Modelling and pa ame e es ima ion o gene
exp ession and cell g ow h in ba ch cul u es 1
R. Cuba si
Dep . Ma em`a ica Aplicada i Telem`a ica
Uni e si a Poli `ecnica de Ca alunya, Ba celona, Spain
J. L. Co che o, P. Vila and A. Villa e de
Ins i u de Biologia Fonamen al
Uni e si a Au `onoma de Ba celona, Bella e a, Spain
1This wo k was p esen ed a he 3ecm Thi d Eu opean Cong ess o Ma hema ics, held 10-14 July, 2000, Ba celona, Spain,
and i has been suppo ed by CICYT o Spain unde g an No. BIO95-0801, and pa ially by Gene ali a de Ca alunya
unde G an 1996XT-00030, and CUR (CIRIT) unde g an 1995SGHR 00376.
Modelling and pa ame e es ima ion o gene exp ession and
cell g ow h in ba ch cul u es
R. Cuba si1, J. L. Co che o2, P. Vila2and A. Villa e de2
1Dep . Ma em`a ica Aplicada i Telem`a ica, Uni e si a Poli `ecn ica de Ca alunya,
Ba celona, Spain; 2Ins i u de Biologia Fonamen al and Depa amen de Gen`e ica i
Mic obiologia, Uni e si a Au `onoma de Ba celona, Bella e a, Spain
Abs ac
Expe imen al p ocedu e o CI857 s-con olled ecombinan gene exp ession in bac e ial ba ch cul-
u es is ma hema ically modelled, and he co esponding minimum a iance pa ame e s a e es ima ed
om speci ic s a is ical o nume ical me hods, basically by using a global and ecu si e weigh ed leas
squa es p ocedu e unde some cons ain s induced by he model. Mo eo e he nume ical echniques
p oposed in his wo k ac by accumula ion o da a coming om se e al uns o he expe imen ,
so ha mo e accu acy is ob ained in he pa ame e es ima ion. In pa icula , o he p oduc ion
p ocess, an ex a-model pa ame e depending on an indica o ec o is in oduced o each un o
he expe imen in o de o globalize he da a. The analysis o ob ained da a leads o an in eg a ed
model o bo h cell g ow h and gene exp ession, which desc ibes an asymme ic dynamics be ween
cul u e g ow h and p o ein yield, and can se e o p edic he maximal alue o accumula ed p o ein
and he ime equi ed o i o be achie ed a any s age o he p einducing cell g ow h.
CORRESPONDING AUTHOR:
Ra ael Cuba si
Dep . Ma em`a ica Aplicada i Telem`a ica
Uni e si a Poli `ecnica de Ca alunya, Campus No d
Jo di Gi ona, 1-3; E08034-Ba celona; Spain
Phone: 34-3-401-5995, Fax: 34-3-401-5981
E-mail: [email p o ec ed]c.es
1991 MATHEMATICS SUBJECT CLASSIFICATION: 62, 65, 92.
KEY WORDS: ma hema ical modelling, pa ame e es ima ion, cons ained leas squa es
1
1 In oduc ion
One o he mechanisms commonly used o es imula e he exp ession o he ecombinan genes, and
consequen ly, he p oduc ion o he encoded p o eins, is a apid inc ease o empe a u e a which he
cells a e cul u ed. A he pe missi e empe a u e, 28◦C, he e is no ecombinan gene exp ession,
bu when he empe a u e is shi ed o 42◦C, cells s a he syn hesis o he ecombinan p oduc
while hey a e also g owing in he cul u e. This empe a u e-media ed induc ion o gene exp ession,
which is e y con enien o indus ial pu poses, is achie ed by he use o wo kind o con olle s
o he gene exp ession, which a e also in oduced in o he ecombinan cells. They a e a posi i e
egula o , he lambda pLand/o pRp omo e s, and a nega i e egula o , he ep esso CI857, which
is ac i e below 32◦C bu i becomes e icien ly inac i a ed a 42◦C (Villa e de e al., 1993). We ha e
de eloped ma hema ical p ocedu es o analyze he pe o mance o p o ein p oduc ion in cul u es o
ecombinan E. coli submi ed o hea induc ion (Cuba si e al. 1998). Fo his analysis, wo ypes
o ma hema ical p ocedu es a e equi ed. The i s ype is composed o s a is ical and nume ical
me hods o pa ame e and e o es ima ion, i ing cu es, e c. Bu when a unc ion is app oxima ed
om a se o da a, he p oblem o wha kind o unc ions mus be used always a ises. In ou case he
unc ion mus be in e p e ed om a biological iewpoin , and i mus also desc ibe some biological
p ope ies o he expe imen al sys em. Thus, abo e echniques can be co ec ly used only i he
biological sys em has been modeled, and he sys em p ope ies o be quan i ied ha e been ocussed.
This is he o he ma hema ical aspec o he wo k, in ac o be done p e iously o he i s one.
Unde he assump ion ha cell g ow h is no signi ican ly al e ed by he p esence in he cell o he
ecombinan p o ein β-galac osidase, he dynamics o he gene exp ession is modeled in wo s eps, so
ha he cul u e g owing model is combined wi h he gene exp ession model, a i s o de di e en ial
equa ion ha desc ibes he p o ein p oduc ion in e ms o cell g ow h, in o de o explain he ime
e olu ion o ecombinan p o ein yield along he induc ion phase. Then, he se o pa ame e s o
bo h models is es ima ed by using speci ic leas squa es echniques subjec o cons ain s om he
models, wi h s a is ical e alua ion o e o p opaga ion.
In o de o minimize he e o s, he nume ical algo i hms o pa ame e es ima ion ake ad an age o
wo king wi h a ba ch cul u e p ocedu e wi h mul iple induc ion sequences o he cul u e, whe e da a
om di e en induc ion sequences o he same non-induced cul u e a e p ocessed all oge he , as a
single expe imen . Howe e , da a om se e al uns o he same expe imen al p ocess can be pooled
in a global da a se only unde some speci ic equi emen s. Thus, o he cul u e g owing p ocess
his can be done i he ime in e al be ween wo consecu i e cul u e samples emains cons an along
all he p ocess. Fo he gene exp ession p ocess, some pa ame e s a e cons an o all he sequences,
namely he model pa ame e s, and o he s a e sequence dependen . In his case, o each un o he
expe imen , an ex a-model pa ame e depending on an indica o ec o is in oduced, such as an
ini ial condi ion o he p oduc ion p ocess. Hence he o al numbe o es ima ion pa ame e s is
inc eased by he numbe o uns o he expe imen .
The esul ing model o cell g ow h and syn hesis o ecombinan p o eins e eals an asymme ic
dis ibu ion o he biosyn he ic po en ial o he cell, which is mani es ed by a p e e en ial syn hesis o
ecombinan p o eins in aged, slowly g owing cul u es. In o he wo ds, bo h g owing and p oduc ion
capabili ies o cul u e cells a e no equidis ibu ed, since when he cul u e g ow h eloci y dec eases,
he p o ein p oduc ion eloci y s ill inc eases up o i s maximum alue. Mo eo e , he p oposed
model also allows a p edic ion o he op imal op ical densi y o a ba ch cul u e o be empe a u e-
induced in o de o ge a p ede e mined amoun o ecombinan p o ein wi h he minimum induc ion
ime.
2
2 Basic no a ion
The ma hema ical no a ion used in he wo k is now in oduced by desc ibing he expe imen al
p ocedu e. An ini ial amoun o cul u e y0, measu ed om i s op ical densi y a he wa e leng h o
550 nm (OD), is g owing a 28◦C (ini ial s age). A his s age he cul u e g ow h can be desc ibed
in e ms o a ime pa ame e by a unc ion y( ), which is he solu ion o an au onomous i s o de
di e en ial equa ion gene a ed by a phase eloci y ield y ha , as we shall see in he ollowing
sec ion, will depend on wo pa ame e s A0and B0:
dy( )
d = y(y( ), A0, B0) (1)
Hence he solu ion o his equa ion may be explici ly exp essed depending on he pa ame e s, and
he ini ial alue y0=y(0), as
y=y( , y0, A0, B0) (2)
A a ime om he beginning o he expe imen , a sample o cul u e wi h OD y( ) is ans e ed o
a p ewa med ba h a 42◦C (induc ion s age). Then he g ow h a e o he cul u e changes and he
p oduc ion o β-galac osidase p o ein begins. The p oduc ion is measu ed in enzyma ic uni s pe
ml, e e ed as β-gal in his wo k.
The g ow h p ocess unde he induc ion condi ions has a simila beha iou as in he ini ial s age,
bu wi h o he model pa ame e alues, namely A1and B1. A e a ime xin he induc ion s age,
he OD o cul u e y( ), ha had been induced a a ime om he beginning o he expe imen ,
a ies along wha we shall call he -induc ion sequence acco ding o a unc ion y (x), which is he
solu ion o a di e en ial equa ion, simila o Eq. 1, such as
dy (x)
dx = y(y (x), A1, B1) (3)
Hence, he solu ion can be w i en explici ly as a unc ion o he model pa ame e s, and he ini ial
alue y( ) = y (0), in he o m
y =y (x, y( ), A1, B1) (4)
On he o he hand, o he gene exp ession p ocess along he induc ion s age, ha is he ecombinan
p o ein p oduc ion, we assume ha he p o ein has no any oxic e ec nei he o i sel , no o he
cul u e, and depends only on he OD o he g owing cul u e. De ails o expe imen al p ocedu e a e
gi en by Co che o e al. (1994), whe e he unc ional dependence o β-gal p oduc ion in e ms o
he OD o he induced cul u e has been p o ed.
Thus, along he -induc ion sequence, i y (x) is he OD o an induced cul u e, o a gi en induc ion
ime x, we can e alua e he p oduc ion p ocess by means o a unc ion β (x) = β(y (x)), depending
on whe he he ime e olu ion o he cul u e OD dependency o he p oduc is emphasized. Thus
β (x) ep esen s he yield o β-gal o he induc ion ime xin he same induc ion sequence. The
unc ional dependence β(y ) is s udied in he ollowing sec ions om an app oxima ion model gi en
by a i s o de di e en ial equa ion depending also on wo pa ame e s c1and c2:
dβ
dy
=φ(y , c1, c2) (5)
whe e φis an a bi a y unc ion o he speci ied a gumen s, whose solu ion can be w i en o each
-induc ion sequence om an ini ial condi ion c
0, so ha c
0=β(0), in he o m
β=β(y , c
0, c1, c2) (6)
3
No ice ha c
0is a unc ion o y( ), ha can be implici ly gi en by β(y( ), c
0, c1, c2) = 0, since a he
begining o he induc ion sequence he e is no p o ein yield.
The sub-index e e ed o he -induc ion sequence will be omi ed when he con ex p o ides su i-
cien in o ma ion.
Finally, he p oduc ion kine ics, he ime e olu ion o he p o ein p oduc ion, can be s udied by
composi ion o he di e en ial p ocesses exp essed in Eq. 3 and Eq. 5, so ha he co esponding
gene a ing ield is dβ
dx =dβ
dy
dy
dx =φ(y (x), c1, c2) y(y (x), A1, B1) (7)
Then, he uc ion ha desc ibes he p o ein p oduc ion in e ms o he induc ion ime xcan be
exp essed by ecu si e subs i u ion o Eq. 2 and Eq. 4 in Eq. 6.
3 Ma hema ical model
In he wo king condi ions, and o bo h expe imen s ( (a)E42, Table 4, wi h induc ion empe a u e
o cul u e a 42◦C, and (b)E40, Table 5, wi h induc ion empe a u e a 40◦C) he g ow h a e o
cul u e can be sa is ac o y desc ibed, be o e and du ing he induc ion phase, by using he equa ion
o limi ed g ow h o popula ion models (see e.g. Hi sch & Smale, 1974):
dy(x)
dx =y(x)(A+By(x)) (8)
wi h Aand Ba bi a y cons an s (A > 0 and B < 0).
A low OD’s, he g owing a e is nea ly cons an bu , a he same ime as he biomass is inc easing,
he exponen ial g ow h s ops and he OD o he cul u e ends o he asymp o ic alue
l=−A
B(9)
ha depends on he g ow h condi ions.
On he o he hand, acco ding o Co che o e al. (1994), a g owing cul u e which has been induced
o e a ime xp oduces an amoun o β-gal β(x), ha depends nea ly in a quad a ic way on he
biomass y(x) o ha cul u e. Thus, non cons an a e o p oduc ion, wi h e e ence o he cul u e
g ow h, may be e lec ed along he induc ion s age, e en hough oxic e ec s o he ecombinan
p o ein a e excluded om his wo k. Then he ela ionship be ween he amoun o ecombinan
p o ein β-gal and he OD o cul u e can be w i en as ollows,
β(y(x)) = c0+c1y(x) + c2y(x)2(10)
The co esponding di e en ial beha iou , acco ding o Eq. 5, will hen ha e he o m
dβ
dy =c1+ 2c2y(11)
The meaning o his ela ionship, om a biological iewpoin , is now in es iga ed by assuming ha
cell di ision is no in luenced by β-galac osidase p o ein, and ha c1and c2a e pa ame e s o he
model.
Along he induc ion s age he cul u e is g owing acco ding o Eq. 8, wi h a g ow h eloci y ygi en
by
y(y) = y(A+By) (12)
Thus, he unc ion y(y) is a pa abola wi h e ex a y=−A
2B, co esponding o an OD he hal o
he limi alue gi en by Eq. 9, and also co esponding o he maximum g ow h eloci y. Simila ly,
4

o he p oduc ion phase, a i s app oach could assume he same beha iou o he p o eins as o
he cul u e, ha is, du ing he induc ion s age he inc ease o β-gal is p opo ional o he inc ease
o biomass, i non-nega i e. Hence he p oduc ion eloci y o Eq. 7, namely β, could be exp essed
in his simple model as
β(y) = k y(y) (13)
wi h ka posi i e cons an . In ac , da a om Tables 4 and 5 sugges ha he inc easing o β-gal is
always associa ed wi h he inc easing o biomass (Flickinge & Rouse, 1993). Fu he mo e, no ice
ha he condi ion o inc easing biomass leads o a wo king in e al 0 ≤y≤l o he cul u e.
Ne e heless a mo e complex beha iou , consis en wi h Eq. 10, mus be adop ed, since he p oduc-
ion p o ein a e wi h espec o he cul u e g ow h is no cons an . Mo e speci ically, he OD o
cul u e co esponding o he maximum p oduc ion eloci y, namely m, could be di e en om he
OD o cul u e o he maximum g ow h eloci y, y=l/2.
The e o e, he gene al case o Eq. 7 mus be conside ed, acco ding o
β(y) = dβ
dy y(y); 0 < y < l (14)
aking in o accoun ha , i oxici y phenomena a e no p esen in he induc ion s age he ollowing
inequali y mus be sa is ied, in he wo king in e al:
dβ
dy ≥0 (15)
This si ua ion is s udied in he i s o de app oxima ion gi en by Eq. 11, which enable us o explain
in a simple way he asymme y ha he p oduc ion eloci y cu e may ha e wi h espec o he
cul u e g ow h eloci y cu e.
4 Asymme y be ween p oduc ion and g ow h
In o de o compa e bo h eloci y cu es o Eq. 14, he unc ion in he igh hand side membe o
Eq. 11 will be deno ed, acco ding o Eq. 5, as
φ(y) = c1+ 2c2y(16)
Then Eq. 14 becomes
β(y) = φ(y) y(y) (17)
The condi ion exp essed by Eq. 15 implies φ(y)≥0 in he wo king in e al, and hen i is easy o
deduce ha : (a) In any case c1mus be posi i e. (b) I c2= 0 bo h eloci ies a e p opo ional and
hey ha e a common maximum a m=l/2. (c) I c2>0 he maximum p oduc ion eloci y is eached
a e he maximum g ow h eloci y o he cul u e, and m > l/2. (d) I c2<0, since c1≥ −2c2yis
ul illed in he in e al 0 ≤y≤l, and he maximum alue o he igh hand side membe is held a
y=l, hen he ollowing inequali y mus be sa is ied:
c1≥ −2c2l(18)
Thus he maximum p oduc ion eloci y is eached be o e he maximum g ow h eloci y o he cul u e,
and m < l/2 is also held.
The h ee unc ions in ol ed in Eq. 17 a e non-nega i e o alues o he OD wi hin he wo king
in e al, and in i s bounds β(0) and β(l) a e null. Fu he mo e, aking in o accoun he polynomial
o m o β(y), i is easy o see ha he e is a single maximum on his in e al. Thus, by assuming
5
Exp. l/2ǫ
E42 1.38 ±0.05 0
E40 1.71 ±0.03 0.36 ±0.03
Table 1: Pa ame e s desc ibing he asymme y be ween β-gal p oduc ion and cul u e g ow h om Eq. 20.
c26= 0, he ela i e posi ion ǫo he abscissa m e e ed o he alue l/2, co esponding o he
maximum g ow h eloci y o cul u e y(y), is in oduced
m=l/2 + ǫ(19)
Then he abcissa o he maximum can be w i en, in e ms o an auxilia y pa ame e α=l+c1
c2, as
ollows
ǫ=sign(c2)1
|α|+q|α|2+ 3l2
l2
2(20)
Abo e exp ession is also use ul in o de o see how a can he maximum mo es a ound he cen al
alue y=l/2, being consis en wi h he condi ion o Eq. 15. No ice ha |ǫ|is a dec easing mono onic
unc ion o |α|, and om Eq. 18 i is easy o see ha |α| ≥ l. The e o e, om Eq. 20, by subs i u ion
o his minimum alue o |α|, we ge he admissible ange |ǫ| ≤ l
6. Also, aking in o accoun Eq.
19, we can conclude ha ou model enable us o explain a maximum p oduc ion eloci iy in he
ollowing ange o alues 1
3l≤m≤2
3l(21)
No ice ha i ǫ > 0 he age o signi ican p oduc ion is delayed owa ds high alues o OD, while
o low OD’s he p oduc ion would be insigni ican . I ǫ < 0 he beha iou is in he opposi e way.
5 P oduc ion kine ics
In his sec ion, o a gi en -induc ion sequence, we s udy he ime e olu ion o he p oduc con en
β (x) = β(y (x)) ha is p esen in he cul u e a an age xo he induc ion s age. Remembe ha he
cul u e wi h OD y (x) has been induced a a ime om he beginning o he expe imen . Following
he p oposed app oxima ion, acco ding o Eq. 8 and Eq. 11, we can w i e Eq. 7 as ollows
dβ
dx = (c1+ 2c2y (x))(A+By (x))y (x) (22)
whe e c1>0, A > 0 and B < 0. Then, when he induc ion ime x→ ∞, he OD o cul u e ends o
he limi lgi en by Eq. 9. Hence he unc ion β (x) ends o he asymp o ic alue
lim
x→∞ β (x) = β(l) (23)
Tha is, om a su icien la ge in e al o ime, he p oduc concen a ion becomes nea ly s a iona y.
Mo eo e , since he ac o c1+ 2c2y o Eq. 22 is non-nega i e in he wo king in e al 0 ≤y≤l, his
asymp o ic alue is he maximum yield ha can be eached.
Thus he unc ion β (x) does no ha e any maximum be o e eaching hei asymp o ic alue, and,
o any induc ion sequence, he kine ic o he p oduc has a mono onic inc easing cu e along all he
induc ion p ocess.
In o de o ob ain he unc ion β (x) we mus ake in o accoun how he cul u e is g owing be o e
and du ing he induc ion s age, since he unc ion y (x) depends also on he OD o cul u e jus a
he beginning o he -induc ion sequence, y( ) = y (0). Thus we w i e, acco ding o he solu ion o
6
Eq. 8, he ela ionship desc ibing he biomass e olu ion o a cul u e ha has been induced o e a
ime x,
y (x) = A1y( )eA1x
A1+ (1 −eA1x)B1y( )(24)
The sub-index 1 is used o dis inguish he induc ion s age, and he alue y( ) ep esen s he OD o
he cul u e a he beginning o he induc ion sequence, acco ding o
y( ) = A0y0eA0
A0+ (1 −eA0 )B0y0
(25)
In he la e equa ion y0is he ini ial amoun o cul u e a he beginning o he expe imen , and he
sub-index 0 is used o dis inguish he g owing s age be o e he induc ion.
Finally, he p oduc ion o β-gal in e ms o he induc ion ime xis ob ained om Eq. 10, also
combined wi h Eq. 24 and Eq. 25, by assuming ha in he beginning o he -induc ion sequence
he e is no any signi ica i e amoun o p oduc ,
β (x) = c1(y (x)−y( )) + c2(y (x)2−y( )2) (26)
Some imes he o egoing equa ion will be used in o de o desc ibe he ime e olu ion o β-gal, and
some imes he p oduc ion in e ms o OD o he induced cul u e. Some consequences and ea u es
o hese equa ions will be poin ed ou in he las sec ion.
6 Cul u e g ow h pa ame e s
The algo i hm o es ima e he g ow h pa ame e s Aand Bo Eq. 8 o he induc ion s age, as well
as in he ini ial phase, is based on he ime equidis ibu ion o he OD samples, as i is shown in
Table 4 and Table 5 o bo h expe imen s.
By in e ing ha exp ession, a linea dependence be ween he in e se o he OD o wo consecu i e
cul u e samples wi h an a bi a y ime sepa a ion ∆x=x−x0is ob ained,
1
y(x)=e−A(x−x0)1
y(x0)−B
A(1 −e−A(x−x0)) (27)
Hence, by de ining
a=e−A∆x
b=−B
A(1 −e−A∆x)
zk=1
yk
(28)
and main aining he ime in e al ∆xcons an o any couple o consecu i e samples, Eq. 27 can be
w i en in a simple and ecu si e way as ollows:
(ξk, ηk) = (zk−1, zk), ηk=aξk+b;k= 1,...,n−1 (29)
Thus, i all he poin s (ξk, ηk) a e g aphically ep esen ed, a s aigh line is ob ained. This equa ion
will be used in o de o compu e he auxilia y pa ame e s aand b o he cul u e g ow h by means
o a linea leas squa es app oxima ion, and by aking in o accoun he co a iance ma ix o e o s.
The p ocedu e can be b ie ly desc ibed as ollows. The OD measu emen s yka e ob ained wi h
independen e o s ∆yk, wi h ze o means and common a iance σ2
OD. The e o s o zka e e alua ed
acco ding o he linea app oxima ion om Eq. 28, ∆zk≃ −∆yk/y2
k, so ha he accu acy o zkis
gi en by he a iance
V(∆zk) = z4
kσ2
OD (30)
7
expe imen E42 E40
s age ini ial induc ion ini ial induc ion
a0.373 ±0.036 0.189 ±0.046 0.725 ±0.024 0.525 ±0.011
b0.279 ±0.029 0.295 ±0.024 0.089 ±0.046 0.139 ±0.005
co (a, b) -0.0009 -0.0011 -0.0011 -0.00005
A0.987 ±0.095 1.665 ±0.244 0.643 ±0.066 1.290 ±0.042
B-0.438 ±0.062 -0.605 ±0.103 -0.208 ±0.1114 -0.377 ±0.017
co (A, B) -0.0057 -0.0249 -0.0071 -0.0007
sOD 0.09 0.25 0.03 0.13
Table 2: Cul u e g ow h pa ame e s, es ima ed e o s and co a iances o he es ima es, om Eq. 38 and Eq. 8, wi h
weigh ed RMS e o sOD o cul u e OD om Eq. 42. The pa ame e s aand ba e ela ed o he s aigh lines o
Figu es 1 and 2.
Thus Eq. 29, a sys em o n−1 equa ions may be explici ly w i en wi h he co esponding e o s as
ollows
zk=a zk−1+b+δk;δk=a∆zk−1−∆zk;k= 1,...,n−1 (31)
Hence he e o associa ed wi h each equa ion depends on he unknown pa ame e aand, aking in o
accoun Eq. 30 and Eq. 31, he e o s o Eq. 31 sys em a e co ela ed acco ding o he ollowing
co a iance ma ix o e o ec o ~
δ,
E(~
δ~
δ ) = σ2
ODVz=σ2
OD









z4
1+a2z4
0−az4
10... 0
−az4
1z4
2+a2z4
1−az4
2... .
.
.
0−az4
2... 0
.
.
....−az4
n−2
0... 0−az4
n−2z4
n−1+a2z4
n−2









(32)
whe e Edeno es he expec a ion and he symbol means anspose.
I is well known ha he alue σ2
OD is no necessa y in o de o es ima e he pa ame e s aand b.
Howe e , since he ma ix Vzdepends on a, he es ima ion mus be done i e a i ely, by e alua ing
Vzin each s ep, and by assuming he ini ial co a iance ma ix as he iden i y ma ix. On he o he
hand, σ2
OD mus be known in o de o e alua e he co a iance ma ix o es ima o s V(a,b), hen an
unbiased es ima o s2
OD o σ2
OD (S ua & O d, 1991, pp.723 and 737) is gi en by he ollowing inne
p oduc , whe e ~η is he ec o o expe imen al alues, and ~η ∗is he p edic ed alues ec o ,
s2
OD =1
n−3(~η −a~η ∗) V−1
z(~η −a~η ∗) (33)
No e ha his is equi alen o e alua e a weigh ed oo mean squa e (RMS) e o wi h espec o he
in e ed co a iance ma ix, o e he numbe o deg ees o eedom, n−3, o he p oblem. Finally, he
co a iance ma ix o pa ame e s Aand B,V(A,B), is ob ained by e o p opaga ion app oxima ion
om he jacobian ma ix J, and he co a iance ma ix V(a,b)(see e.g. Ba low, 1989)
V(A,B)=J V(a,b)J ;J=∂(A, B)
∂(a, b)(34)
The es ima es and co esponding e o s o aand b, as well as o he pa ame e s Aand B,
a e lis ed in Table 2. Also he weigh ed RMS e o s2
OD is gi en o OD es ima ions. No e ha
leas squa es app oxima ion p o ides us wi h unbiased es ima o s o he pa ame e s and o sampling
a iances and co a iances o he es ima o s, wi hou assump ions conce ning he o ms o he e o
dis ibu ion (S ua & O d, 1991, p.716). This is only necessa y when es ing hypo heses abou he
8