 So, welcome back to today's lecture, we were discussing polarization transfer where we started to discuss steady-state and transient NOE, so we will continue from there. So now what previously we have looked at, what is the origin of NOE and what is the physical principle behind NOE that is nuclear relaxation, so how it is done, so what we are doing, we have a two signal which are somehow coupled through space and we saturate one signal, so by saturation what we mean by we equalize the population across the transition by radiating with a weak field, so that means there are two levels here and we saturate we apply a weak RF pulse here and now this is equalizing the population between those two states and then while we are observing what is happening to other signal, so that is what is NOE, two coupled system one is perturbed and perturbed means equalize the population between these two states of these signal and then we look at the effect of that perturbation on other signal, so what we are observing what is happening on another signal, so NOE is a manifestation of the attempt of the system to come back to the equilibrium, so that means how it comes come back to equilibrium, so because of perturbation the population equalizes and then what is happening actually it is trying to come back by some mechanism called relaxation, so relaxation is trying to equalize that sorry re-store the equilibrium and in attempt to store that there is a signal enhancement that is given by an eta, eta that is a NOE signal, so that is the signal because of perturbation minus the original signal divided by I0 is the original signal that is called NOE enhancement. So essentially NOE is manifestation of the attempt of the system to come back to the equilibrium and that is given by the symbols where I is the signal obtained because of this perturbation minus I0 is original signal that is the ratio that is given as a NOE enhancement. So next what we looked at the two kind of NOE, one was steady state NOE and another was transient NOE, we looked at what is the difference between the steady state and transient NOE and then we looked at the positive NOE and negative NOE, in some case you get a positive enhancement, some case you get a negative enhancement, so we looked at that. So here we looked at there are four states of two spins, so here there are two spin I and spin S and we can notify this as alpha, alpha state, alpha beta state, beta alpha state and beta beta state, so two transitions belongs to I, this and this and two transitions belongs to S, this and this. Now that is what transition we have written, so actually if we perturb one of the spin either S or I because of perturbation the population equalizes and then we are looking at how by different relaxation mechanism it is coming back and what is the effect of that relaxation on the other spin. So we looked at positive and negative NOE and we have seen that actually this double transition and this zero transition probability actually that plays important role in giving the NOE enhancement, so the ratio of this with the omega 0, omega 1 and omega 2 is given as NOE enhancement and then depending upon what kind of NOE signal we get that will be enhancement for that. So we have seen for generally for small molecule it is positive NOE and for large molecule it is negative NOE and we have also looked what is the reason. So for positive NOE it is omega 2 which majorly contributes and for negative NOE omega 0 that majorly contributes. So we have done rigorous analysis of population distribution and we have looked at the effect of the relaxation when it equalizes the population and we have developed a master equation to explain the perturbation in the population. So up to here we have seen in the previous class. Now we go back and do little more rigorous analysis to explain the steady state NOE. So let us see for a steady state NOE what we are doing, we are looking at the spin X. On spin X what is the effect to irradiation of spin A? So we are irradiating say spin A and looking the effect of that irradiation on a spin X. So like MA can be 0 and we are looking at how MX is dealing with time to 0. So we put that in master equation and what we can get the change of population of X spin with time will be given by rho and sigma so MX M0 so that is population of X magnetization of X spin and this initial magnetization of X spin. So this is one rate and there is another rate here so this is called auto relaxation rate and this is called cross relaxation rate. So cross relaxation rate is happening because A and X spin are correlated. So in that case if you do the analysis we get an enhancement of NOE of X that will be given by MX magnetization of X at any time t minus magnetization of X at time 0 divided by magnetization of X at time 0. So we can do the simple algebra to get this relation which will give the magnetization of A at time 0 divided by magnetization of X at time 0 and ratio of these two rates the sigma AX divided by rho X. So that is a kind of steady state enhancement we are getting and that we have seen earlier. So why this is happening because both spins are actually dipole and there is a dipole-dipole interaction because though both spins are connected by the dipolar interaction in space. So suppose this is X A spin and X spin they are both are dipole and connected by dipolar interaction so that is causing relaxation. So when it causes relaxation there is auto relaxation rate and there is a cross relaxation rate and that can be calculated by measuring the transition probability that transition probability we have looked in the previous slides. So that is omega 0, omega 1 and omega 2 if you can calculate this one can find it out what is the omega 0 for this correlation. So that will be given by K divided by 20 to tau c 1 plus omega A that is a frequency of A, omega X that is frequency of X and tau c is the correlation time of this molecule. So we can give for omega 0 similarly we can have for omega 1 for A, omega X for A and that we can get the correlation by doing simple algebraic equation. So we can get omega A, omega X and omega 2 that is transition probability for double quantum transition. So we can get and the K here can be defined as this relation which takes care of the gyromagnetic ratio of A and X and also distance if you look at here distance dependent. So this interaction or transition probability has important term the separation between these two spins. So this is R A X. So R X is the distance between two spins. So if you put it we can get the transition probability of zero quantum transition the single quantum transition and double quantum transition and there is a term correlation time. So if we put everything together now we know that NOE depends upon rate of molecular motion because the tumbling time tau c is important factor here tau c. So that is a molecular motion. So now spectral density are actually dependent upon what is the reorientational time. So if we put up how much time it takes to reorient itself that is reorientational correlation time tau c that is a relation. So here is the enhancement factor of NOE NAX and as you can see for a small molecule we have positive NOE and for large molecule as we go we have a negative NOE all the way up to minus 1. And the correlation time is increasing so that means for shorter molecule which has a faster correlation time you have a positive NOE that is 0.5 and for a smaller molecule you have a negative NOE. So that is a dependence of correlation time with the NOE enhancement factor and for small molecule and larger molecule that is the relation. So what happens in extreme narrowing condition for a small molecule this is the case where extreme narrowing condition comes because they can tumble very fast. So the spectral density becomes equal to the transition probability. So we can write it omega 0 omega a that is 0 quantum transition single quantum transition probability of a spin and the double quantum transition probability is in ratio of 2 to 3 to 12 and that if you get it do the algebra so ratio of this cross relaxation rate and then auto relaxation rate one can get it half and that is what we were saying. So equilibrium magnetization of a spin ma 0 and x spin are proportional to individual nuclear spins and they are guided magnetic ratio. So if we put that all the equation so the magnetization of a is proportional to gamma a which is gyromagnetic ratio similarly this x spin magnetization proportional to gamma x. So for a half spin system suppose it is a proton proton or proton carbon that is a half spin system what we have is NOE enhancement will be half gamma a by gamma x. Now if you take this so for a carbon proton NOE pair the enhancement is 2 why because gamma of proton is 4 times more than the carbon so this will be 4 divided by 2 that is 2 for a nitrogen this will be half 10 by 1 that is 5 and for proton proton because both spins are now proton so 1 divided by 1 and multiplied with half that is 0.5. So if you look at clearly here for proton proton like a small molecules where we are looking the NOE enhancement between 2 proton we have only 50 percent enhancement if it is carbon we have 2 times enhancement if we have nitrogen we have 5 times enhancement. So now this is for fast tumbling molecules small molecules what happens in the throat motion so for a large molecule suppose biological molecule protein the transition probability mostly will be governed by the zero transition and that will be given by K by 10 tau c. So here and the first quantum transition is zero zero quantum transition is also zero. So NOE for such system we will be given by this relation so ultimately it comes gamma a by gamma x. So now if we are taking the for large molecule 2 proton pairs so this will be 1 by 1 so that is 1 enhancement and that is negative and that is what exactly we saw for the large molecule. For large molecule the NOE enhancement in fairly broad correlation time 1 to 100 enhancement is minus 1 so that means 100 percent can remember but it will be negative enhancement. So for large molecule say negative NOE up to 1 so that is what is for slow motion limit so for that enhancement we get for biological molecules. So that is steady state NOE now coming back to transient NOE. Just to refresh your memory what is transient NOE that we have 2 spins like previously I spin and S spin we are applying one selective pulse on a spin here which is inverting the population and then we are waiting for some time which is called mixing time then we apply a 90 degree pulse and we detect what is happening after this that is one dimensional experiment and as we discussed this TM is the mixing time during this spin mixes. So transfer of magnetization occurs during this mixing time the inverted spin to other spin via dipolar coupling so that is what happening here we have 2 spins spin S spin I we are applying a 180 degree pulse and then we are waiting and during this time by dipolar coupling the spin-spin mixing is happening. So we can do the same analysis of change of magnetization of spin with respect to time and that will be given by this master equation. So m is now representing multiple spin in column vector like m1, m2, m3, mn and that mi at any i is the difference of the z magnetization of high spin minus the equilibrium magnetization. So we can write the R is various rates the auto relaxation rate and cross relaxation rate. So if you look at the diagonal element is correlating with self that is auto relaxation rate and then you have a cross relaxation rate. So that is the R matrix and then we have a column vector given by m so if we take this solution for any spin magnetization at any time t will be given by e to the power minus RT and equilibrium magnetization m0. So for any time tm because we are mixing for time tm the magnetization will be given by this series multiplied with equilibrium magnetization. So if you do that we consider the short tm so then second and higher order term can be neglected. So we can find an explicit solution which will look like magnetization of any spin at time t will be given by this and then will be equilibrium magnetization for any spin j not equal to i. So one can give it a simple formula of mi magnetization of any spin at time t will be given by this auto relaxation rate and multiplied with the time ta and summation of cross relaxation rate with tm and mz. So if you do that one can get the contribution of the NOE enhancement and that NOE enhancement at any time tm can be given by this formula. So we can sum over the time that is there and for selective inversion of say we are doing of on jth spin one can find it out what is the NOE enhancement so that will be given by say for eta at time t that will be minus 1 by mi0 minus 2 mj and this is cross correlation rate into time tm. So if you look at we can simplify this and one can get essentially 2 sigma ij tm. So it depends upon the tm NOE enhancement at the end we can conclude that it depends upon tm. So if you have short mixing time the enhancement can be one and if you have long mixing time enhancement can be other. So that means if you keep increasing your NOE can maybe probably increase. So that is how for short mixing time the transient NOE at spin i is due to inversion of spin j and cross correlation rate between 2 spins is also linearly dependent upon mixing time that is what I was saying. So relaxation rate between 2 spins is linearly dependent upon mixing time. So if you increase mixing time the relaxation rate will vary. So cross relaxation rate is inversely proportional to the inverse power of the inter nuclear distance. What it says here we have 2 spins and r to the power 6 that we have seen. So relaxation rate depends upon 1 to the power r6 so that is actually very important factor. So the NOE ability between 2 spins is distance dependent and that depends upon the dipolar interaction between these 2 spins and that dependency is 1 divided by r to the power 6. So if you keep increasing the distance the effect is going to be minimized. So this transient NOE experiment actually provides the powerful tool for estimate the inter nuclear distance in coupled spins. So here suppose these 2 spins there how much NOE enhancement that depends upon r. Shorter you have more enhancement longer you have less enhancement and therefore looking at the cross peaks coming because of cross relaxation rate if you can measure the intensity of that cross peak we can get an estimate of the distance between these 2 spins. So actually this is very important and this actually opens a new avenue for measuring the inter nuclear distance between these 2 spins and that is what essentially was used to develop an experiment which is called nuclear overhouser effect spectroscopy NOG and these actually opened a new avenue in structural biology because we can precisely or maybe up to a great accuracy we can measure the distance between 2 coupled spin. So then what is short mixing time I was saying that here we mentioned for a short mixing time the transient NOE of spin i due to inversion of spin j depends upon cross relaxation rate. So how do we know that what is short and what is long it is a relative term. So short mixing time is essentially it should be much shorter compared to the spin lattice relaxation time of spin j. So you know spin lattice relaxation time it is a inversion recovery if you invert the spin j how much time it takes to come back to equilibrium that is called T1 time or spin lattice relaxation time. So that the short mixing time has to be less compared to the spin lattice relaxation time and then this will hold true. Typically what we have is like a for protein suppose the relaxation time is spin lattice relaxation time is say 1 second. So your short mixing time will be few millisecond that will be compared to less. So therefore for say small molecules you can have a short mixing time up to few 100 millisecond like 500 millisecond or 800 millisecond however protein has relatively short spin lattice relaxation time therefore mixing time which is used in case of protein is generally 100 millisecond to 200 millisecond or 250 millisecond. For a small organic molecule you can go all the way up to 500 millisecond to 800 millisecond. So that is a short mixing time that is a relative for this is for protein biobig biomolecule this is for a small organic molecule. So these concepts are used in NOGIE so what I am going to do in next class I will take you these concept of magnetization transfer or polarization transfer how this is going to use as a measurement of distance by few experiment called NOGIE and NOGIE briefly I will introduce you and then we go to the heteronuclear polarization transfer by an concept called inept transfer. So we will continue with this. Now if you have any questions please write to us or ask us we will try to solve it. Thank you very much.