PHYS CHEM BIOC II
PHYS CHEM BIOC II CHEM 453
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This 11 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 453 at University of Washington taught by Gabriele Varani in Fall. Since its upload, it has received 10 views. For similar materials see /class/192549/chem-453-university-of-washington in Chemistry at University of Washington.
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Date Created: 09/09/15
University of Washington Department of Chemistry Chemistry 453 Lecture 11 Text Reading Ch 24 p 636643 Equilibrium centrifugation Summary of lecture 10 Sedimentation can be used very effectively to separate purify and analyze all kind of cellular components It can be understood using the mechanical analogy with ow under gravitation The steady state velocity at which a particle move under centrifugation is determined by the balance between the angular velocity at which centrifugation occurs and the opposing buoyancy and frictional forces F g F F f 0 d 1 1 V1szrzswzr dt f We have introduced the concept of sedimentation coef cient Its dimensions are sec but a more convenient unit is the Svedberg 8103913 s Values for s are usually referred to pure water at 293K20 C Under these conditions the sedimentation coefficient is indicated as follows m 1 V2 p20w S20w 6m720wR A sedimentation coefficient measured under other conditions ie in a buffered aqueous solution b and or at another temperature T can be related to standard conditions by the equation 1 V sz W L 7717 1 PVZ 7720w Boundary sedimentation is an equilibrium technique that can be used to separate and analyze macromolecules by sedimentation S20w S Equilibrium Centrifugation A macromolecular solution subjected to a centrifugal eld will quickly attain a steady state condition in which transport of solute mass occurs at constant velocity and a concentration gradient will be generated If we spin very fast eventually the entire macromolecular population will deposit at the bottom of the tube at a rate that depends on the centrifugal speed and the density and viscosity of the solvent However if we do not spin too fast centrifugation and diffusion will balance each other out so that the system will attain equilibrium at which point net transport will cease and transport velocity is zero This is because transport by centrifugation and by diffusion will oppose and balance each other centrifugation will generate a density gradient and diffusion will try and eliminate such gradient The rate at which equilibrium is reached depends on kinetic properties diffusion coefficient angular velocity etc but the equilibrium state does not the concentration at equilibrium is only determined by thermodynamic properties of the system and not by sedimentation coefficients diffusion etc The equilibrium concentration can be derived from Boltzmann s distribution Let us consider again the analogy with quot quot M 39 39 in a quot quot 39 field will have different energy depending on whether they are higher or lower the distribution of molecules in different energy levels is given by Boltzmann s expression exp E1 EjkT J The probability ratio is equivalent to the ratio of concentrations if we express the energy per mole instead of per particle by multiplying by Avogadro s number we find 5 exp E RT Let us return to centrifugation the centrifugal acceleration for a centrifuge spinning with angular speed a is wzr the force acting on the particle is ml Vzpa2r mlrl 2 U0quot J Fbuoyancy F 2n2r1 4galyr J m m0 mzrdr m m0 Therefore the energy is E g Zpyfrz Substituting this expression we immediately find the expression for the concentration as a function of the axial distance r C 2M 1 V ln rw 2 2p r213 C 2RT 0 Another derivation reintroduces the chemical potential The condition of equilibrium requires that the free energy ie the system s chemical potential which is the sum of the chemical and centrifugal potential is at a minimum at equilibroum If you think about how centrifugation works what we would like to know is really the concentration pro le with respect to the radius r not as a function of time as in nonequilibrium centrifugation but rather at equilibrium We can then impose that the derivative of the chemical potential with respect to r is 0 at equilibrium In lecture 6 we have introduced the chemical potential in analogy to the classical concept of force as of a potential gradient to express differences in free energy that induce diffusion Ifthe concentration of solute C 2 is a function of x the chemical potential has the general form G 2x Gf RT 111C x A difference in chemical potential exercises a force on the solute molecules and the force that induces solute ow is related to the chemical potential by the diffusion equation In the case of centrifugation we have to add a term that describes the centrifugal field so that the total chemical potential of the system is 202 Iuchzm Czntrl gal Gfolutz RT 1n C0quot U0quot Here C is the solute concentration at position r in the centrifuge tube and U is the centrifugal potential at position r in the centrifuge tube At equilibrium 611 0 a1 d j E lchzrmml luczntrxfugal 2 0 0 dGSOW RT dlnCr dUr 0 dr dr dr We can substitute for dUdr the expression for the centrifugal force provided in the previous lecture remember a force is a gradient of a potential then multiply by Avogadro s number to obtain the centrifugal potential per mole of solute which is what the chemical potential will be expressed in and equate the derivative of the concentration with the derivative in centrifugal potential to obtain d 1 C d 1 C n rkBT n rdUr N dr dr dr A F 2 czntrx gal m mo a r Integrating that equation lnC MZ 2 2M21 l72 r2r2 r r0 0 C0 ZkBT 2RT And finally C 2RT 1n ZMZ1 V7 r2 4 0 This equation means that when a solution reaches equilibrium in a centrifugal eld generated by spinning the sample at an angular frequency on a concentration gradient will be generated of the shape given above This experiment can be used to measure macromolecular masses or separate components of a mixture Clearly a plot of 171C vs r2 is a straight line with slope proportional to M Notice that this method provides absolute molecular weight while electrophoresis for example only provides relative molecular weights Example Calculate the weight of carboxyhemoglobin ch using the following data obtained from an equilibrium centrifugation experiment At r46l cm Cchl220 weight At r456 cm Cchl06l weight T2933K spinning at 8703 revolutionsminute The specific volume of ch is 0749 cm3g The density of water is lgcm3 Rearrange the expression for the equilibrium centrifugation experiment 2RT 1nC C0 l VZ p r2 r02 2 2x831x107 ergsmolexKx2933lenl22l06l 1 0749cm3 gxlgcri1327239x8703min 1 mir 6032 461cm2 456ch 70 900g mole Example Consider the centrifugal separation of two gases with molecular weights of 349 gmole l and 352 gmole 2 How fast do you have to spin the sample to enrich molecule 1 to a level of 1 at r3cm if its level is 07 at r10cm mi sz2 IVZ prz r02 2RT Use the equation 10 Set the buoyancy correction to l for gases and subtract the two expressions for C 2 and C1 to obtain CZCI 2M2 M1 m2r2 r2 2YOCI 2RT 0 From which we can solve for a 2 ZRT C2C10 1 M2 M1r2 r02inC20C1 And 2 2x831x107 ergs molexKx273K I 001 0813x1083 2 352gmole 349gmole10Z 32 cm2 0007 a 901 x 103 radians sec 3 V 21 144 X 103 revolutions sec 7239 Equilibrium Sedimentation in a Density Gradient Atype of equilibrium centrifugation involves spinning of a concentrated salt solution at very high speed to generate a density gradient the density of the solution increases with the salt concentration and has proven very useful in the study of nucleic acids If a macromolecule is also present it will form a boundary at a point in the salt gradient where the macromolecules are buoyant Suppose a solution of a macromolecule eg DNA also contains a salt such as CsCl Initially the salt and the DNA have uniform concentrations Once the centrifugation has commenced the salt quickly reaches equilibrium the concentration of CsCl will reach equilibrium as described by the equilibrium centrifugation equation In C0501 Masclwz 1 VCSCZpr2 r2 C ZR 0 ccz0 Because of the equilibrium condition the density of the solution will vary as a function of l r the d1stance from the sp1nn1ng aX1s Suppose at r the solutlon has a dens1ty p V 2 where T72 is the specific volume of the macromolecule cm cm Conwnlrniun A on rum DNA T lt on Kinks llnm IIIVX lupullube a bum Mlube 1 Attltwthe density othe soluhon is less than andthe DNA smksquot to the 2 bottom of the tube bemg pulled downwardquot by the cenmfugal force Atrgtr39 the density ofthe solutmh ls greaterthan V1 and the DNAquot oatsquot 2 upwards toward the top ofthe eehtnfuge tube ALFY the DNA density mereases suppose the solutmh density gradnent is roughly hhear wtth Ifthe soluhon density m the buoyancy eoneetmh is the funehon of de ned above then MM 17VW I dl M d 2R7 d h CW m2 7 t d Integrate the equahon to nd and a er some simple calculus 2 M lnwi39 zm vii5 07 Cmo 2RT dt H t w h tem 397 sttead obit so it dAffers The loganthm can be removedto obtam CmrCMr exp 2 m2 4 2RT d Or Cmrexp C J This is a bellshaped curve with standard deviation 2 RT 039 mzr MDMVDM dpdr Notice that the standard deviation increases as the salt gradient dpdr decrease it also depends on the inverse mass so that the higher the mass the sharper the gradient The utility of this technique is that it has very high resolution with respect to molecular mass Many macromolecules have buoyant densities su iciently different that they can be separated by this technique Hybrid DNARNA was discovered using equilibrium centrifugation in a salt gradient for example although they differ in mass by very little End of Lecture 11 University of Washington 7 Department of Chemistry Chemistr 3 Lecture 10 Text Reading Ch 24 636643 U39ltracentrifugation N Ultracentrifugation and Sedimentation Sedimentation is a technique used to separate purify and analyze all kind ofcellular rnmnnnent 4quot 39 I I 39 anal DEV unsure F5 mg a buoyant force F mug m Epg and thefrictional force F fv In these equations m r the particle p is the density ofthe tluidfis the frictional coef cient ofthe falling body andv at M A 39 39 39 39 39 39 At eadv state the terminal velocity is obtained from the force balance F5 F F 0 From which it follows Vm Vjpk f The term m1 7 Ep is calledbuoyancy sninn39 uh cenrrimre mbe Sedimenulian lree ml on lurllde Em rimenl distance from the solute particle to the axis and air is the c acceleration F roman 41er if is the buoyant force all terms de ned as above F fv is the frictional force F iner is the centrifugal force where wis the angular velocity risthe entrifugal By direct analogy the steady state velocity of a solute particle being spun in a centrifuge tube can be obtained by balancing all forces once again EEF0 1 i7 l v m f 2p m2rsw2r Here we have introduced the concept of a sedimentation coefficient Its dimensions are sec but a more convenient unit of measure for s is the Svedberg 13 810 s Standard Sedimentation Coef cient If we remember the definition of frictional coefficient from Stoke s law then rw T mhih S f 67T77R Of course the quantities p and 11 are dependent on solvent and temperature while R brings about again the molecular properties of the molecule undergoing sedimentation Values for s are usually reported in Svedberg and referred to a pure water solvent at 293K20 C It is useful to use these conditions to standardize the measured sedimentation coefficients 3 Under these conditions the sedimentation coefficient is indicated as follows m1 V2p20w 20w 6 anR A sedimentation coefficient measured under other conditions ie in a buffered aqueous solution b andor at another temperature T can be related to standard conditions by the equation 1 szow S 77b 1 sz 7720w 20w S This relationship only holds true if changing conditions temperature or buffer conditions have not significantly affected the shape or hydration property of the molecule Conversely if it is found that the sedimentation coefficient changes then it can be concluded that one of those properties has changed Sedimentation measurements can be used to estimate the size of macromolecules or especially their assemblies By measuring the sedimentation coefficient diffusion coefficient and partial specific volume we can calculate the molecular mass of any particle under any experimental conditions and follow how it changes e g dimerization Example 16 The following data have been gathered for ribosomes obtained from a paramecium szoyw 826SD207W 152 X 10 7 cm2 sV2 06lcm3 g We can calculate the molar weight of the ribosome by substituting the diffusion coefficient in place of the frictional coefficient k T B m S 20w j 1 l 2p20w 20w 138 10 23JK 293K 826x10 l3s x 7 2x 2 34x106gmole 1 061x1152x10 cm slmlOOcm careful with this expression to get the final answer you will need to convert from kg to g and use Avogadro number to calculate the weight of a mole Boundary sedimentation In a first method to measure sedimentation coefficients a homogeneous solution is spun in a ultracentrifuge As the macromolecule moves down the centrifugal field a solution solvent boundary is generated We can estimate 3 by following the movement of the boundary with time Bear in mind that by generating a boundary we also generate a concentration gradient and therefore we would expect the molecule to begin diffusing however if the macromolecule is large or the field very large high spinning speed then the boundary will be very sharp because transport by sedimentation will be much larger than transport by diffusion If diffusion is significant then the boundary broadens as it shifts towards the bottom of the cell with time If we assume transport by diffusion can be neglected and rewrite dr 2 M r dt Then we can find a solution ln r0 swz I to V0 to At zero time the concentration is uniform throughout the cell as time increases a sharp boundary is generated with solvent to the left and solute to the right the concentration of solute will be constant on either side of the boundary This equation means that as centrifugation proceeds the solute concentration boundary with position r relative to the spinning axis proceeds from an initial position rto to rt Rearranging the equation for rt we obtain lnrtlnr0t0 sm2t t0 A plot ofln rt versus elapsed time t tg is a straight line With slope soa2 The concentration of solute at the right of the boundary is not the same as the st 39ng concentration uniform Cg It can also be shoWn that cm rm 2 Cu 5 spinning ms bunndarv r meniscus r0 Zone Sedimentation A second Way to measure sedimentation is called zone sedimentation A thin layer ofa macromolecular solution is placed at the top of the solvent at the beginning of the centrifugation le below As L 39 39 progresses the mo e through the solvent as a band or zone right below To prevent mixing of the dense macromolecular band With the solvent a density gradient is created so that the net density increases in the direction of the centri igal eld Typically a linear sucrose concentmtion gradient is used Alternatively one could use a concentrated salt solution eg CsCl in Which case the density gradient Will be generated by the sedimentation of the salt itself spinningxi mnnmgun lnirill mininn u rrmnin Lam Cenlrilllsl nib The macromolecule band can become broadened due to dif ision thus reducing resolution of the macromolecular bands End ofLect39ure 10
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