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, 67-684 (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.0 0. 0.0 0 3 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 0. 0.0 0 3 0.3333 BKEB Lec 8: Optmal degn 6
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, 697-35 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, 697-35, g. 4. lght catterng..0 0.9 0.7 [Hc70] µm lght catterng.5.0.05.00 0.95 0.90 5 0 0.00 0.0 0.04 0.06 0.08 0.0 tme, ec t 0 to 5 ec: tep ze 0 mec 0.5 0 4 6 8 0 tme, ec lght catterng.5.0.05.00 0.95 0.90 5 0 5.00 5.0 5.04 5.06 5.08 5.0 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
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?? (0.0.. 00) 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 0.0000 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, 640-646 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, 697-35, 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
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, 697-35 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, 697-35 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, 697-35 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, 697-35 ACTUAL EXPERIMENTAL DATA: 0.5 0 [Hc70] x aay / experment 90 nm clathrn 00 nm auxln 4 0.5 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?? (0.00.. 00) fle run0 concentraton CA 0., T?? (0.00.. 00) fle run03 concentraton CA 0., T?? (0.00.. 00) fle run04 concentraton CA 0., T?? (0.00.. 00) fle run05 concentraton CA 0., T?? (0.00.. 00) fle run06 concentraton CA 0., T?? (0.00.. 00) fle run07 concentraton CA 0., T?? (0.00.. 00) fle run08 concentraton CA 0., T?? (0.00.. 00) 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
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 30-90 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