Lecture outline. Optimal Experimental Design: Where to find basic information. Theory of D-optimal design

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1 v I N N O V A T I O N L E C T U R E (I N N O l E C) Lecture outlne Bndng and Knetc for Expermental Bologt Lecture 8 Optmal degn of experment The problem: How hould we plan an experment uch we learn the mot from t? Petr Kuzmč, Ph.D. BoKn, Ltd. WATERTOWN, MAACHUETT, U..A. The oluton: Ue the Optmal Degn Theory of tattc An mplementaton: oftware Dynat An example: Knetc of clathrn cage daembly BKEB Lec 8: Optmal degn Optmal Expermental Degn: Where to fnd bac nformaton DOZEN O BOOK Theory of D-optmal degn MAXIMIZE THE DETERMINANT ( D ) O IHER INORMATION MATRIX edorov, V.V. (97) Theory of Optmal Experment edorov, V.V. & Hacl, P. (997) Model-Orented Degn of Experment Atnon, A.C & Donev, A.N. (99) Optmum Expermental Degn Endreny, L., Ed. (98) Degn and Analy of Enzyme and Pharmaconetc Experment y f (, p) f algebrac fttng functon x x ndependent varable, th data pont (,,, N) y dependent varable, th data pont p vector of M model parameter f ( x, p) p, entvty of f wth repect to th parameter, th data pont, N,,,,, (,)th element of the, her nformaton matrx M M, D-Optmal Degn:, L, M, L, M O M M, L M, M M max x, x,, xn det Chooe the ndependent varable x,, x N (e.g., total or ntal concentraton of reagent) uch that the determnant of maxmzed. BKEB Lec 8: Optmal degn 3 BKEB Lec 8: Optmal degn 4 D-Optmal degn example: Mchael-Menten equaton RONALD DUGGLEBY UNIVERITY O QUEENLAND, AUTRALIA (979) J. Theor. Bol. 8, (979) v V + K v, V V + K v ntal rate of enzyme reacton, th data pont ubtrate concentraton, th data pont (,,, N) V, K vector of model parameter K Mchael contant, V maxmum rate v V K ( + ), K K entvty functon Realtc degn for the Mchael-Menten equaton ININITE UBTRATE CONCENTRATION (TO GET V max) I IMPOIBLE TO ACHIEVE Box-Luca two-pont degn wth one pont ( max) already gven: aume K 0.5, max.0 max det max, V, V max, K, K.0 Box-Luca two-pont degn: max det,, V, V K, K, K.0 K 0.5 V BKEB Lec 8: Optmal degn 5 max max K max + K RECIPE In the determnaton of K M, alway nclude a ubtrate concentraton that correpond to a reacton rate approxmately one half of maxmum achevable rate. v BKEB Lec 8: Optmal degn 6

2 Theory of D-optmal degn: The chcen-and-egg problem PROBLEM: TO DEIGN AN EXPERIMENT WE MUT IRT GUE THE INAL ANWER! EQUENTIAL OPTIMAL DEIGN perform a mallet poble experment refne parameter etmate repeat pecal cae: Tme-coure experment wth fxed tme-pont TART HERE gue model parameter (lterature, hunche, ) optmal degn for very few data pont optmal degn theory ue here BKEB Lec 8: Optmal degn 7 BKEB Lec 8: Optmal degn 8 A typcal netc experment: xed meh of tme-pont PROBLEM: WE DO NOT ALWAY HAVE A CHOICE O INDEPENDENT VARIABLE VALUE Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, topped flow expermentaton: xed tep ze AT BET, WE CAN CHANGE THE (IXED) TEP IZE OR A GIVEN INTERVAL Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, , g. 4. lght catterng [Hc70] µm lght catterng tme, ec t 0 to 5 ec: tep ze 0 mec tme, ec lght catterng tme, ec t 5 to 0 ec: tep ze 00 mec BKEB Lec 8: Optmal degn 9 BKEB Lec 8: Optmal degn 0 Varaton of the degn problem: Optmze ntal condton I WE CAN T CHOOE OBERVATION TIME, AT LEAT WE CAN CHOOE INITIAL CONCENTRATION Theory of D-optmal degn: Intal condton n ODE ytem MAXIMIZE THE DETERMINANT ( D ) O IHER INORMATION MATRIX BAIC PRINCIPLE: In netc tude -thendependent varable tme; - ntal concentraton are condered parameter of the model. D-Optmal degn theory concerned wth optmal choce of ndependent varable - n th cae the obervaton tme. Unfortunately, n the real-world we cannot chooe partcular obervaton tme: - uual ntrument are offerng u only a fxed meh of output pont. But we can turn thng around and - treat ntal concentraton a ndependent varable. - Then we can optmze the choce of ntal concentraton, ung the uual formalm of the D-Optmal Degn theory. dc / dt f ( c,) t 0 : c c 0 ntal value problem (frt-order ordnary dfferental equaton) c vector of concentraton vector of rate contant c 0 concentraton at tme zero y g( c t ),r) y expermental gnal at th data pont (tme t ), N ( g( c (, t ), r), p,, c r, concentraton at tme t vector of molar repone and/or offet on gnal ax entvty of f wth repect to th parameter, th data pont p model parameter: vector and r combned D-Optmal Degn: max det c 0 BKEB Lec 8: Optmal degn BKEB Lec 8: Optmal degn

3 Optmze ntal condton: Dynat notaton THE OTWARE TAKE CARE O ALL THE MATH Optmze ntal condton: Algorthm and Dynat ettng THE DIERENTIAL EVOLUTION ALGORITHM REQUIRE PECIAL ETTING ta degn [mechanm] yntax otherwe ued for confdence nterval [data] et concentraton X?? ( ) th value gnored (preent for yntactcal reaon only) lower and upper bound mut be gven ta degn copy thee ettng from one of the [mechanm] dtrbuted example problem [ettng] {DfferentalEvoluton} Populatonzexed 300 populaton not too large MaxmumEvoluton MnmumEvoluton perform the optmzaton only once TetParameterRange TetParameterRangeAll TetParameterRangeull topparameterrange 0. TetCotunctonRange y relatvely wea convergence crtera topcotunctonrange 0.0 TetCotunctonChange y topcotunctonchange TetCotunctonChangeCount 5 BKEB Lec 8: Optmal degn 3 BKEB Lec 8: Optmal degn 4 Clathrn tructure: trelon and cage CLATHRIN CAGE ARE LARGE ENOUGH TO VIIBLE IN MICROCOPY AND LIGHT CATTERING Cae tudy: Knetc of clathrn cage daembly clathrn trelon clathrn cage BKEB Lec 8: Optmal degn 5 BKEB Lec 8: Optmal degn 6 Clathrn bology: Role n endocyto CLATHRIN I INVOLVED IN INTRACELLULAR TRAICKING Eenberg et al. (007) Traffc 8, In vtro netc of clathrn daembly: Expermental data WATCHING CLATHRIN CAGE TO ALL APART BY PERPENDICULAR LIGHT CATTERING Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, , g. 4 µm Hc70 ATP-dependent uncoatng clathrn-coated vecle Clathrn cage (0.09 μm trela) premxed wth 0. μm auxln were mxed wth Hc70 (concentraton n μm hown on graph) and 500 μm ATP, and perpendcular lght catterng wa meaured ung topped-flow BKEB Lec 8: Optmal degn 7 BKEB Lec 8: Optmal degn 8 3

4 In vtro netc of clathrn daembly: Theoretcal model MODEL ELECTION UING THE AKAIKE INORMATION CRITERION (DYNAIT) Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, In vtro netc of clathrn daembly: Dynat notaton THE MOT PLAUIBLE MODEL: THREE TEP EQUENTIAL Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, THREE-TEP EQUENTIAL: DYNAIT INPUT: AUTOMATICALLY GENERATED MATH MODEL: TWO-TEP EQUENTIAL: THREE-TEP CONCERTED: ta ft model AHAHAH? [mechanm] CA + T > CAT CAT > CAD + P : r CAD + T > CADT CADT > CADD + P : r CADD + T > CADDT CADDT > CADDD + P : r CADDD > Prod : d ta ft model AHAH? Etc. In total fve dfferent model were evaluated. BKEB Lec 8: Optmal degn 9 BKEB Lec 8: Optmal degn 0 In vtro netc of clathrn daembly: Preferred mechanm CONCLUION: THREE ATP MOLECULE MUT BE HYDROLYZED BEORE THE CAGE ALL APART Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, In vtro netc of clathrn daembly: Raw data THI WA A VERY EXPENIVE EXPERIMENT TO PERORM Rothne, Kuzmc, et al. (0) Proc. Natl. Acad. c. UA 08, ACTUAL EXPERIMENTAL DATA: [Hc70] x aay / experment 90 nm clathrn 00 nm auxln up to 4 µm Hc70 a lot of materal expenve and/or tme conumng to obtan BKEB Lec 8: Optmal degn BKEB Lec 8: Optmal degn How many aay are actually needed? D-OPTIMAL DEIGN IN DYNAIT ta degn [mechanm] CA + T > CAT CAT > CAD + P CAD + T > CADT CADT > CADD + P CADD + T > CADDT CADDT > CADDD + P CADDD > Prod : r : r : r : d [contant] a 9? r 6.5? d 0.38? Chooe eght ntal concentraton of T uch that the rate contant a, r, d are determned mot precely. [data] fle run0 concentraton CA 0., T?? ( ) fle run0 concentraton CA 0., T?? ( ) fle run03 concentraton CA 0., T?? ( ) fle run04 concentraton CA 0., T?? ( ) fle run05 concentraton CA 0., T?? ( ) fle run06 concentraton CA 0., T?? ( ) fle run07 concentraton CA 0., T?? ( ) fle run08 concentraton CA 0., T?? ( ) Optmal Expermental Degn: Dynat reult URPRIE: WE DID TOO MUCH WORK OR THE INORMATION GAINED IMULATED DATA OPTIMAL EXPERIMENT: D-Optmal ntal concentraton: [T] 0.70 µm, 0.73 µm [T].4 µm,.5 µm,.5 µm [T] 76 µm, 8 µm, 90 µm maxmum feable concentraton upwng phae no longer een Jut three experment would be uffcent for follow-up! One half of the materal compared to the orgnal experment. BKEB Lec 8: Optmal degn 3 BKEB Lec 8: Optmal degn 4 4

5 Optmal Expermental Degn: Dynat reult - dcuon EACH O THE THREE UNIQUE AAY TELL A DIERENT TORY Optmal Expermental Degn n Dynat: ummary NOT A ILVER BULLET! motly ATP aocaton ( a) [mechanm] CA + T > CAT CAT > CAD + P : r CAD + T > CADT CADT > CADD + P : r CADD + T > CADDT CADDT > CADDD + P : r CADDD > Prod : d Ueful for follow-up (verfcaton) experment only - Mechantc model mut be nown already - Parameter etmate mut alo be nown Tae a very long tme to compute - Contraned global optmzaton: Dfferental Evoluton algorthm - Clathrn degn too mnute - Many degn problem tae multple hour of computaton Crtcally depend on aumpton about varance motly daembly ( d) aocaton ( upwng ) no longer vble BKEB Lec 8: Optmal degn 5 - Uually we aume contant varance ( noe ) of the gnal - Mut verfy th by plottng redual agant gnal (not the uual way) BKEB Lec 8: Optmal degn 6 5