation data you can calculate the hydrodynamic radio radii in a iterative manner. So, this is kind of a workflow for selecting a suitable correlation function in a iterative manner and one can do in a iterative manner to find it out. So, start with a s square or s square rex s square tau e internal motion or s square tau e rex. So, all those slow or fast what is happening is it fitting it. So, if no then reject this modal take another modal and go and keep iterating to find the generalized order parameter that we get. That is where consideration is that we have the isotropic motion in a protein polyglobular protein. But you know the proteins are not always globular, there is fair bit of anisotropy in a protein. So, if there is anisotropy which is commonly observed in many proteins which are disorder region or even a disorder protein, this Lipari-Zabo modal free approach gets complicated. So, in those case where there is anisotropy, Peng and Wagner suggested the calculation of power spectral density function using 6 proton and nitrogen relaxation rate right. So, what are those 6 we are going to come in a moment. Now, this is called a spectral density mapping or spectral density function. It does not depend upon any modal, so it is just free of modal and no dependent on any form of time dependence or autocorrelation function nor does it require any form of rotational diffuser tensor of the molecule, it is just does not require any of these. So, only thing you need a spectral density mapping at different frequencies. So, for NH bond that are directly sampled at several relevant frequency or say 0 frequency or omega n frequency or omega H minus omega n frequency like if you add the 2 frequency and subtract the 2 frequency, so these are 6, 0 frequency J 0 spectral density function is 0 frequency, J omega n the frequency of nitrogen, J omega H frequency of proton, J omega H minus n the difference of proton and nitrogen frequency and sum of proton and nitrogen frequency. So, these are the 6, 1, 2, 3, 4, 5 spectral density functions that are there which needs to be considered in this spectral density function right. So, in this spectral density function the main goal is to understand the spectral density and several different methods has been derived to model the protein as it is a spectral density function with different parameters are fit to experimental data that is what proposed by Peng and Wagner. So, this is called spectral density mapping that we are discussing. So, three mathematical relation between the relaxation parameter T1 and T2 and J omega that we had seen in previous slide can be used for calculating this spectral density function. So, they come up with a much simpler method for interpreting this relaxation data and that was provided in terms of reduce spectral density function. So, what is reduced that we are going to just see it. So, if you consider this J omega H more or less equal to J omega H plus n and more or less equal to J omega n pi because spectral density functions say we are doing at 600 megahertz. So, proton spectral frequency is at 600 and nitrogen is at 60 right 60. So, 600 plus minus 60 is in the same range right. So, one can like roughly say that J omega H is equivalent to J omega H plus J omega n or J omega H minus J omega n. So, this one can club into one. So, J omega H then third one can be 0 frequency J 0 J omega H and third one will be J omega n. So, instead of 5 we can just do with a three spectral density function and these three as we have seen in the previous slide can be expressed in turn of the three parameter that we are calculating what R1, R2 and NOE. So, these are three measurable parameter R1, R2 and NOE we have we can measure in a residue a specific manner and using these three we can calculate the spectral density of J omega 0, J omega H and J omega n using this spec reduce the spectral density function right. So, these rates like whatever we can measure for proton and nitrogen at any frequency like say we are doing on 500 megahertz which means 11.76 tesla as a allows measurement happening at 0 megahertz, 50 megahertz or 500 megahertz and these two like omega H minus omega n means 450 or omega H plus omega n means 550 right or 7.6 tesla 750 megahertz it means 0 frequency 750 frequency 750 megahertz, 675 and 825, but these three are in the same range. So, we can do with only three of the spectral density functions right ok. So, now we have come up to a spectral density function and then using these three parameters that we have measured R1, R2 and NOE or J omega 0 can be calculated or similarly J omega H or J omega n can be calculated three spectral density function can be calculated right. So, as we know the R1, R2 and NOE, RNOE so what is RNOE? It can be calculated from our NOE experiment that we did. So, it takes NOE experiment R1 omega n by sorry gamma n by gamma H it is a ratio of the ratio of the gyromagnetic ratio. Now, for calculating this we need two other constant called C square and D square and the value of these R have been calculated. So, this is the typical value of C square and D square at 600 megahertz, relatively it can change little bit at 750 or 500 megahertz. Now, using these all parameter if you have measured the R1, R2 and NOE you can put all these value and can calculate the G 0, you can calculate the J omega H and J omega n. So, one thing you notice here that I am going to explain you in next slide NOE which is fast amplitude motion is reporting more towards omega H. Similarly, the J 0 is contributed by R1, R2 and NOE and here J omega n which is at intermediate frequency is contributed by R1 and NOE. So, what does this employs? So, omega H high amplitude motion like spectral density function at higher frequency let us go here. So, we are saying J omega H at higher frequency like here it is a fast motion it captures. Fast motion is captured by J omega H, the slower motion is captured by J 0, yeah. So, J omega H is largely determined by the heteronuclear NOE that is a fast amplitude motion and most sensitive to high frequency motion of the protein backbone like the motion happening at picosecond time scale that will be given by this parameter J omega H, ok. Similarly, J omega n, J omega n which is happening here is what contributes here R1 and NOE, right. So, J omega n is determined by R1 whereas, J 0 is contributed from R1, R2 as well as NOE. So, J 0 is given by everything it is a spectral density at lower frequency is given by R1, R2 and NOE. So, J 0 is sensitive to nanosecond time scale motion as well as exchange phenomena that is happening at slower time scale microsecond to millisecond. So, if we are measuring spectral density function at 0 frequency it is a sensitive to slower time scale motion nanosecond as well as microsecond to millisecond. The J omega n is given by mostly nanosecond time scale motion because this is contributed by R1 and some also contributes from R2 and NOE. So, nanosecond to kind of picosecond time scale motion you can say here this is nanosecond and microsecond to millisecond time scale motion this J 0 is given and this giving picosecond time scale motion. So, if you calculate this spectral density from the R1, R2 and NOE data we know exactly where the how the spectral density is mapped across the protein sequence and that gives the amplitude of the motion happening across the protein sequence and that is what one can calculate it. So, you see this the protein that we were doing sumo 1 which has a n terminal scale and a folded region and some loops again. So, here we had calculated in previous slide that if you remember the R1 was like this here for flexible tail and this was for kind of a globular domain alpha helix beta sheet over there and same little bit of the tails were here that is a R1. The R2 here R2 was low, NOE if you come to NOE was negative here right. So, if we calculate J omega H using this NOE value you can see here for the flexible portion the J omega H is very high and for the all order it is relatively low. So, now picosecond time scale motion in this scale in this protein for the flexible tail is very high picosecond time scale motion is higher here and lower here. Now if we go to J omega N which is dominated by R1 you can see there is a 1 to 1 correlation between these two like a very large correlation higher J omega N because this is capturing at nanosecond time scale motion and all these are lower the loops are again higher ok. Now J omega 0 which is contributed by R1, R2 and NOE everything has a impact here. So, you can see this is quite mirroring with our R2 because it is capturing like a lot more from the slower time scale motion. So, spectral density mapping telling it how the protein is sampling various amplitude motion across the backbone which are calculated from three experimentally determined parameter R1, R2 and NOE fantastic. So, we understood how the spectral density function can be mapped. So, what next we find it out from the R1, R2 and NOE. Now the R2 you know that it has also REX R2 has a REX. So, if you map this it says that along this line there seems to be exchange phenomena happening in this region and that is why you see lots of variation. So, the spectral density function is also capturing this because the slower time scale motion microsecond to millisecond time scale motion. The exchange or microsecond time scale motion the exchange is happening and that essentially is nicely captured if you measure the spectral density function at a 0 frequency at J 0 great. So, now using these if you like there is a linear relation that exists between J 0 and J omega n and between J 0 and J omega h. You can write this equation so, J of i for any residue J omega n or J omega h is alpha some factor alpha of n or alpha of h J of i residue at 0 frequency and some parameter called beta n or h. So, this linear relation many a time this linearity is affected by contribution of this exchange that is happening which affects the J 0 value, but if you solve this rearrange this equation what we can find it out one can find it out the tau right you can find a tau and omega is a Larmor Larmor frequency for n and h right. So, if you solve this equation linearly if you plot J omega n or J omega h versus J omega 0 you can find the alpha and beta and alpha and beta you put it here and solve this equation you can find the tau c sorry tau and that is a correlation time right. So, correlation time in a residue specific manner now we are finding it wonderful protein tumbling and you can find the correlation time happening at different. So, same data we are going to take it here like J omega h J omega n and 0 and plot it a linear function. So, here is our J omega h J omega n and J omega 0 this is my protein this is n terminus this is c terminus n terminus is here c terminus is here and that is what we had calculated experimentally. Now, can we use this equation to plot it a linear equation and find it out what is happening. So, we calculated for each of these amino acid J omega n in nanosecond per radian versus J omega 0 nanosecond per radian we plotted it and you can see if you plot it it looks like it is not very linear there are some residue which deviates from non-linear. So, you need to fit two curve one curve for these black residue one curve for the red sorry the open residue if you plot these two now they are looking more linear right. Similarly if you plot J omega h versus J omega 0 again very nicely one portion here black and one portion for open circle residue. Now, this training is very interesting phenomena because we have started spectral density function understanding that anisotropy exist. So, if you go in this protein structure you can see that rest that the amino acid that are there will have a different kind of motion than a globular protein this you can consider as a sphere spheroid or all those, but these guys are very freely tumbling in solution and because of that freely tumbling there is anisotropy existing and that anisotropy has is being captured in the spectral density function. So, if you do that if you do that if you plot this linear equation and deduce the parameter that is coming. So, if you fit this equation J omega 0 versus J omega n what we are seeing fitting by this equation separately and fitting this equation separately we can calculate from the previous equation that we had alpha beta and tau we can calculate this three parameter. So, alpha beta and tau is coming here now just look at this interestingly important thing. For a low amplitude motion what we are finding two motions are in nanosecond, but there is a motion in microsecond this is a slower motion happening for a globular protein. Some motions are in nanosecond and some motion is in microsecond, however for this high amplitude you are getting a nanosecond time scale motion of 5 nanosecond typically for a protein. If we plot this J omega 0 versus J omega h here again we are getting some tau c of nanosecond, but for a high amplitude motion for the flexible portion we are getting some of the motion as fast as picosecond. Now, that is a high amplitude motion we can see for the flexible portion and some picosecond and all those because now you see here is an interesting phenomena happening right. If you remember it here this guy is a flexible tail right, flexible tail and will that have a correlation time of few nanosecond yes of course typical globular protein have 5 to 6 nanosecond, but one of the parameter that came here is 21 nanosecond quite high. And because it is a long open rope is tumbling in solution that of course will have little longer correlation time. So, now doing this spectral density mapping solving these equations finding it out what are the alpha beta and tau we are deducing now all sorts of correlation time that can be seen in the protein and that gives a wide range of the tumbling time that we do and that is the power of NMR dynamics. If you use any other technique like a dynamic light scattering it is a complementary technique for measuring the correlation time of a molecule you mostly see one or two correlation time depending upon how it decays and gives you one correlation time of a protein whether it is a globular or elongated, but NMR precisely gives you all the correlation time that is possible measuring the dynamics in a like in a residue a specific manner. So, measure this three experiment basic three experiment R1, R2 and NOE from there we deduce the spectral density function j omega h j omega n and j 0 fitted those equation and solve this quadratic equation to find it out what kind of correlation time we are getting and now we can deduce it that why some of the most some of the correlation time are longer. So, it is a open chain and that are expected to show higher collier. So, that is all about time and motion in protein and that is what we in this week we wanted to give an impression to you that NMR is a beautiful tool to understand the protein dynamics because dynamics dictate the function. Dynamics dictate the function that protein do and we can measure the dynamics in a residue a specific manner in more elegant way than possible from any other complementary technique. So, that is a strong point of NMR. With this I am going to close the lecture for this week and next week I would like to see you in the discussion where we are going to discuss more about protein-protein interactions and how we can use NMR to understand protein-protein interactions. So, looking forward to have you in the next class. Thank you very much.