 So, this week we are discussing protein dynamics as proved by NMR spectroscopy. So, in last lecture I mentioned why dynamics, why it is important, what why the dynamics is important for understanding the biological phenomena, biological function, protein function and all those and what are the techniques that I summarized can be used for understanding the protein dynamics and then we went ahead and looked at the how NMR can be used for understanding the protein dynamics. So, it is all about time and motion and we will continue from there. So, NMR appears to be a versatile tool for the atomic structure of various time scales of transition ranging from say picosecond to second time scale motion or in frequency term from terahertz to hertz right. So, there are some which can called is a internal dynamics or it is some called be molecular diffusion. So, local time scale motion or a global time scale motion and those can be proved by various NMR experiments that we mentioned. So, relaxation in laboratory time frame if we do something like a T1, T2 or NOE that can prove basically the first time scale motion ranging from picosecond to sub micro second time scale motion. Then we can do the relaxation in rotating frame like T1 row, T2 row or ROE and that actually captures the micro second to millisecond time scale motion and then even fast even slower motions can be proved by exchange NMR like one of them is hydrogen deuterium exchange can be proved. So, this basically the faster time scale comes from average anisotropic interaction and this can be like a slower time scale motion can be also inferred from line shape analysis. So, essentially all these utilizing some of the NMR experiment can be proved. Now, for in solid since molecule tumbles slowly, so in solid state generally you have a slower time scale motion. So, what actually relaxation means? So, relaxation a process by which any spins returns to their equilibrium position equilibrium population. So, that means, generally in a B0 magnetic field you have a spins aligned to this you create some perturbation and the time that takes to come to the equilibrium position is essentially relaxation. So, it is governed by the fluctuation that happens in local field and that local field fluctuation in is experienced by these nuclear spins. So, generally what is happening are spins are oriented in a stronger magnetic field and some fluctuation happening in this magnetic field and that basically causes a relaxation process. So, because of the fluctuation that is happening these spins will be reoriented right and that cause the variation in their interactions between two spins or their chemical shift anisotropy. So, these basically these are contributing towards the relaxation phenomena chemical shift anisotropy and dipolar coupling that we are going to look little detail what these are. So, typically we also looked at these heteronuclear are well suited for understanding the relaxation mechanism heteronuclear such as 13 C 15 N. Proton has slightly complex relaxation behavior, but yes it can be used. In one case you can see you can spin dilute it change with deuterium many of the proton can be changed with deuterons and then few protons can be actually proved in an elegant way to understand the relaxation mechanism, but typically all for all simplistic calculation in protein 13 C or N 15 or both are exploited to understand the relaxation mechanism in protein. So, relaxation mechanism actually influenced by two of the major interactions one is called dipolar coupling another is called chemical shift anisotropy. There are some other which influence the relaxation mechanism is called spin-spin coupling or J coupling or it can be even quadruple or coupling or the exchange happening between between the spins. So, major population and minor population how they are exchanging. So, all these essentially contributes towards the relaxation mechanism. Dipolar coupling are between two spins chemical shift anisotropy how the spins are oriented in the magnetic field what is the anisotropic interactions in them, J coupling is a scalar coupling between two spins and quadrupolar like one dipole and one quadrupole interacts and chemical exchange as I said if it is exchanging between two states those all contributes to the relaxation mechanism. So, some basic theory of spin relaxation in protein. So, one of the major contributor is this dipolar interaction. Another one is chemical shift anisotropy. So, suppose in the magnetic field these are two spins spin 1 and spin 2 and they are separated by some distance called r 1 2. So, this is distance and there is some angle with the main magnetic field which is theta. So, there will be dipolar interaction that depends upon this distance and also on the orientational angle and that basically contributes towards the relaxation phenomena. So, in liquid what happens this the tumbling most of the time averaged out this dipolar interaction in solid that tumbling does not happen. Therefore, dipolar interactions are there and that is how lines in solids are broader which we are going to look at in the next weeks. But to understand this is one of the cause even if averaging happen this is the one of the cause for dipolar coupling. So, if you look at if dipolar coupling is present you can see the there is a line splitting and that can come and line becomes broader because of this dipolar coupling. Another important phenomena that contributes to the dipolar relaxation phenomena is called chemical shift anisotropy. So, if spins are not tumbling and they are in the magnetic field and they. So, they are actually oriented in various direction right. So, each of this direction will have one resonance frequency which is shown here. And if you take envelope of all these resonance frequency because of this different orientation you get a really broad line something like this and this is a essentially chemical shift anisotropy. So, because of like the orientation depend sorry chemical shift anisotropy is essentially orientation dependence of the chemical shift right. So, when we start like when we start tumbling of these spins these anisotropic interactions essentially averaged out and you have a isotropic chemical shift generally we see these sharp peaks in liquid state NMR spectrum. So, now in solid basically you spin very fast to make these anisotropic interaction look like a isotropic which again we are going to look at the next slides. But this anisotropy is present all over in liquid it is quite a based averaged out because of the Brownian motion that spins can take, but this is one of the again major contributor to the relaxation phenomena. So, these two DD and CSA are dominant source of relaxation. Now, yes so, this is the essentially spin relaxation or nuclear spin relaxation depends upon two phenomena these are the two dominant contributing phenomena the dipolar relaxation and chemical shift anisotropy. So, suppose these two spins are in the main magnetic field which is B 0 they oriented along the magnetic field spin 1 and spin 2 and there is a distance between them which is R 12 and the angle of orientation is a theta. That so, the dipolar interaction depends upon this angle which is 3 cos square theta minus 1 and also the distance between them which is say R i s or R 12. So, it is a 3 to the power like a 1 to the power 1 divided by R to the power 3 and this gamma 1 and gamma 2 are the gyromagnetic ratio of these two spins this is permittivity pi and Planck's constant. So, these are the phenomena that contributes to dipolar coupling. Now, one thing you notice this 3 cos square theta minus 1. So, typically these this is the main contributor of the spin. So, if the spins are quite close that means, if this distance is short the interaction dipolar interaction is large if it is long then dipolar interaction is weaker right. So, when we do spin dilution that means, we make spins talk to like a talk to each other in a less dominant way and therefore, we reduce this spectral the dipolar interaction that I was talking that proton has a complicated relaxation phenomena. So, we can spin dilute it by putting deuterons of proton-proton dipolar coupling can be reduced in that sense or in solid this is a trick that we are going to look at in the last week of this course. In solid state we always try or tends to make this term 0 by setting or spinning the or sample at certain angle which is called magic angle. So, this 3 cos square theta minus 1 becomes 0 and that is how we try to reduce the dipolar interaction between those. The another dominant interaction that we talked is a chemical shift anisotropy this is again an orientation dependent interactions. So, in magnetic field spins can be oriented in various fashion and since they tumble most of the time this interactions is 0 or it tends towards 0 because of their tumbling brownian motion, but suppose there is some orientation is remaining some orientation is still there. So, they will cause the chemical shift anisotropic that anisotropic interaction will arise because of this will be oriented in different dimension. So, like suppose they are isotropic we see one peak and that is what we see in liquid state, but when there is a restriction in motion it is not tumbling very fast you will see some kind of anisotropic interaction emerges out and you see many lines are there and if you take an envelope of all those line you see a really broad peaks, yet so this is chemical shift anisotropic. Even in solution quite a bit of those are not there average out, but these causes relaxation DD and CSA are major source of relaxation phenomenon. Therefore, typically an isolated system is chosen sorry isolated is X H spin system is chosen for relaxation rate constant, where X spin like a 13 C and N 15 are chosen. And the dipolar interaction between that X spin and proton is considered and also the CSA originating from X spin is contributed. So, relaxation rates that can arise because of these anisotropic interactions CSA or DD can be expressed in something called spectral density function. So, we are going to look at what is essentially the spectral density function. So, essentially all the relaxation rate that we are talking can be expressed in this term spectral density function. So, I am going to explain you soon what is spectral density function. But before I go to spectral density function let me define something, one is called correlation function right. So, correlation functions and then we will come to spectral density functions. So, correlation function in any case so the correlation function can be given G of T with time 1 by 5 exponential T by tau C. Now, this tau C essentially is the correlation time. So, correlation function for an isotropic diffusion of a rigid rotor we call let us explain this spin as a reduced rotor can be given in this term G of T 1 by 5 exponential T by tau C. Now, this correlation time which is tau C is a time constant right for an exponential decay of the function tau C is approximate amount that molecule take to make rotation by one radian. So, how much time it takes to make rotation by one radian that is a tau C correlation time ok. So, short correlation time essentially short correlation time causes the correlation functions to decay rapidly whereas, a long correlation time, long correlation time makes function to decay slowly and these correlation time essentially depends upon the molecular weight of a molecule, what is the shape, what is the solvent viscosity, what is the temperature. So, let me simplify this, if a molecule is bigger right, molecule is bigger that means in solution it will tumble slowly. If the molecule is smaller in solution it will tumble fast, if the solvent is viscous that means the molecular tumbling will be also slow. If you rise the temperature the same molecule can now like can make a rotation fast. So, it depends upon various the shape and size of a molecule, molecular weight of a molecule, the solvent viscosity whether it is more viscous or less viscous solvent and what is the temperature. So, correlation like a correlation function with the time can be expressed like this. So, suppose a molecule has the correlation time of 15 nanosecond, it is a correlation function will decay slowly and if it has a correlation time of 1 nanosecond you can say it decays very fast right, so very fast. So, that is what here we were saying, it is time exponential decay of a function, it is approximately the amount of time molecule takes to make 1 radion and if the correlation function is decaying rapidly here, it takes long time to cause function to decay like a if it is decaying rapidly it takes long time ok. So, short correlation time like this function decay rapidly, long correlation time function decay slowly. So, that is a correlation time. The another one we were talking the spectral density function. So, it is essentially the power right, so power it is connected with the correlation time. So, suppose here the correlation time is 100 nanosecond. That means the molecule is 100 nanosecond molecule is solely tumbling, so you can see the spectral density function which is j omega right. So, omega is a frequency j is a spectral density function with the frequency it dies or dies very rapidly. If the correlation time is shorter like 1 nanosecond, this is very very slowly decaying. So, spectral density function decays very slowly, so j omega j of omega is given by this function where tau c is the correlation time 1 omega is the frequency and again tau c. So, it is store the power of a molecule how rapidly or how quickly or how slowly it decays how the power it dissipates that is what the spectral density function is saying. So, for a longer correlation time it can dissipates power very fast. If a shorter correlation time it dissipates power very slowly with the frequency that is the spectral density function tells about and we said that we can express our relaxation parameters in terms of a spectral density function. So, spectral density function j omega is a Fourier transform of correlation function just as rapidly accessing domain signal give rise to a broader line. If something is rapidly decaying it gives the broader line, if something is slowly decaying it gives the sharper line right. So, give rise to broader line short correlation time like have a broader spectral density function. So, this makes molecule sense that molecule tumbles very rapidly can sample a wide range of frequency and molecule that tumbles slowly have a very long correlation time and only samples fewer frequency. So, let me explain again. So, a small molecule a small protein a small bio a small peptide and all those tumbles very fast a bigger molecule tumbles slowly. So, if the molecule is tumbling rapidly that means it can sample a wide range of frequency that it can essentially wide range of frequency say here in this correlation function the molecule which is a shorter correlation time can essentially samples the all frequency right. So, a smaller molecule can basically samples the wide range of frequency or is bigger molecule a protein of 2 Kd can sample many frequency a molecule of 100 Kd which correlation time is about 100 nanosecond samples only few frequency right. So, that is the molecule tumbles very rapidly can sample a wide range of frequency and molecule that tumbles slowly like a bigger protein have a very long correlation time and only can sample few frequency that is essentially a spectral density function tells about itself Fourier transform of a correlation function right. So, now if we know this what is correlation function now we can express our relaxation parameters in terms of this spectral density function. So, R 1 which is longitudinally relaxation rate can come from the R 1 because of D dipolar coupling R 1 because of CSA and you can explain this R 1 in terms of this formulae where the spectral density function of the H spin and X spin is given. So, D square divided by 4 6 j omega H plus omega X. So, this is the joint frequency that H and X are evolving that is a spectral density function for proton and carbon 13. This is the difference in the spectral density function or and then individual spectral density function of X as well as H spin. So, that all contributes towards the relaxation of these spins in R 1 longitudinally relaxation rate. Similarly R 2 can be given by these formulae R 2 of dipolar coupling and R 2 due to CSA and it again depends upon the spectral density function of proton and carbon given by these formulae omega H plus omega X here individual spectral density difference of a spectral density and the spectral density at a 0 frequency. So, that is what R 2 and then this is cross relaxation. So, these are individual relaxation and this is cross relaxation this again will be given by the sum of these two frequency and difference of these two frequency. So, R 1, R 2 and sigma X H are the rate constant for spin lattice relaxation as well as the spin relaxation and this one is cross relaxation right how the H and X spin cross relaxation with each other that is given by sigma X H. So, these now we can see that these simple relaxation parameter the individual R 1 and R 2 spin and spin lattice relaxation can be given by the spectral density function of the individual spins X and H their joint frequency their difference frequency and the 0 frequency. So, in terms of these spectral density function we can explain our R 1 and R 2. So, dependent of a spectral density function can be evaluated on this 5 frequency. What are those 5 frequency? The joint frequency omega H and omega X. So, omega H plus omega X, omega H, omega H minus omega X, omega X and 0. So, these are 5 different frequency which can contributes towards this spectral density function right. Some of the parameters that I had given in the previous slides like a D is essentially these parameter which also depends upon the distance between the X and H and rest are the gyromagnetic ratio or Planck constant and permeability of the vacuum the X H is essentially the bond length and omega X, omega H are gyromagnetic ratio and C is a constant right. So, now delta delta is the CSA of the X spin right. So, and you consider that the chemical shift tensor is axially symmetric. Now, you can tell it CSA for different nuclei which are typically given for N 15 this chemical shift anisotropy is about 170 ppm references from here and for carbonyl about delta delta is about 35 ppm for C alpha it is about 30 ppm. So, these are various constant that were clocked in can be clocked in here to find it out their contribution coming from different relaxation rate ok. So, now the R 1 and R 2 rate constant are determined experimentally we are going to look at how we can determine on proteins and the cross relaxation rate whereas, sigma X H is determined from the steady state NOE that also we are going to look at how we can determine from the how we can design an experiment to do this heteronuclear NOE. So, as we mentioned this cross relaxation rate depends upon D and the value of D is given here the spectral density function can be coming from NOE minus 1. So, R 1 multiplied NOE minus 1 and NOE can be given as 1 plus the sigma X H by R 1 you can just rejig and do little bit of algebraic conclusion. So, you can find it out the NOE that we calculate experimentally can basically come from these numbers spectral density function. So, NOE is this cross correlation rate. So, if we do the three basic experiment R 1, R 2 and heteronuclear NOE we can essentially determine the all spectral density function and tau C and that is what typically is done in the protein NMR. So, spectral density function at the phi frequency cannot be determined from three exponentially determined relaxation rate constant by just measuring T 1, T 2 and NOE. So, assumption must be made so that only three unknown need to be determined from these three value right. So, we are want to determine the phi frequency omega at omega H, omega X, omega H plus omega X, omega H minus omega H and 0. So, those are phi frequency we wanted to determine just from three rates which is not possible. So, we need to make some assumptions. So, that three equation and three unknowns are there. So, there are various mathematical models that maps the spectral density function and one of them is modal free analysis widely known as LeParis-Jabot modal free analysis and that basically gives this site specific internal motion of protein will be towards the end of this week we were going to briefly touch upon what is the modal free analysis, but essentially it can be determined from this relaxation rate by making some assumptions and we will be looking at that. So, today I am going to give you a glimpse and we can continue over it what experiments are done for measurement of relaxation parameter. So, typically whatever we have learned right HSQC or HMQC based experiment can be utilized for understanding the relaxation rate like a T1 and T2 or even NOE. So, concept is that we plugged in these parameters where we can we can determine the T1 or T2 or NOE from these HSQC or HMQC based experiment doing a heteronuclear relaxation doing the heteronuclear correlation experiment for determining the heteronuclear relaxation. So, essentially we start with a preparation of desired coherence and then we invoke this T1 delay for a autocorrelation or cross correlation and then we encode the frequency. So, here T1 period I mean coding the frequency and then we transfer the magnetization to proton nuclei and then you acquire on proton and then we give some delay like a D1 delay between the scans. So, typically this is the pulse sequence design we are going to start with. A preparation phase invoking the delay so that we can encode the T1 or T1 relaxation time or T2 relaxation time, then we encode the frequency in direct dimension, then we transfer back magnetization to proton, acquire proton and encode the T2 frequency. So, that we can do a like this delay time dependent HSQC and finally, towards end of pulse sequence we give the relaxation delay so that magnetization returns to the equilibrium state. So, these are typical design of an HSQC HMQC based T1 and T2 relaxation. Next class I am going to discuss how we can basically design a pulse sequence to measure the T1 and T2 and what did how data comes and how we can interpret this data for understanding the relaxation mechanism in proton. So, with this I let me close it today and looking forward to have you in the next class. Thank you very much.