Bayesian Designs for Michaelis-Menten kinetics

Transcription

1 Bayeian Deign for ichaeli-enen kineic John ahew and Gilly Allcock Deparen of Saiic Univeriy of Newcale upon Reference ec. on hp://

2 Enzyology any biocheical reacion would, of heir own accord, proceed a a rae ha i far oo low o be of ue. Enzye are naural caaly which grealy increae he rae of reacion. e.g. Subrae Enzye e. g. G6Pae Produc glucoe phophae e.g. glucoe

3 ichaeli-enen equaion For any enzye he rae of reacion i deerined by he ichaeli-enen equaion v ax Here ax i he axiu rae a which ubrae i urned ino produc and i he ichaeli paraeer, he ubrae concenraion a which he rae of reacion i 50% of i axiu. Enzyologi are inereed in he value of hee paraeer, and alo in derived quaniie uch a he pecificiy conan ax /. 3

4 Paraeer Eiaion The enzyologi oberve he value of v, v i, a a erie of ubrae concenraion, i, i,..,n. Paraeer are eiaed by fiing he ichaeli-enen equaion o hee daa Will ar wih he odel v i ax wih ε i a reidual wih zero ean and conan variance. Subanial hiory o fiing hi odel, and alo oe concern over he ue of hi odel Rupper, Creie and Carroll, 989; Nelder, 99; alo Cornih-Bowden 995 i i ε i 4

5 Deign Proble How hould he experiener chooe he ubrae concenraion? Soe work on hi: Currie 98 in Bioeric, alo Duggleby 979 and Endrenyi & Chan 98 in enzyology lieraure Depend on he ai of he experien Will be aued ha he ai i o eiae he paraeer and o do hi wih axial preciion. Will no conider udie where he ai i o differeniae beween differen ype of reacion. 5

6 6 Expeced Inforaion arix For he above odel he expeced inforaion arix i proporional o N N 4 ax 3 ax σ σ We aue ha N obervaion are ade a diinc ubrae concenraion. The nuber of obervaion a i N, where 0,.

7 7 Locally D-opial deign The log of he deerinan of he above can be wrien a he log of : 3 4 where er no involving he deign poin ξ, have been oied Depend on hough no on ax. For wriing y / give he above a y y y y The opial deign ha ½ and y ½ and y, i.e.,. Currie, Duggleby, Endrenyi

8 Bayeian D-Opial deign Find deign by axiiing E prior log deσ - N Specify knowledge abou hrough a prior. Obecive facor ino fnfσf ax f, deign So no need o pecify a prior for ax, only arginal for Convenien o aue prior ha finie uppor on L, U. Thee o be pecified by inveigaor. Soe pariony achieved by caling: wrie U, U wih L / U L < <. Two prior:. unifor over i range. log unifor over i range. 8

9 Opial Bayeian -poin deign A bi of an indulgence, bu analyical progre can be ade here. Deign all give equal weigh o boh poin. Larger concenraion i a infiniy Saller concenraion i a he oluion o E π 0 An approxiae oluion i herefore Eπ, which fi wih locally opial oluion. Alo, Jenen inequaliy how ha in fac Eπ. For prior, i L 0; for prior, L. 9

10 Opial Bayeian deign Search nuerically for opial deign for 3, 4, Ue NAG ofware for quadraure and opiiaion. Search for 0 T, and 0,, where T i u oe large caled concenraion, arbirarily e a 0 eniiviy o choice can be explored 0

11 Opial deign L Unifor on Unifor on log E-5 4.9E-4 3.8E-3.9E-.9E All of hee can be confired o be opial fro he derivaive plo d ξ* Eπ [ r ξ*, ], i if ξ* i opial and only a poin in

12 Alernaive crieria There ay be inere in iply finding deign which are good for eiaing or alernaively ax /. For forer, crierion i o iniie log log f ; ξ, where f.;. denoe he deerinan in he preceding crierion. Locally opiu deign, give /, ; /. For pecificiy raio, ax / opial deign are baed on

13 log log Locally opiu deign ha ae deign poin a for bu differen weigh. / / -/ ax / ½ / ½ -/

14 Opial Deign L Opial deign for var ˆ E-5 4.5E-4 3.6E-3.7E-.4E Deign need greaer weigh a lower concenraion han for D-opial deign. Inuiively reaonable a he relaive iporance of inforaion abou ax i le iporan. 4

15 Why he poin a 0? In all deign found o far, oe weigh ha been given o a poin a he upper lii of he range for he caled ubrae concenraion. Thi give inforaion abou ax : eenial even when inere i focued olely on. Alo, deign apply o all prior on ax,, including hoe wih very pecific prior knowledge abou ax. If here i good prior knowledge abou ax, why he poin a 0? 5

16 6 Anwer i ha crierion N E 4 ax 3 ax logde σ π doe no ake prior inforaion ino accoun in he analyi. To do o require crierion o be odified o: R N E 4 ax 3 ax logde σ π R - being he diperion arix of he prior.

17 7 Prior Preciion arix, R Reaonable o ake he prior for ax and o be independen. ax var 0 var / 0 var / σ U R Opial deign now depend on σ and N. However, wrie R* a: N N N N R N R U / var 0 / var / 0 / * σ λ σ λ σ where σ σ λ i he prior variance of ax in uni of he RS. New crierion i expecaion over prior of log of 3 4 var ~ N N λ λ where ~ i ax caled by i prior SD.

18 0 Prior pecificaion Prior for ax i N σ, σ, {o for ~ i N 0,}. Prior for i eiher he prior of he aociaed unifor din. or iproper, var - 0. Noe ha if iproper prior ued for hen obecive funcion doe no depend on, excep hrough /Nλ, o expecaion i a one-dienional inegral. 8

19 Deign obained for Iproper Prior All have N λ λ

20 A Glipe of oher error odel Rupper e al. 989 dicued conan variance aupion and a weighing/ranforaion approach. Nelder 99 uggeed applicaion of exended quai-likelihood o explore odel ς wih ar y σ µ wih a daa-deerined value for ζ in Nelder exaple a value beween and wa obained We explore he cae ζ and Inforaion arix i σ µ β T diag µ ς i µ β 0

21 Inforaion arix 4 ax 3 ax ax ς ς ς ς ς ς ς ς ς D-Opial deign for ζ and earching uing T 0 give L Unifor on Unifor on log E-5 5.7E-4 7.7E-3.E

22 For ζ deerinan of inforaion arix becoe w z z z Thi i axiied by a wo-poin deign, wih concenraion a 0 and T, equally weighed for any prior

23 Soe Efficiencie Duggleby uggeed equal nuber of obervaion a each of /4, /,,, 4. Wha i efficiency of hi deign? We have a prior for, and i ee reaonable o ue he ean of he prior o copare Bayeian deign wih Duggleby deign. Scaling hi ugge coparing opial deign wih equal o: /4, /,,, 4 Crierion i expe/p where p i no. paraeer and [ ] [ log de ξ* E log de ξ ] E Eπ π Duggleby 3

24 L D-opial Unifor Log-Unifor Aended deign L D-opial Unifor Log-Unifor

25 General reark Opial deign can have few poin Relian on idea ha here i a ingle purpoe behind he udy Uing a prior diribuion increae he nuber of poin in he deign, a a hedge again he uncerainy around he value of he paraeer 5