Statstcs of Radom DNA Aruma Ray Aaro Youg SUNY Geeseo Bomathematcs Group
The Am To obta the epectato ad varaces for the ethalpy chage ΔH, etropy chage ΔS ad the free eergy chage ΔG for a radom -mer of DNA
Backgroud Why do we care? Kowledge of the thermodyamc propertes of DNA s essetal all applcatos of DNA volvg structure-depedet hybrdzato Is a measure of the stablty of DNA Useful for uderstadg bologcal processes. Useful for devsg accurate usage of DNA applcatos such as PCR, mcroarrays, etc.
Backgroud Some thermodyamcs thalpy ΔH s a measure of the "useful" work obtaable from a closed system uder costat pressure. Ths s sometmes referred to as the heat of a reacto. tropy ΔS s a measure of the degree of dsorder a thermodyamc system. Gbbs Free ergy ΔG s defed as ΔH - TΔS. It provdes a measure of the spotaety of a reacto ad stablty of the product. The more egatve the value of ΔG, the more stable the product.
Backgroud Duple DNA s formed due to complemetary H-bods betwee trogeous bases. Stablty of DNA s mataed by teractos betwee pars of cosecutve bases P-ways due to crease etropy as a result of stackg. Ths s the bass of the earest-eghbor model
Nearest Neghbor Model -The Bascs Poeered by Zmm, ad by Toco ad coworkers The total ΔH, ΔS ad ΔG of a strad s equal to the sum of the ΔH, ΔS ad ΔG of the stack pars preset the strad For olgoucleotdes, there are pealtes for dfferet types of eds, symmetry, etc. For detals, see A Ufed Vew of Polymer, Dumbbell ad Olgoucleotde DNA Nearest-Neghbor Thermodyamcs, by Joh Sataluca Jr.
Sequece 5'-3' ΔH kcal/mol ΔS cal/mol.k AA -7.9 -. AC -8.4 -.4 AG -7.8 - AT -7. -0.4 Ufed Olgoucleotde Parameters M NaCl Allaw, H. T. ad Sataluca, J. Jr., 997, Bochemstry 36, 058-0594 CA -8.5 -.7 CC -8-9.9 CG -0.6-7. CT -7.8 - GA -8. -. GC -9.8-4.4 GG -8-9.9 GT -8.4 -.4 TA -7. -.3 TC -8. -. TG -8.5 -.7 TT -7.9 -. Termal GC 0. -.8 Termal AT.3 4. Symmetry Correcto 0 -.4 p wth [.5,.5,.5,.5] -8.75 -.3 p wth [.7,.,.,.] -7.954 -.997 p wth [.,.,.7,.] -7.85-0.5
Nearest Neghbor Calculatos Cosder the 8-mer 3 -T-C-C-G-A-A-G-T-5 5 -A-G-G-C-T-T-C-A-3 ΔH of the total strad s gve by ΔHduple = ΔHAG+ ΔHGG + ΔHGC+ ΔHCT + ΔHTT+ ΔHTC +ΔHCA = -7.8-8.0-9.8-7.8-7.9-8. -8.5kcal/mol = -58.0 kcal/mol
At Preset We ca compute ΔH, ΔS ad ΔG for a sgle strad wth some accuracy usg the N-N model calculatos However May applcatos cocer lbrares of DNA as opposed to a sgle type of strad Computg ΔH, ΔS ad ΔG for each strad the lbrary s costly ad tme-tesve
So I applcatos volvg the use of large umbers of DNA molecules t s desrable to have a dea about the rage of ΔH, ΔS ad ΔG Such calculatos should Oly volve the probablty dstrbuto of the four bases DNA Be possble for ay legth of the molecule, ad ay temperature.
How? We kow the values of ΔH ad ΔS for ay gve stack par. It has bee epermetally show that these values do ot chage wth temperature a rage of above 00 degrees.
How? Let be the radom varable represetg the ΔH for the stack par at the -th posto a -mer of DNA Gve the probablty dstrbuto for the bases DNA, P = [p A, p C, p G, p T ], we ca fd the probablty of fdg ay stack par. Oce we kow ths, we ca calculate
How? Frst we compute the epectato for ΔH For a sgle strad ths ca be obtaed by addg the ΔHs for the stack pars preset. 3 4... 3... [ ]
Summary Ths ca be easly computed usg publshed values of ΔH sce we kow the probablty of fdg each stack par. For a uform dstrbuto of bases, ΔH for a 8 mer of DNA s 8-* = -57.95 kcal/mol. We have a procedure Maple 0, whch takes the dstrbuto of bases ad legth of DNA, ad computes ΔH, ad smlarly ΔS ad ΔG
How? Now we eed to obta the varace for the ΔH of a -mer of DNA For a sgle strad ths ca be obtaed by addg the ΔHs for the stack pars preset Therefore we wsh to obta Var 3 4... Note that the calculatos should volve cosderato of term ad symmetry, but these ca be eglected for our purposes
How? 4 3 4 3......... 4 3 Var 4 3... j j 3... j j
How? The mportat step : If -j>, ad j are depedet. Thus j = j Thus j - j = 0
How? 4 3... j j 3... j j...
How?... Var Var
How? We thus have a formula : Var 3 4... Var We used Maple 0 to compute the varace ΔH usg the values for each stack par as gve by Sataluca The same procedure ca be used to compute the varace of ΔS
How? Also we kow that ΔG=ΔH-TΔS Sce ths s a lear combato, Var G Var H T Var S
Results Probablty Dstrbuto [p A,p C,p G,p T ] ΔH kcal/mol ΔS cal/mol.k ΔG at 30 K kcal/mol p Var p Var p Var [0.5,0.5,0.5,0.5] -57.95 6.598-54.99.375-9.900.45 [0.7,0.,0.,0.] -55.678.47-53.979 6. -7.945.04 [0.,0.,0.7,0.] -54.964 6.35-40.875 43.45 -.93.33 Table : The epectato ad varaces for the ΔH, ΔS ad ΔG at 30 K are show for a radom 8-mer, geerated from the probablty dstrbutos show.
What Now? We are curretly tryg to compute the epectato ad varaces the meltg temperatures of a lbrary of DNA molecules.
Summary We have a procedure Maple 0. Iputs Outputs Probablty Dstrbuto of bases Temperature Legth of DNA strad pectato of ΔG, ΔH ad ΔS Varaces ΔG, ΔH ad ΔS
Ackowledgemets We would lke to ackowledge the support of Athoy Macula, Mathematcs, SUNY Geeseo Wedy Pogozelsk, Chemstry, SUNY Geeseo. Ths research was fuded by grats from the NSF ad the AFOSR.
Questos?