405
Clinical Science (1982) 63,405-414
EDITORIAL REVIEW
Kinetic analysis of transport processes in the intestine and
other tissues
M . L. G. G A R D N E R A N D G . L. A T K I N S
Department of Biochemistry, University of Edinburgh Medical School, Edinburgh, Scotland, U.K.
Introduction: aims and scope of kinetic analysis
Kinetic analysis, similar to that used in enzymology, is often applied to membrane and
epithelial transport processes. Essentially, it
entails the derivation of an equation relating the
transport rate of a solute to the solute concentration or to the concentration of a second
substrate or an inhibitor. Thus one can characterize the transport process mathematically (i.e.
create a mathematical model) and then mechanistically (i.e. create a physiological model).
Kinetic analysis can help to answer questions
such as: (i) is there evidence for carriermediation? (ii) is more than one carrier involved?
(iii) is the carrier@) shared by another substrate?
It can help to explore the nature of interactions
between transport of two solutes, and another
application is to correct for extracellular space
without the use of a marker. Where two transport
processes are to be compared (e.g. substrate 1 vs
substrate 2; healthy vs pathological; experimental
vs control) it is generally meaningless to make the
comparison at a single concentration; a full
kinetic analysis is much more useful. If solute
transport is affected in particular conditions, it
may be possible to attribute this to a change in
the number of carrier sites or in their affinity for
substrate-binding. Similar questions are asked as
to how transport varies between different regions
of the intestine.
Kinetic analysis has been applied to many
tissues, including erythrocytes, yeasts, bacteria,
kidney, lung, placenta, salivary gland, brain,
intestine etc. The present review is mainly
Key words: intestinal transport, kinetic analysis,
kinetics, transport.
Correspondence: Dr M. L. G. Gardner, Department of Biochemistry, University of Edinburgh
Medical School, Hugh Robson Building, George
Square, Edinburgh EH8 9XD, Scotland, U.K.
0143-5221/82/110405-10%01.50
concerned with transport across the intestine, a
particularly complex example in view of the
heterogeneity of the cell population, the polar
(asymmetric) nature of the cells, the presence of
intercellular (shunt) pathways and the villous
morphology. However, the general principles
apply to other systems, and those aspects
concerned with curve-fitting and assessment of
goodness of fit are also relevant to enzyme
kinetics, from which field many of our examples
are taken.
The three equations commonly used to characterize intestinal transport are [ 1I :
Michaelis-Menten equation :
Michaelis-Menten equation plus a linear term:
Double Michaelis-Menten equation:
where v is the rate of transport, [Sl is the solute
concentration and V,,,,, K,, k, etc. are parameters to be estimated.
Eqn. (l), widely applied to many transport
processes [21 as well as to enzymic reactions, was
applied to intestinal transport first by Fisher &
Parsons [31; it reflects a reversible binding of
substrate molecules to a site on a membrane
carrier. Several subsequent authors found that a
@ 1982 T h e Biochemical Society and the Medical Research Society
406
M . L. G. Gardner and G. L. Atkins
second, non-inhibitable and non-saturable, component was apparently present; addition of a
linear term in [S] was more appropriate for their
data [4, 51. Eqn. (2) is also applicable where
permeation of solute into the extracellular space
has not been corrected for by use of a marker [e.g.
61. In other instances, eqn. (3) appears to be more
appropriate, and this is often interpreted in terms
of two independent carriers operating in parallel
[7, 81. In a survey of published data on intestinal
absorption, we noted that eqn. (1) was often more
appropriate than eqn. (2) or eqn. (3), although the
fit was often rather poor; however, there were
clear exceptions [ 11. Inhibition equations analogous to those used in enzyme kinetics are also
often used to characterize interactions between
substrates that may compete for common carrier
sites [21.
Originally, graphical analysis was used [2], but
better methods, including non-linear regression
and non-parametric methods, have now become
available, and the linear forms of the MichaelisMenten equation have now become somewhat
discredited [91. Aspects of kinetic analysis that
need special consideration are: (i) methods for
curve-fitting and parameter estimation, (ii) assessment of goodness of fit and selection of “the
best-fitting model”, and (iii) the design of experiments that will facilitate (i) and (ii). We cannot be
comprehensive in a brief review, but we aim to
illustrate at an elementary level the scope of
kinetic analysis, to draw attention to some
procedures that are not as widely adopted as they
perhaps should be, and to emphasize some of the
pitfalls. We do not discuss experimental methods,
but it must be remembered that good data
analysis cannot compensate for poor experimental design or execution.
Competition and inhibition studies
Studies on the mutual inhibition by two substrates of each other’s transport provide the main
approach to test whether both substrates share a
common carrier site and, if so, whether a second
transport mechanism operates for one or both
substrates.
If two substrates share a common carrier site,
then each should behave as a competitive
inhibitor of the transport of the other. Further, if
only one carrier is involved, the inhibition
qonstant, Ki, for a substrate acting as an inhibitor
of transport of another substrate should be the
same as the K, for its own transport [2, 101. If
K, # K,, then a second transport mechanism is
probably present.
For a one-carrier system with inhibitor (I)
present, the simple Michaelis-Menten equation is
modified to:
Graphical methods can be applied to this
equation (see, e.g. [2]), but they suffer from the
same disadvantages as other linear transformations (see below). A considerable improvement is provided by non-linear regression, which
can fit the equation directly to sets of observations of u, [S] and [I]. A non-parametric
method is also available, although its results are
not as reliable (i.e. as precise or accurate) as
those obtained by non-linear least squares [ 1 1I.
Two approaches to test whether two substrates
share a single common carrier are provided by
the Preston-Schaeffer-Curran plot [ 12; see also
131 and the ‘Inui constant-ratio test’ [ 10, p. 1411.
Where there is doubt whether the behaviour of
a competing or inhibiting substrate obeys
rigorously the criteria for competitive inhibition,
the common carrier-site hypothesis must be
rejected as there are alternative causes of mutual
inhibition (see below).
Unstirred water layers
Probably the greatest single obstacle to valid
experiments arises from the ‘unstirred layer’.
Whenever cells are exposed to an incubation or
perfusion medium there is a relatively stagnant
layer of fluid adjacent to the cell surface.
Inevitably, the absorption of solutes leads to the
solute concentration in this ‘microclimate’ being
lower than that in the bulk fluid. The consequences of this are crucial, since the composition of the fluid in contact with the membrane is unknown. The effective thickness of this
layer can be decreased, but not eliminated,
experimentally by shaking or stirring in incubation experiments 114-171 or by a ‘segmented flow’ in perfusion experiments [ 18-191.
Unless one of these technical manipulations is
adopted or allowance is made mathematically for
the unstirred layer, kinetic analysis may be
seriously invalidated. Diffusion through the unstirred layer may even become rate-limiting, in
which case the kinetics of transport will not give
any information about the membrane transport
step. Although this phenomenom is widely
recognized, failure to take it into account degrades the reliability of much work. The unstirred
layer may well be responsible for the disquieting
variation in kinetic parameters reported between
407
Kinetic analysis of transport processes
laboratories and for the conflicting data on the
variation in K, and V,,,,,. between different
regions of intestine. The classic paper by Dainty
& House on unstirred layers in frog skin is of
interest 1201; Levin gives an excellent brief
account with references relating to intestine 1211.
The simple analysis by Fisher (221 has been
considerably extended, notably by Dietschy and
co-workers 123-251 and Winne 126-301, who
have derived complex equations that are,
nevertheless, approximations. It should be noted
that the effective thickness of an unstirred layer
depends not only on the physical properties and
velocities of the solution but also on the diffusion
coefficient of the solute [201. The situation in the
intestine is particularly complex, owing to the
convolutions (hence the unstirred layer cannot be
uniform [301) and also to the presence of mucus,
which affects the effective ‘resistance’ of the
unstirred layer. Because the basic MichaelisMenten equation is invalid in the presence of a
significant unstirred layer, it follows that all the
linearized versions of it are also invalid.
Model or curve fitting
Initial inspection of data
It is helpful to plot and inspect the raw data,
although this is very unlikely to give any clue as
to which equation is best. A hyperbola flattens
out only at very high concentrations, many times
the K , value. Thus at [Sl = 5 x K, the rate is
still only 83% of V,,,.. Hence, if a curve does not
flatten out, it is imprudent to presume that a
linear component is necessarily present. Conversely, if it does flatten out unduly, this may suggest
that substrate inhibition is present or that some
other model is appropriate. Also, deviations from
linearity become apparent only at concentrations
equal to several times the K,; hence apparent
linearity or lack of saturation is poor evidence for
a passive diffusive process. If there is substrate
inhibition at high concentrations, the analysis
must take it into account.
It is valuable next to examine various linear
plots. Hofstee [311 and Walter [321 advocated a
plot of v vs v / [ S ] to distinguish between a single
and a double Michaelis-Menten equation; Childs
8z Bardsley [33] argued that all three linear plots
(l/v vs l/[Sl, u vs ul[Sl and [Sllu vs [Sl) should
be used to test for departure from the MichaelisMenten equation because they exaggerate different
ranges of the data. However, although nonlinearity is good evidence for departure from
Michaelis-type kinetics, the converse is not true
[ l , 10, 341. Even if the plot is biphasic or curved,
nothing can be deduced as to what model would
be better. For example, an unstirred water layer
can sometimes cause these plots to be curved [26,
35-371. It should also be noted that simple
extrapolation of a biphasic plot is not a valid
method for evaluation of the parameters in a
double Michaelis-Menten equation [ 10,381.
Linear regression
Straight or curved lines can be fitted ‘by eye’,
but this is always subjective and bias is likely to
be introduced (see, e.g., [391). Thus the ‘best’ line
must be obtained by calculation to avoid subjective bias; normally a least-squares method is
used, i.e. a sum of squares of residuals (SSR):
SSR = l’(yobs.- ycaJ is minimized, where yobs.
is the dependent variable and y,,,,. is its calculated value. Alternatively, a non-parametric
(distribution-free) method is available [40-421.
Some equations can be transformed into a linear
form; linear regression should then be used. Since
real data contain error, the regressions must be
correctly weighted, otherwise the linear transformations can introduce serious bias into the
parameter estimates. The best-known example is
the notorious Lineweaver-Burk plot for the
Michaelis-Menten equation. Unless correctly
weighted it gives inaccurate estimates for K, and
V,,,. because the linear transformation itself can
introduce serious bias i9’43-461.
More complex equations have been fitted by
various ‘replot’, ‘peeling’ and ‘alternating’
methods, all of which require straight lines to be
fitted. Thus linear regression is necessary for
these, and it is essential that the regressions be
correctly weighted-a point frequently ignored.
Examples (although not weighted!), with explanations of the methods, are: (i) competitive
inhibition, eqn. (4) (replot method [471), (ii)
Michaelis-Menten equation plus linear term, eqn.
(2) (peeling method [2]), and (iii) double
Michaelis-Menten equation, eqn. (3) (alternating
method [481).
Non-linear regression
Simple and attractive as the above methods
based on linear regression seem, it has been
shown that more reliable values for most models
are obtained by non-linear regression, where all
the parameters are estimated simultaneously [9,
45, 46, 49, 501. Cornish-Bowden 191 goes so far
as to state that ‘for definitive work it is unwise to
use any plot, linear or otherwise, for estimating
the parameters .. .’ and ‘.. the continued
widespread use of the Lineweaver-Burk plot is
.
408
M. L. G. Gardner and G. L. Atkins
evidence of the laziness of the majority who
cannot be bothered to discover the most basic
information about data analysis’ ! Neverthless,
linear plots are useful for obtaining initial
estimates of parameters. Also, they may show
departures from linearity that should cause one to
reject forthwith the model in question, although
the converse, an apparently straight line, is not
strong evidence in support of a model.
The general strategy for non-linear regression
[511 requires initial estimates for the parameters
from which an initial sum of squares of residuals
is calculated. A second set of parameters yielding
a smaller sum of squares is calculated, and these
values are used in a second cycle of calculations;
the whole process is repeated until the sum of
squares is a minimum and there is no further
change in the parameters. Many methods are
available for these iterations [521. The simplest in
concept are the search methods in which many
sets of parameter values are searched systematically in order to find the set giving the
minimum sum of squares. The ‘Simplex’ method
[531 is the one most widely used. There are also
many methods based on a Taylor expansion of
the function in terms of both the initial estimates
of the parameters and the first partial derivatives
of the function with respect to the parameters.
Wilkinson [431 and Johansen & Lumry [541
successfully used this approach for the
Michaelis-Menten equation, and Cleland 155,561
extended it to other equations used in enzyme
kinetics. Alternatively, there are many gradient
methods in which a decrease in the sum of
squares is sought along a specified direction (as
opposed to the search methods where the search
is less directed). Most popular of these is the one
of Davidson 1571, as modified by Fletcher &
Powell 1581. The method- of Marquardt [59],
now becoming the most frequently used one,
combines the principles of both the Taylor
expansion and the gradient methods. Note that
no one method is of universal application and,
when complex models are being fitted, it may be
necessary to try several methods in order to find
a satisfactory one [ 601. Non-linear regression is
versatile in that it can be used for a wide variety
of complex functions; several methods are particularly easy to adapt for different equations,
including integrated rate equations.
Error structure and weighting
Two fundamental requirements for all leastsquares methods are (i) the data error must be
normally distributed and (ii) the variance at each
data point must be known and the regression
weighted accordingly, i.e. each point should be
weighted by the reciprocal of its variance. If the
regression is performed with incorrect weighting,
as often is the case, then this introduces bias into
the values estimated. Cornish-Bowden [9, p. 18 11
gives an example where a Michaelis-Menten
equation was weighted (i) assuming constant
variance and (ii) assuming variance proportional
to u. The two fitted curves look almost identical,
but V,,,,,, and K , differed by 28% and 43%
respectively. Surprisingly little is known about the
error structure (i.e. the distribution of the error
and how it varies as a function of the dependent
variable) of intestinal transport data, although the
situation is slightly better for enzyme kinetics
[61-641 and immunoassays t651.
Another problem arises when a data point is an
‘outlier’, i.e. produces a residual that is significantly larger than its fellows. Least-squares
methods are sensitive to outliers because they
make a large contribution to the sum of squares
of residuals. It is tempting to reject such values,
but, since occasional outliers are present in a
normal distribution, this must be resisted unless
they are definitely caused by experimental
mistake.
Robust methods
There is increasing interest in ‘robust’ methods
for curve fitting, i.e. ones that are tolerant of
unknown error structures and the presence of
outliers. The best-known example is the nonparametric method of Eisenthal & CornishBowden for the Michaelis-Menten equation [ 66,
671. A non-parametric method is also useful for
fitting a straight line [401, especially where both
variables are subject to error [421. Although the
same approach can be extended to some more
complex models, it is not always successful [ 1 11.
Other robust methods include M-estimation [ 68,
421, the jackknife [42, 691 and bi-weight regression -170, 711; these are based on leastsquares methods. It is claimed that they may
protect against outliers or wrong weighting, but
more work is needed before any of them can be
recommended.
ConJidencelimits of estimated parameters
All programs for least-squares methods should
be able to calculate a variance-covariance
matrix. Provided that the errors are normally
distributed and that the sum-of-squares contours
are elliptical near the minimum, then approximate
standard errors for each of the parameters can be
calculated from this matrix (see, e.g., 1551).
However, if these assumptions are invalid, the
Kinetic analysis of transport processes
409
standard errors are seriously in error (see, e.g.,
[721). Therefore these standard errors give only a
rough guide to the precision of parameters and
should not be used in statistical tests. The only
reliable way to obtain confidence limits is to
perform replicate experiments each giving an
estimate of the parameters. Then confidence
limits can be calculated, preferably by a nonparametric method so as to be independent of
outliers or a non-normal distribution of the
parameters [ 731.
tests for absolute goodness of fit. However, it is
better if the models can be compared directly.
There are three methods available. (i) For many
years the only method used was an F-test to
compare the residual variances of the two fitted
models (see, e.g., [77, 801). However, since
F-tests are not very powerful, it is often difficult
to distinguish between two models that are close
fits. (ii) A new statistic, the ‘Akaike information
theoretic criterion’ (‘AIC’), has been introduced
[81]. For normally distributed data this is given
by :
Testing the adequacy of a model
AIC = constant + residual variance
of parameters)
Testsfor goodness off7t
There are many simple qualitative tests available, and it is surprising that few people use them
routinely. It is valuable first to plot the data and
the fitted curve in order to visualize the goodness
of fit [381. Note should be taken of how well the
computer program for non-linear regression has
converged, whether the standard error calculated
for each parameter is unreasonably large
(although caution is .necessary; see above) and
whether the values of the parameters are algebraically and biologically plausible (see, e.g., [ 1,
74, 751). Useful information can be obtained by
plotting the residuals against several different
variables in order to inspect their distribution [51,
76, 771. Re-fitting the equation with selective
omission of ranges of data may be helpful [781.
Quantitative tests, which calculate a statistic, can
also be performed. Simplest of these is the Runs
test [51]. Tukey shows how to analyse quantitatively the plots of residuals, but, since his
statistics are difficult to interpret [741, they have
seldom been used. Two forms of variance-ratio
test (F-test) have been used, although such tests
are acknowledged not to be very powerful. The
first compares the variance of the residuals with
either the sampling variance or the expected
variance of the data (see, e.g., [511), and the
second compares the residual sum of squares
with the sum of squares due to regression [791. It
has also been suggested that a coefficient of
correlation between the observed and calculated
values may be useful [791. However, these last
two tests have rarely been used, and they need to
be explored further. Since individual tests can
give conflicting results, one should use several
tests together, especially if any doubt remains 1,
741.
+ 2 (number
and whichever model gives the lower AIC is the
better fit. Although often used in the physical
sciences, this test has been used only occasionally
in biology and then not always correctly 182,831.
(iii) The third test calculates for each model the
ratio of the residual sum of squares to the total
variability, i.e. 2 = SSR/SST where SST =
(yobs.- j)’ and j = mean of yobs..The model
giving the lower value for Z is the better fit [79,
84,851.
Design of experiments
Experimental design should include the deliberate
selection of the particular concentrations of
substrates, inhibitors etc. that best enable one
either to discriminate between two rival models
(‘discrimination designs’) or to estimate a
parameter within given precision with the
minimum number of experiments (‘estimation
designs’). These two types of design are normally
quite different. See reviews by Cleland [551 and
Mannervik [751.
Discrimination designs
Cleland [ 551 discussed both discrimination and
estimation designs, but his recommendations are
based on subjective ideas rather than on
theoretical work. Federov & Pazman [861 and
Bartfai & Mannervik [871 have proposed two
different functions to aid in the design of
experiments for distinguishing between two rival
models. Markus & Plesser compared them and,
for their chosen progress curve, found the former
function to be the better [881.
Estimation designs
Comparison between models
The fit of two models can be compared by the
information obtained from the several individual
The most common method minimizes the
determinant of the variance-covariance matrix
(‘D-optimization criterion’). The method shares
410
M . L. G. Gardner and G. L . Atkins
the assumptions and limitations of least squares
and, for non-linear applications, is dependent on
prior estimates of the parameters. Thus it
provides guidelines rather than rules for experimental design. The most efficient designs require,
for n parameters, that replicate measurements are
made at n sets of concentrations. Good examples,
both from enzyme kinetics, are given by Duggleby [891 and Endrenyi [901. Duggleby also
shows how ‘D-optimization’ can be used with
non-parametric methods and how to calculate the
minimum number of replicates required [89l.
Software available and implementation
Non-linear regression is widely used in enzyme
kinetics, and its advantages are now being
exploited in a number of laboratories engaged in
transport studies (see, e.g., [ l , 78, 91, 921). There
should be no difficulty in obtaining programs in
the major computer languages: many, such as
Cleland‘s package [471 and NONLIN [931, are
available from their authors. Alternatively, most
local computing services should be able to
provide programs and advice. Packages distributed widely include the Genstat programs
(Rothamsted Experimental Station, Statistics
Department, Harpenden, Herts., U.K.), NAG
library (Numerical Algorithms Group, 7 Banbury Road, Oxford, U.K.), the Harwell SubRoutine Library [941 and the BMDP package
[951. Some micro-computers have large enough
memories for non-linear regression, and two
suitable programs in BASIC have been published
[96,971. The simplest programs, including one for
Wilkinson’s method [431 and others for weighted
linear regression and various statistical tests, can
readily be run on a programmable calculator.
Thus there is no reason why these techniques
should be beyond the reach of any laboratory.
Those without any computing facilities may be
able to arrange collaborative studies.
Limitations of kinetic analysis
Although kinetic analysis can help to resolve the
questions outlined in the Introduction, its pitfalls
and limitations must not be overlooked. Whereas
these do not necessarily negate the value of
kinetic analysis, they do emphasize that kinetic
parameters must be interpreted circumspectly.
Whenever possible this approach should be used
as a complement to other methods rather than
relied upon solely. It must be recognized that
some plausible models of transport are probably
not amenable to kinetic analysis. For example, if
amino acids were transported by many multiple
carriers with overlapping specificities, this
would not necessarily be deducible from kinetic
data.
Unstirred layers
The unstirred layer, if neglected or accorded
only lip-service, can seriously bias results (see
above).
Non-uniqueness of models
A major weakness is that two or more quite
different models often appear to fit well to the
same data: this emphasizes the need for rigorous
tests for goodness of fit. We previously discussed
an instance where a single Michaelis-Menten, a
Michaelis-Menten plus linear term and a double
Michaelis-Menten equation could give curves so
similar that it would probably be impossible to
distinguish between the three models even with
very precise data [381, and we have often seen
this phenomenon in data from various
laboratories. Also, Paterson, Sepulveda & Smith
initially found all three models to be good fits for
their data on amino acid absorption, and warned
against the use of kinetic analysis to support one
or other of the models [781. However, they then
examined the goodness of fit more objectively
and became able to reject two of the potential
models. It must be stressed that, if ‘the wrong’
model is fitted, the parameters obtained bear no
relation to ‘the true’ ones: hence the consequences of fitting a ‘wrong’ model are serious.
Initial or steady-state rates?
Strictly, the Michaelis-Menten equation and
its variants apply only to initial rates of transport,
corresponding to unidirectional flux [2, 10, 981.
In incubation experiments, many workers test the
validity of this assumption by comparing parameters obtained from experiments of, e.g., 1 min
and 2 min duration (see, e.g., [61). Perfusion
experiments yield steady-state rates, the full
kinetic analysis of which is theoretically very
complex.
Interpretation of interactions
Although it may appear that one substrate
competitively inhibits absorption of another
substrate, caution must be exercised before this is
interpreted as evidence for binding at a common
carrier site. Thus substrates may compete for
energy supplies for their, transport; a substrate
can alter the transmembrane potential, which in
Kinetic analysis of transport processes
turn affects the transport of all other charged
substrates [e.g. 991; likewise, a substrate may
alter the microclimate pH, which in turn affects
transport of other dissociable or associable
species; alterations in fluid movement caused by
one substrate may in turn affect transport of
other substrates (see, e.g., [ 1001); interactions of
an allosteric nature may occur and, then, the
whole membrane should perhaps be regarded as a
multi-carrier complex. Allosteric interactions are
one explanation of ‘mixed-inhibition’ kinetics
(see, e.g., [ 101-1031), but see reference [991.
Hence, before competitive inhibition is claimed,
membrane-potential and water-flux effects must
be eliminated; then kinetic analysis must use
strict criteria to prove that the V,,,, is unaffected
by the inhibitor.
Interpretation of kinetic parameters
Kinetic analysis is a very indirect method, and
transport is a complex multi-stage process that
may involve binding and translocation and at
least two membranes plus a paracellular route:
thus considerable caution is needed in attributing
physical interpretation to kinetic parameters. In
particular, K , must not automatically be regarded as an inverse affinity constant for carriersubstrate binding; this approximation would be
valid only in the special case (which cannot
readily be identified) where there is a single rate
constant for translocation and it is negligible
relative to that for binding. Hence Riggs argued
that K , should be regarded as no more than ‘a
constant of convenience’ [ 1041. Likewise, k, must
not be interpreted as a diffusion constant:
Christensen & Liang provide an example where
k, showed structure specificity, a high Q,, and a
pH-dependence, properties inconsistent with diffusion [ 1051.
Heterogeneity of tissue
A further problem arises because the absorbing
cells of the intestine themselves constitute a
heterogeneous population. Cells at different
heights on a villus are at different stages of
maturity; also, there clearly are functional differences between regions of intestine. Thus,
though transport kinetics at a particular locus
may conform to one of the above equations, it
does not follow that the overall process will do so.
Furthermore, if replicate observations are made
on different animals, the mean data will not
necessarily be fitted well by the model that would
be best for the data from each individual animal,
since the actual values for K , etc. are likely to
41 1
differ between animals. The difficulty where the
values for K,,
k, etc. are members of a
variable population is generally ignored (but see
reference [61).
Vmax..,
Practical aspects
Studies in which both mucosal and serosal
surfaces are exposed to a single incubation
medium pose a special problem, since these two
membranes are functionally different, and the
assumption that access to the serosal membrane
is negligible during a brief incubation is not
necessarily proven. Further, overall uptake of
solute into the tissue (‘accumulation’) includes
entry into both cells and extracellular spaces;
correction for the latter can be included in the
kinetic analysis (see, e.g., 16, 1061) (in which case
information about any linear component of entry
into the cells is lost) or can rely on extracellular
markers provided that their use is properly
validated. It must also be noted that experimental manipulations producing cut cells or
temporary hypoxia can affect transport activities
in vitro. The kinetic parameters reported for
particular absorption processes vary widely
among laboratories, and the differences are often
so large that one must question whether the
kinetics reflect the characteristics of a physiological process rather than those of the experimental system. Some variability can be attributed
to differences in technique; for example, workers
in one laboratory observed changes in kinetic
parameters for peptide uptake concomitant with
a change in the size and shape of the incubation
vessels [ 1071.
Conclusions
The conclusions are as follows.
1. Kinetic analysis is a valuable tool for
characterization of transport mechanisms and
especially for detecting heterogeneity in transport systems. However, great caution must be
exercised in extrapolation from mathematical
models to physiological mechanisms.
2. Non-linear regression, both by least-squares
and non-parametric methods, is available for
fitting the equations typically encountered in
kinetic analysis. These are versatile and greatly
preferable to linearized plots fitted by leastsquares, and more use should be made of them.
3. Parametric methods, including least-squares
ones, require correct data-weighting based on the
known error structure of the data.
4. Curve-fitting should always be accompanied
by objective tests for goodness of fit. The
M . L. G. Gardner and G. L. Atkins
412
possibility that alternative models may fit the
data equally well should be seriously considered,
and conformity to classical Michaelis-Menten
kinetics must not be assumed.
5. Where possible, experimental designs should
be chosen carefully (and modified on the basis of
trial experiments if necessary) to obtain precise
and accurate parameters and to discriminate
between alternative models.
6. Unstirred water layers can invalidate kinetic
analysis unless they are minimized experimentally
or allowed for mathematically.
1161 DUGAS,M.C., RAMASWAMY,
K. & CRANE,R.K. (1975) An
analysis of the D-glucose influx kinetics of in vitro hamster
jejunum, based on considerations of the mass-transfer
coefficient. Biochimica et Eiophysica Acra, 382,576-589.
1171 L‘HERMINIER,M. & ALVARADO.F. (1981) Virtual
elimination of the interference of unstirred water layers on
intestinal sugar transport kinetics by use of the tissue
accumulation method at appropriate shaking rates. PJugers
Archiv, European Journalof Physiology, 389,155-158.
1181 FISHER, R.B. & GARDNER.M.L.G. (1974) A kinetic
approach to the study of absorption of solutes by isolated
perfused small intestine. Journal of Physiology. 241, 2 I I 234.
I191 WINNE, D., KOPF,
1201
Acknowledgment
1211
M. L. G . G . thanks the Medical Research
Council for support.
1221
References
I I I ATKINS,G.L. & GARDNER,
M.L.G.
1231
(1977) The computation
of saturable and linear components of intestinal and other
transport kinetics. Biochimica el Biophysica Acta, 468,
127-145.
121 NEAME, K.D. & RICHARDS,T.G.
(1972) Elementary
Kinefics of Membrane Carrier Transport. Blackwell Scientific Publications, Oxford.
131 FISHER,
R.B. & PARSONS,
D.S. (1953) Glucose movements
across the wall of the rat small intestine. Journal of
Physiology, 119,210-223.
141 FORSTER,H. & MENZEL,B. (1972) Zur Bestimmung der
Michaelis-Konstanten fur die intestinale Glucoseresorption
bei Untersuchungen in vivo. Zeitschrfl fur Ernahrungswissenschaff,11,24-39.
151 DEBNAM,
E.S. & LEVIN,R.J. (1975) An experimental method
of identifying and quantifying the active transfer electrogenic
component from the diffusive component during sugar
absorption measured in uiuo. Journal of Physiology, 246,
181-196.
J.W.L., ANTONIOLI,
J.-A. & JOHANSEN,
S. (1980)
161 ROBINSON,
171
181
191
I101
IIII
I121
1131
I141
1151
The kinetics of sodium-dependent phenylalanine influx in the
intestine of the dog: a comparison between ileum and colon.
Journal de PhysioloRie, 16,637-645.
MUNCK,B.G. & S~HULTZ,
S.G. (1969) Interactions between
leucine and lysine transport in rabbit ileum. Biochimica et
Biophysica Acra, 183, 182-193.
HONNEGER,
P. & SEMENZA,
G. (1973) Multiplicity of carriers
Tor free glucalogues in hamster small intestine. Biochimica et
Eiophysica A d a . 318,39&4 10.
CORNISH-BOWDEN,
A. (1976) Principles of Enzyme Kinefics.
Butterworths, London.
CHRISTENSEN,
H.N. (1975) Biological Transport, 2nd edn.
Benjamin-Cummings, Mento Park, California,
ATKINS,G.L. (1982) Fitting biological equations to data
using non-parametric methods. Compufers in Biology and
Medicine (In press).
PRESTON.R.L., SCHAEFFER,J.F. & CURRAN,P.F. (1974)
Structure-affinity relationships of substrates for the neutral
amino acid transport system in rabbit ileum. Journal of
General Physiology, 64,443-467.
MATTHEWS,
D.M., CANDY,
R.H., TAYLOR,E. & BURSTON,
D. (1979) Influx of two dipeptides, glycylsarcosine and
L-glutamyl-L-glutamic acid, into hamster jejunum in vitro.
Clinical Science, 56, 15-23.
LUKIE,B.E., WESTERGAARD,
H. & DIETSCHY,
J.M. (1974)
Validation of a chamber that allows measurement of both
tissue uptake rates and unstirred layer thickness in the
intestine under conditions of controlled
stirring.
Gasfroenrerology.61,652-66 1,
WILSON,F.A. & DIETSCHY,J.M. (1974) The intestinal
unstirred layer: its surface area and effect on active transport
kinetics. Biochimica et Biophysica Acfa, 363, 112-126.
s. & ULMER. M.L. (1979) Role Of
unstirred layer in intestinal absorption of phenylalanine in
uiuo. Biochimica et Biophysica Acla. 550, 120-130.
DAINTY,
J. & HOUSE,C.R. (1966) ‘Unstirred layers’ in frog
skin. Journal of Physiology, 182.66-78.
LEVIN, R.J. (1979) Fundamental concepts of structure and
function of the intestinal epithelium. In: Scienrifc Basis of
Gastroenterology. pp. 308-337. Ed. Duthie, H.L. &
Wormsley, K.G. Churchill-Livingstone, Edinburgh.
FISHER,R.B. (1964) The mutability of K,. In: Oxygen in the
Animal Organism, pp. 339-348. Ed. Dickens, F. & Neil, E.
Pergamon Press, Oxford.
J.M. (1974) Delineation of
WESTERGAARD,
H. & DIETSCHY,
the dimensions and permeability characteristics of the two
major diffusion barriers to passive mucosal uptake in the
rabbit intestine. Journal of Clinical Invesfigafion, 54,
718-732.
J.M. & WESTERGAARD,
H. (1975) The effect of
1241 DIETSCHY,
unstirred water layers on various transport processes in the
intestine. In: Intestinal Absorption and Malabsorption, pp.
197-206. Ed. Csakv, T.Z. Raven Press, New York.
1251 THOMSON,
A.B.R.-& DIETSCHY,J.M. (1977) Derivation of
the equations that describe the effects of unstirred water
layers on the kinetic parameters of active transport
processes in the intestine. Journal of Theorerical Biology.
64,277-294.
1261 WINNE,D. (1973) Unstirred layer, source of biased Michaelis
constant in membrane transport. Biochimica er Biophvsica
Acfa, 298,27-3 1.
1271 WINNE,D. (1976) Unstirred layer thickness in perfused rat
jejunum in vivo. Experienlia, 32, 1278-1279.
1281 WINNE, D. (1977) The influence of unstirred layers on
intestinal absorption. In: Infestinal Permeation, pp. 58-64.
Ed. Kramer, M. & Lauterbach, F. Excerpta Medica,
Amsterdam.
1291 WINN, D. (1977) Correction of the apparent Michaelis
constant, biased by an unstirred layer, if a passive transport
component is present. Biochimica et Biophysica Acra, 464,
118-126.
[301 WINNE,D. (1978) The permeability coefficient of the wall of
a villous membrane. Journal of Mathematical Biology, 6,
95-108.
1311 HOFSTEE,B.H.J. (1952) On the evaluation of the constants
V,, and K , in enzyme reactions. Science. 116.329-33 I.
1321 WALTER,C. (1974) Graphical procedures for the detection of
deviations from the classical model of enzyme kinetics.
Journal of Biological Chemistry, 249,699-703.
L331 CHILDS,R.E. & BARDSLEY,W.G. (1976) An analysis of
non-linear Eadie-Hofstee-Scatchard
representations of
ligand-binding and initial-rate data for allosteric and other
complex enzyme mechanisms. Journal of Theoretical
Biology, 63, 1-18.
1341 FISHER,R.B. & GILBERT, J.C. (1970) The effect of insulin on
the kinetics of pentose permeation of the rat heart. Journal of
Physiology, 210,297-304.
A.B.R. (1979) Limitations of Michaelis-Menten
1351 THOMSON,
kinetics in presence of intestinal unstirred layers. American
Journal of Physiology. 236, E701-E709.
A.B.R. (1979) Limitations of the Eadie-Hofstee
1361 THOMSON,
plot to estimate kinetic parameters of intestinal transport in
the presence of an unstirred water layer. Journal of
Membrane Biology. 41,39-57.
1371 THOMSON,
A.B.R. (1981) A theoretical discussion of the use
of the Lineweaver-Burk plot to estimate kinetic parameters of
intestinal transport in the presence of unstirred water layers.
Kinetic analysis of transport processes
Canadian Journal of Physiology and Pharmacology, 59,
932-948.
M.L.G. (1980) Data handling
1381 ATKINS, G.L. & GARDNER,
and interpretation of intestinal transport kinetics. Clinical
Science, 59,301-303.
1391 NIMMO,I.A. & ATKINS, G.L. (1978) An evaluation of
methods for determining the initial velocities of enzymecatalysed reactions from progress curves. Biochemical
Society Transactions, 6,548-550.
1401 SEN,P.K. (1968) Estimates ofthe regression coefficient based
on Kendall's tau. Journal of the American Statistical
Association, 63,1379-1389.
1411 ATKINS,G.L. & NIMMO,I.A. (1980) Current trends in the
estimation of Michaelis-Menten parameters. Analytical
Biochemistry, 104,l-9.
1421 ATKINS,G.L. & NIMMO, LA. (1981) Robust alternatives to
least-squares curve-fitting. In: Kinetic Data Analysis: Design
and Analysis of Enzyme and Pharmacokinetic Experiments,
pp. 121-135. Ed. Endrenyi, L. Plenum Press, New York.
1431 WILKINSON,
G.N. (1961) Statistical estimations in enzyme
kinetics. Biochemical Journal. 80,324-332.
1441 DOWD,J.E. & RIGCS,D.S. (1965) A comparison ofestimates
of Michaelis-Menten kinetic constants from various linear
transformations. Journal of Biological Chemistry, 240,
863-869.
1451 COLQUHOUN,D. (1969) A comparison of estimators for a
two-parameter hyperbola. Applied Statistics, 18, 130-140.
1461 COLQUHOUN,D. (1971) Lectures on Bioslatistics, pp.
257-272. Clarendon Press, Oxford.
1471 CLELAND,W.W. (1963) Computer programmes for processing enzyme kinetic data. Nature, 198,463-465.
H. (1971) Intestinal
1481 RUEINO,A., FIELD,M. & SHWACHMAN,
transport of amino acid residues of dipeptides. 1. Influx of the
glycine residue of glycyl-L-proline across mucosal border.
Journal of Biological Chemistry, 246,3542-3548.
1491 ATKINS, G.L. (1973) A simple digital-computer program for
estimating the parameters of the Hill equation. European
Journal of Biochemistry, 33,175-180.
[Sol NIMMO,LA. & ATKINS,G.L. (1976) Methods for fitting
equations with two or more non-linear parameters.
Biochemical Journal, 157,489-492.
1511 DRAPER,N.R. & SMITH, H. u981) Applied Regression
Analysis, 2nd edn. John Wiley and Sons, New York.
I521 BEVERIDCE, G.S.G.
& SCHECHTER, R.S. (1970)
Oplimization: Theory and Practice. McGraw-Hill, New
York.
1531 NELDER,J.A. & MEAD,R. (1965) A Simplex method for
function minimization. Computer Journal, 7,308-3 13.
G. & LUMRY,R. (1961) Statistical analysis of
1541 JOHANSEN,
enzymic steady-state rate data. Comptes Rendus des Travaux
du Laboratoire Carlsberg, 32, 185-214.
W.W. (1967) The statistical analysis of enzyme
I551 CLELAND,
kinetic data. Advances in Enzymology and Relaled Areas of
Molecular Biology, 29, 1-32.
W.W. (1979) Statistical analysis of enzyme kinetic
1561 CLELAND,
data. Methods in Enzymology, 63A, 103-138.
I571 DAVIDON,W.C. (1959) Variable metric method for minimisation. ANL-5390 (Revised). Argonne National Laboratory,
Argonne, Illinois.
1581 FLETCHER,R. & POWELL,M.J.D. (1963) A rapidly convergent descent method for minimization. Computer
Journal, 6,163-168.
D.W. (1963) An algorithm for least-squares
I591 MARQUARDT,
estimation of nonlinear parameters. Journal of rhe Sociery
for Industrial and Applied Mathematics, 11,43 1-44 I.
1601 ATKINS,G.L. (1980) Simulation studies on the kinetics of
intestinal absorption. Biochimica et Biophysica Acta, 596,
426-438.
I611 STORER,A.C., DARLISON,M.G. ,& CORNISH-BOWDEN,
A.
(1975) The nature of experimental error in enzyme kinetic
measurements. Biochemical Journal, 151,361-367.
I621 SIANO,D.B., ZYSKIND,J.W. & FROMM,H.J. (1975) A
computer program for fitting and statistically analyzing initial
rate data applied to bovine hexokinase type I11 isozyme.
Archives of Biochemistry and Biophysics, 170,587-600.
B. (1976)
I631 ASKELBF,R., KORSFELDT,M. & MANNERVIK,
Error structure of enzyme kinetic experiments. European
Journal of Biochemistry, 69,6 1-67.
413
1641 NIMMO,LA. & MAEOOD,S.F. (1979) The nature of the
random experimental error encountered when acetylcholine
hydrolase and alcohol dehydrogenase are assayed. Analytical
Biochemistry, 94,265-269.
1651 RODBARD,D., LENOX,R.H., WRAY,H.L. & RAMSETH,D.
(1976) Statistical characterization of the random errors in the
radioimmunoassay
dose-response
variable.
Clinical
Chemistry, 22,350-358.
1661 EISENTHAL,
R. & CORNISH-BOWOEN,
A. (1974) The direct
linear plot. A new graphical procedure for estimating enzyme
kinetic parameters. Biochemical Journal, 139.7 15-720.
A. & EISENTHAL,R. (1978) Estimation
I671 CORNISH-BOWDEN,
of Michaelis constant and maximum velocity from the direct
linear plot. Biochimica et Biophysica Acta, 523,268-272.
J. (1979) The application of robust non-linear
I681 WAHRENDORF,
regression methods for fitting hyperbolic Scatchard plots.
International Journal of Bio-Medical Computing, 10.75-87.
A. & WONC,J.T.-F. (1978) Evaluation of
[691 CORNISH-BOWDEN,
rate constants for enzyme-catalysed reactions by the
jackknife technique. Biochemical Journal, 175,969-976.
1701 CORNISH-BOWDEN, A. (1981) Robust estimation in enzyme
kinetics. In: Kinetic Data Analysis: Design and Analysis of
Enzyme and Pharmacokinetic Experiments, pp. 105-1 19.
Ed. Endrenyi, L. Plenum Press, New York.
1711 CORNISH-BOWDEN, A. & ENDRENYI,
L. (1981) Fitting Of
enzyme kinetic data without prior knowledge of weights.
Biochemical Journal, 193,1005-1008.
A,, PORTER,W.R. & TRACER,W.F.
1721 CORNISH-BOWDEN,
(1978) Evaluation of distribution-free confidence limits for
enzyme kinetic parameters. Journal of Theoretical Biology,
74, 163-175.
1731 NIMMO,I.A. & ATKINS,G.L. (1979) The statistical analysis
of non-normal (real?) data. Trends in Biochemical Sciences,
4,236-239.
[741 ATKINS,G.L. (1976) Tests for the goodness of fit of models.
Biochemical Society Transactions, 4,357-36 I.
[75] MANNERVIK,B. (1981) Design and analysis of kinetic
experiments for discrimination between rival models. In:
Kinetic Data Analysis: Design and Analysis of Enzyme and
Pharmacokinetic Experiments, pp. 235-270. Ed. Endrenyi,
L. Plenum Press, New York.
1761 MANNERVIK,
B. & BARTFAI, T. (1973) Discrimination
between mathematical models of biological systems exemplified by enzyme steady-state kinetics. Acta Biologica et
Medica Germanica, 31,203-215.
R.G. (1978) What happens when
I771 ELLIS,K.J. & DUGGLEBY,
data are fitted to the wrong equation? Biochemical Journal,
171,s 13-5 17.
J.Y.F., SEPULVEDA,
F.V. &SMITH,M.W. (1979)
1781 PATERSON,
Two-carrier influx of neutral amino acids into rabbit ileal
mucosa. Journal of Physiology, 292,339-350.
1791 GOMENI,R. & GOMENI,C. (1979) AUTOMOD, a polyalgorithm for an integrated analysis of linear pharmacokinetic models. Computers in Biology and Medicine, 9,
39-48.
I801 HURST, R.O. (1974) Evaluation of the Hill coefficient for
yeast aspartate transcarbamylase by nonlinear regression
analysis. Canadian Journal of Biochemistry, 52, 1137-1 142.
1811 AKAIKE, H. (1974) A new look at the statistical model
identification. IEEE Transactions on Automatic Control, 19,
7 16-723.
1821 TONG,H. (1977) Some comments on the Canadian Lynx
data. Journal ofthe Royal Slatistical Society, Series A , 140,
432-436.
R.N., IDER,Y.Z., BOWDEN,C.R. & COBELLI,C.
1831 BERGMAN,
( 1 979) Quantitative estimation of insulin sensitivity.
American Journal of Physiology, 236, E667-E677.
1841 DAMMKOEHLER,
R.A. (1966) A computational procedure for
parameter estimation applicable to certain non-linear models
of enzyme kinetics. Journal of Biological Chemistry, 241,
1955-1957.
1851 HURST, R., PINCOCK,A. & BROCKHOVEN,
L.H. (1973)
Model discrimination and nonlinear parameter estimation in
the analysis of the mechanism of action of phydroxyRhodopseudomonas
butyrate
dehydrogenase
from
spheroides. Biochimica et Biophysica Acta, 321,l-26.
V.V. & PAZMAN,A. (1969) Design of physical
1861 FEDEROV,
experiments. Fortschritte der Physik, 24,325-355.
414
M . L. G. Gardner and G. L. Atkins
1871 BARTFAI,T. & MANNERVIK.
B. (1972) An attempt to
discriminate between mathematical models in enzyme steadystate kinetics. In: Analvsis and Simulation of Biochemical
Systems (Federation of European Biochemical Societies, 8th
meeting, vol. 25), pp. 197-209. Ed. Hemker, H.C. & Hess, B.
North-Holland, Amsterdam.
1881 MARKUS.M. & PLESSER,T. (1981) Progress curves in
enzyme kinetics: design and analysis of experiments. In:
Kinetic Data Analysis: Design and Analysis of Enzyme and
Pharmacokinefic Experiments, pp. 3 17-339. Ed. Endrenyi,
L. Plenum Press, New York.
1891 DUGGLEBY,
R.G. (1979) Experimental designs for estimating
the kinetic parameters for enzyme-catalysed reactions.
Journal of Theoretical Biologv. 81.67 1-684.
L. (1981) Design of experiments for estimating
1901 ENDRENYI,
enzyme and pharmacokinetic parameters. In: Kinetic Data
Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments. pp. 137-167. Ed. Endrenyi, L. Plenum
Press. New York.
1911 ANTONIOLI,
J.-A., JOSEPH,CL. & ROBINSON,
J.W.L. (1978)
Kinetics of the absorption of amino acids by the rat intestine
in uiuo. Biochimica el Biophysica Acla, 512,172-191.
1921 ROSENBERG,
R. (1981) L-Leucine transport in human red
blood cells: a detailed kinetic analysis. Journal of Membrane
Biolog.v, 62,79-93.
1931 METZLER,
C.M., ELFRING,G.L. & MCEWEN,A.J. (1974) A
Users' Manual f o r NONLIN and Associaled Programs.
Upjohn Co., Kalamazoo, Michigan.
1941 HOPPER, M.J. (1979) Harwell Subroutine Library. A
Catalogue of Subroutines. Computer Science and Systems
Division, AERE Harwell, Oxfordshire.
1951 DIXON,W.J. & BROWN,M.D. (Eds.) (1979) BMDP-79.
Biomedical Computer Programs. P-Series. University of
California Press, Berkeley, California.
1961 DUGGLEBY,
R.G. (198 I) A non linear regression program for
small computers. Analytical Biochemistry, I10,9- 18.
1971 KNACK, 1. & ROHM, K:H.
(1981) Microcomputers in
enzymology. A versatile BASIC program for analyzing
kinetic data. Hoppe-Seyler's Zeitschrp fur Physiologische
Chemie, 362, I 1 19- 1 130.
W. & ROSENBERG,
T. (1961) The concept of
1981 WILBRANDT,
carrier transport and its corollaries in pharmacology.
Pharmacological Reviewis, 13, 109-182.
1991 MUNCK,
B.G. (1981) Intestinal absorption of amino acids. In:
Physiology of the Gastrointestinal Tract. volume 2, pp.
1097-1 122. Ed. Johnson, L. R. Raven Press, New York.
I1001 MUNCK,B.G. (1968) Amino acid transport by the small
intestine of the rat. Effects of glucose on transintestinal
transport of proline and valine. Biochimica el Biophysica
Acta, 150.82-91.
I1011 ALVARADO,
F. (1966) Transport of sugars and amino acids
in the intestine: evidence for a common carrier. Science, 151,
1010-1013.
I1021 ROBINSON,
J.W.L. & ALVARADO,F. (1977) Comparative
aspects of the interactions between sugar and amino acid
transport systems. In: Inlesfinal Permeation, pp. 145-163.
Ed. Kramer, M. & Lauterbach, F. Excerpta Medica,
Amsterdam.
I1031 HIMUKAI,
M. & HOSHI,T. (1980) Mechanisms of glycylL-leucine uptake by guinea pig small intestine: relative
importance of intact peptide transport. Journal ofphysiology,
302,155-169.
I1041 RIGGS,D.S. (1970) The Mathematical Approach to Physio-
logical Problems. M.I.T. Press, Cambridge, Massachusetts.
I1051 CHRISTENSEN,
H.N. & LIANG,M. (1966) On the nature of
the 'non-saturable' migration of amino acids into Ehrlich cells
and into rat jejunum. Biochimica el Biophysica Acta, 112,
524-5 3 I.
11061 SCHEDL,H.P., BURSTON. D., TAYLOR,E. & MATTHEWS,
D.M. (1979) Kinetics of uptake of an amino acid and a
dipeptide into hamster jejunum and ileum: the effect of
semistarvation and starvation. Clinical Science, 56,487-492.
I1071 TAYLOR,
E., BURSTON,
D. & MATTHEWS,
D.M. (1980) Influx
of glycylsarcosine and L-lysyl-L-lysine into hamster jejunum
in uitro. Clinical Science, 58,221-225.