 Okay, so students this week we are discussing protein dynamics as proved by NMR spectroscopy and we discussed why protein dynamics, how you can measure protein dynamics, what experiments are to be done. So we already discussed the longitudinal relaxation and transverse relaxation. So we will start just with little background like revising the previous class. So experiment for measurement of relaxation parameters in protein, basically we are going for HSQC or HMQC based experiment. As we discussed proton relaxations are complicated therefore we measure on heteronuclei like 13C or N15 and the scheme for all these 2D heteronucleus relaxation measurement is simple like any 2D experiment we have start with a preparation state creating a desirable coherence. Then we introduce a time delay variable length which can captures the auto or cross correlation rate of this selected coherence. Then we introduce a T1 period and this T1 period basically is for frequency encoding in the indirect dimension. And finally, we transfer our magnetization to the proton and we detect on proton while decoupling the heteronuclei. And there has to be a delay between the each scan and this delay should be about 5 times of T1. So therefore typically delay in relaxation experiment is longer and we already discussed longitudinal relaxation rate and transverse relaxation rate R1 and R2. So today we will be focusing on the heteronuclear NOE, but before going to that I just we discussed this transverse relaxation rate is extremely important to understand the protein dynamics the conformational exchange because R2 is R2 intrinsic plus Rx. So it also gives the information of the exchange phenomena that happens in the time regime of millisecond to microsecond. Now just one example I just want to give you before I really move. So this is say transverse relaxation rate measure of a protein at 0 molar urea concentration and we add a mild denaturent and like a 0.9 molar and we measure again the dynamics. So as expected the termini has a lower or slower rates if you can see it. The folded protein region like a folded domain has a higher R2 rates and again termini here have a lower R2 rate. So if you take the difference 0 molar 0.9 molar versus 0 molar what we see here that lots of residue shows higher R2. They are saying that they are already going into some kind of conformational exchange just by doing this simple experiment of recording R2 at two different urea concentration telling a lot about this contribution that is coming because of exchange. So R2 is a good experiment to measure the exchange contribution as well as how protein folding happens, how these stabilisation protein happens. So many things can be simply investigated and studied using the transverse relaxation rate which is essentially R2. Now coming to the heteronuclear NOE, this essentially captures the first amplitude motion that is in picosecond time scale. So this heteronuclear NOE is determined from the change in the intensity of the NMR signal of a heteronuclear say X 13C or N15. Then the equilibrium magnetisation of a proton in vicinity is perturbed by saturating. So the first experiment we do we saturate the proton signal and then we can measure the how much this perturbation is affecting the heteronuclear X when the equilibrium magnetisation of proton is perturbed. So we measure it either as a transient NOE or a steady state NOE. So experiment is something like this you like here 3 channels that we are showing proton channel, the nitrogen 15 channel and the gradient channel. You start perturbing the proton signal exclusively so here are those saturation pulse so we saturated the proton. Now we are looking at the effect of that saturation on the relaxation rate of the nitrogen. So here then after saturating we are starting with a proton magnetisation transferring to the nitrogen and then we are encoding here nitrogen you can see here T1 by 2, T1 by 2 so frequency encoding nitrogen while decoupling the proton and then we transfer back to proton and then the due sensitivity enhancement we record on the proton while decoupling the nitrogen. These are gradient for coherence selection as well as this porous magnetisation suppression. So this is typically pulse sequence we start with a saturation on proton transferring that magnetisation on the nitrogen and then detecting back on proton. So like a first 90 degree pulse that we are applying on proton this is combined with a gradient right if you look at here it is combined with a gradient and that ensures that the magnetisation is the only magnetisation in the experiment. So actually we have saturated started with a 90 degree on proton here and then we have a gradient here so that ensures that magnetisation only goes to nitrogen. And after this after the first 90 degree pulse which is this the chemical shift of the heteronuclear here heteronuclear N15 here in this case is evolving and that is a evolving during the T1 period right and then we as we discussed we transfer the magnetisation back to the proton which can go for detection. So here we decouple so that the does not evolve under coupling during this period so there is a 180 degree pulse here. And finally this orthogonal magnetisation component is generated during the T1 period which is here refocused in the PEP sequence and then we simultaneously detect by inverting the phase 4-5-4 here phase and the phase 5-4 here you invert and do the coherent selection right. So then what typically we do in these experiments we do two experiments. So this is the experiment quadrature detection we do in F1 dimension by shifting the phase of the this pulse first pulse and the receiver together so that we detect in a state TPPi manner. So this is typical way of doing experiment saturating it starting with a proton transferring to nitrogen encoding the T1 frequency transferring back for PEP on the proton detecting proton. And then like here is for TPPi detection you phase shift this together with this so that we select like a quadrature detection for this and the coherent selection is done by cycling this pipe ok. So to measure the NOE we do two experiment a pair of experiment in one case where we are saturating that is called NOE and in one case where we are not saturating that is called no NOE and then we take the intensity of this NOE and no NOE saturated and not saturated. So here if you look at the intensity is given by I z it is some xz magnetization at equilibrium this sigma xh divided by r1 and hz equilibrium. So if you solve this equation you can get I unsaturated 1 plus sigma xh divided by r1 gamma h by gamma x these are gyromagnetic ratio and I z and I unsat right are the intensity of saturated and unsaturated. So sigma sh is the rate constant of the cross correlation and as we discussed gamma h and gamma x are gyromagnetic ratio. So here is the signal for saturated here is the signal for unsaturated. Now the NOE is given by I z divided by I unsat where we saturate for say 3 second and when we do not saturate so that is how to you do. So this comes out to be 1 plus gamma xh this is the cross correlation rate divided by r1 these are two gyromagnetic ratio right. So this is the NOE. Now in the error in the measurement of NOE is given something like this gamma sorry sigma NOE divided by NOE sigma square I z divided by I s square plus sigma s square I unsat divided by I s square unsat. So this is the error in the measurement of NOE. So typically I sigma I z and sigma I unsat represents the standard deviation right. So that can be calculated separately and then we can find it out how much the I z by I unsat that is the NOE value ok. Now typically for N 15 N 15 nuclei the NOE varies between 1 to minus 4 and this is because of the negative gyromagnetic ratio of nitrogen as you know the gyromagnetic ratio of nitrogen is negative whereas for carbon 13 it varies from 1 to 5 right. So 1 means NOE 1 means this is coming from the rigid portion of the protein minus 4 means it is extremely flexible. So as we discussed NOE reports about the fast temperature motion NOE value closer to 1 means this is coming from the rigid portion and minus 4 is for flexible portion ok. So how do we set up the experiment typically we have a recycle delay of about 5 second right. So because we are starting with N 15 allows for longitudinal relaxation. So all the time what we want that before we start our next scan magnetization should go back to equilibrium like we have started from somewhere but it should go to equilibrium. So that is how we have a longer recycle delay and usually NOE and no NOE recorded in an interleaved manner to reduce the artifact. Typically you record 128 complex T 1 point a 5 D can be processed with 0 filling of 512 in direct dimension and about 1 k in the direct dimension then you do processing by just 90 degree shifted square sign bell in both dimension and then you calculate basically I ratio of the peak I set divided by I unset for cross peak intensity. So here we are getting 2 for like we can record in an interleaved manner then do a split NOE it gives me 2 spectrum here it is a I unset right. And then you have another spectrum where intensity are little weaker this is I set. So you measure the intensity of I set say X amount and divide that by corresponding intensity by Y. So this is our NOE value and then error in NOE can be calculated using this formula. So you can have residue a specific NOE value and if you do for this protein that I was discussing with you sumo 1. Now what we see this is these are the NOE value so this is actual NOE value and these are errors basically you can see here these are error bars. So in this protein let us look at the structure like our residue wise the NOE value. If you look at starting from 22, 23 and going up to about 90 we see that all NOE are positive with little variations. And this says that most of the like here the protein is quite rigid in this domain starting from here will be 22 and here will be about 93, 94 right. So all these shows positive NOE values are quite close to 1 and when we look at the termini here and here the value is negative. Here all the way up to minus 4 as we discussed this is N15 NOE so the maximum negative value can go up to minus 4 and here also the flexible protein shows the negative NOE. So all the flexible portion in this protein is showing negative NOE all the rigid portion is showing the positive NOE. That is how typically the heteronuclear NOE value for N15 nuclei one can obtain it right. So experiment is very simple you record an NOE where relaxation delay is 5 second in one case you saturate about 3 second in another case you do not saturate, record 2 HSQC spectrum measure the intensity of each peak and then you calculate the heteronuclear NOE dividing I set divided by I unset that is a residue wise your NOE value then you can calculate the error and then get information of the first temperature motion in a residue specific manner good. So now we summarize R1, R2 and NOE let us look at together what happens when we measure for the same protein this is our N terminals this is our C terminals this protein is called sumo 1 small ubiquitin related modifier 1. So here are the second structure element if you look at here now let us look at all these three R1, R2 and NOE for the flexible portion we have a high R1 value, low R2 value and negative NOE value for all the structure portion you look at the beta sheet position or alpha helix position we have a low R1 value, high R2 value and positive NOE again for the loop you have flexibility. So high R1 value relatively like some of those are low R2 value but because of exchange contribution some of these are still high and positive NOE value. So by comparing it you know where is the fast amplitude motion right this also reports about nanosecond time scale motion this also nanosecond to picosecond time scale motion whereas R2 reports nanosecond to mic like a picosecond to nanosecond and microsecond to millisecond time scale motion right. So that is what we can get information and just to for your ready reference if you denature this protein R2 and R1 in a sequence wise manner down so much variation they are more or less flat. So for a structured protein we know where is the flexibility and that are very well captured in the experiments R1, R2 and NOE that is what is being shown here great. One important thing right the R2 has one important contribution that we had call it REX. Now that is exchange contribution so here is a protein where you see R2, R1 value, R2 value and NOE. In this case you do not see much variation only you see that there are some loops here but not like whatever sumo one we had. Now in this case you see that R2 value for some reason you have a like a relatively high R2 value right. So this high R2 value and here if you look at NOE it is about 0.6 so that means relatively rigid not so flexible few of them NOE are less but protein seems to be quite rigid the R1 value also not so much variation but in R2 there are some portion which shows high R2. Now one can measure the REX which is exchange contribution and you can see the portion where high R2 values are there you see high REX that means the value in R2 are coming from that exchange contribution that we are talking. In a simplistic term this you do not need to really do the REX experiments separately you just if you have measure the R1 and R2 you do R2 by R1 and those portion will be quite exposed. So you know that this portion is coming from this high value is coming because of REX measurement. You can even do the analysis called R2 multiplied with R1 and that also reports about this REX contribution. The another parameter which we are going to talk about S square this the order parameter. Now order parameter also tells about the rigidity in the molecule that I am going to talk in after few slides how it is calculated but this tells about the rigidity of the molecule. If you calculate the order parameter you can also see it is quite flat. So order parameter about 1 is showing the rigidity and about coming towards 0 shows flexibility. So here you can see on average the order parameter is 0.8 which says that protein is quite rigid. So if you do all these experiments R1, R2, NOE will learn a lot about the protein dynamics and possible alternate states that it might be capturing. Great, so we measure R1, R2 and NOE. Now can we go and do more analysis. So you know that if you remember from the first second lectures we had talked about the how 1 by T1 which is R1 is captured in different spectral density function like j omega H the spectral density at proton frequency, nitrogen frequency, sum of this frequency difference of all this frequency. So it can be given by the spectral density of the difference of this frequency, nitrogen frequency and sum of this frequency. So one can like have T1 similarly T2 can be explained in terms of j omega 0, the difference in the frequency and all those and NOE can again be measured or it can be represented in terms of the sum of the frequency and difference in the frequency. So essentially if you measure R1, R2 and NOE we can get the spectral density function of a protein that we are going to discuss in the coming slides. So essentially you need 3 parameters this R1 measurement, R2 measurement and NOE measurement right. So if we like a if we have these then how to interpret the relaxation data that we can now look at. So relaxation data for a folded protein can be analyzed using something called modal free approach of liparisabho. Now liparisabho give a modal free approach where we do not consider a modal. So this is based on assumption the separate that internal or globular motion are separate. In like a globular motion here is a protein how it is tumbling and internal motion how the loops residue in the protein are tumbling. It is based on an assumption that these two can be separated and the dynamics can be described in term of the overall tumbling. So here is my molecule how it is overall tumbling that is Tm, how internal correlation time for say particular this portion is happening that is Te and the generalized order parameter that describes the amplitude of the of the internal motion. So using this experimental parameter and liparisabho model one can describe the motion that are present in the protein. The only thing that and that there are some assumptions in doing the liparisabho calculation. So what it says the when overall tumbling of a protein. So say protein is quite rigid molecule and you are considering that as a globular. So if you are considering globular the overall tumbling in a protein that happens can be defined by single correlation time. So here is my protein and it is tumbling. So you know that there is a only one tau c the correlation time and internal motion can take on a time scale which is happening much faster. What it means say the protein is tumbling at a nanosecond time scale but internal motion is much faster than that right. So that is one of the assumptions that basically this analysis of Nodal field takes. So if this is the case then the correlation of two motion internal and overall can be separated and total correlation time can be given as that. So correlation time at time t will be C0 t and Cit right. So the correlation function of overall motion assuming it is isotropic distribution. So everything is moving in an isotropic manner and that is how you give this correlation time. So one can have a C0 t which will be given by 1 by 5 experimental t by tau m the overall correlation and if you do Fourier transform of this you can get the spectral density function which is 2 by 5 tau c divided by 1 omega square this is the frequency of tau c correlation time. So t m being the rotational correlation time for overall tumbling of a protein that is how you essentially measure in this case considering the isotropic motion. Now the correlation function for internal motion which was this component internal motion this is external motion like overall tumbling and for internal motion you can get it S square that is generalized order parameter 1 minus S square experiment t by tau e that is the internal motion. So effective correlation time. So for a completely restricted motion that is S square like S square completely the S square value goes to 1 that shows the protein is quite rigid and for completely disordered reason that is completely disordered reason the S square is 0. So now coming back to this whatever we discussed few slides back here if you look at the S square for this protein is more or less 8 what it means that pointed sorry pointed what it means that this protein seems to be quite rigid. If the protein has a flexibility you see the order parameter will be 0.2, 0.3 or something right. So that is that is what now for doing this analysis it is considered like this leperizavo has considered that this molecule is quite globular and it is having isotropic motion and that is how you get this get these values of S square 1 for rigid and S square S square equal to 1 for rigid S square for 0 is for flexible. So modal free assumptions it can it is assuming the molecular overall motion is an isotropic right. So that is a bigger assumptions and the method for characterization over all motion overall rotation is priori implying that motion of most of the protein are very fast like if you remember we are saying that here is a my molecule it is considering overall motion to be tau m to be isotropic and internal motion is very fast like a less than 100 picosecond. Now usually the parameter governing the relaxation for N50 nuclei are kept fixed in this predetermined value in modal free analysis. The modal free approach assume that intermolecular motion are independent of overall tumbling. Why because this is happening very fast compared to the overall tumbling and the conventional modal free approach implicitly assume that protein does not aggregate. So at any concentration just protein is well behaved and typically NMR relaxation there is no conformational exchange sorry there is no aggregation because of during the experiment that we are doing. So all these assumptions are there in the modal free analysis and that is how one can do it ok. So now let us see what actually how modal free analysis is done very briefly. If you remember your j omega is 2 by 5 tau c 1 plus omega square tau c and what j omega means how much power it has. So it captures the power with the frequency. So for a fast tumbling and a slow tumbling if the tau c is of 1 nanosecond. So it is a fast tumbling a bigger protein it has less j omega and slow tumbling a smaller protein has a more and it rapidly decays it ok. So this is the j omega that we had seen in the earlier slide. So just to remind it spectral density function j omega is the Fourier transform of correlation function and just as a rapidly relaxing domain signal gives a broad signal like if it is rapidly relaxing it has a broad signal, if it is slowly relaxing it has a sharp signal and that basically is given by the spectral density function. So this makes sense right molecule that tumbles very rapidly samples a wide range of frequency right. So and molecules that some tumbles slowly have a very long correlation times and only samples free frequency. So now with this assumptions can we delve deeper and do the modal free analysis to understand what parameters can be extracted. Remember modal free analysis is for isotropic for globular protein the it considers that the isotropic motion is happening in the protein and this internal motion is very fast. With this I will stop you here I stop here and we will can take the analysis of modal free in the next class and if your protein is not behaving isotropically there is lots of an isotropy if it is not a globular protein there are some portion which is very rigid and some portion which is flexible what needs to be done. These two questions we are going to take it in the next class and we will discuss modal free analysis as well as the reduced spectral density analysis for protein to interpret the data of relaxation. With this I will stop it here for today. Thank you very much.