Biological problems involving sums of exponential functions of time: a mathematical analysis that reduces experimental time

123

Biological Time:

Problems

Involving

A Mathematical

Commnnicatrd

by

Sums of Exponential

Functions of

Analysis That Reduces Experimental

Time

D. G. Kendall

ABSTRACT

z\ recent

1”oblem

of ratllol~iochemistr~

problems

in which experimental

ponential

functions

equation

where

is typical

of a class of biological

of the time, but the number of these functions,

stants, and their coefficients of experimental

[lj

readings are known to be sums of decreasing

are unknown.

readings of a quantity

ex-

their time con-

Thus suppose that we have a number

r(t) known hy biological

theory to satisfy an

of the form

t is the time, p au unknown positive integer, the _4i and ui unl~nown constants,

and

Then the conventional

procedure

is to determine these constants

two at a time, by

plotting log Y against t and estimatin g the straight line to which the graph is asymptotic. This gives d, and n,; and the process is repeated for the function

(y(t) - rl,Cy, \vhich gives .4 *, ~1~; and so on. to estimate the asymptotic

The disadvantage

of this procedure is that, in order

straight lines accurately,

it IS essential to take readings

for large values of the time, that is to carry the experiment Hy contrast,

on as long as possible.

the analysis given in this article, which is based on the theory of

difference equations, enables us to determine the integer p in advance, and then to determine the 2p unknown constants by means of simultaneous linear equations. We shall see that it is actually advantageous for all readmgs to be taken as earl! as possible in the experiment, which is thus conducted as quickly as possible.

Mnthenzatical

Copyright

0

1968 by American

Biosciences

Elsevier

2, 123.-

Publishing

128

Company,

(1968)

Inc.

124 1.

II.

STATEMEKT

Given times then

OF

THE

PARSONS

PKOBLE.ll

a sequence

yl, Ye, . . . , 7% of experimental

t,, t,, . . . , t,, respectively, 2p constants

H.

to determine

readings

a positive

taken

integer

$,

at and

A,, A,, . . ., A,, a,, a2,. . ., ap, such that + ilge- %z + . . . + ‘4pe-apf,l

A,e - G

Y,, =

(2)

forn = 1,2,3,.

. ., where we can suppose that 0 < a, < a2 < a3 < . . . < a, ?

2.

SOLIJTIOK

METHOD

OF

il:e can suppose spaced intervals

that

the experimental

of time.

readings

If this be inconvenient,

are taken

can be read from a simple graph of r(t) against t or obtained Thus,

let the interval

be a total

of

between

hT readings

consecutive

available.

f, = t, + (n -

readings

Then

l)T

at equally

equally spaced readings by interpolation.

be 7‘, and let there

we have

(M-l,%

)...)

nr).

(3)

Let “j

=

P

-ajT

1,2 ,...,

(j=

Bi = Aje-%

$),

(1)

(i=I,2,...,P),

(f,)

so that A4j~-~;‘J‘ = K ..x:”- ’

(?L= 1,. . .,,)’

I I

Then the problem ~5, and then

is reduced

29 constants

to that of determining

Br, B,, . . ., B,,

Y, = N,x;’ - t + I:,,$

x1, x2,.

a positive

integer

., xp, such that

- * + . . . + f&Y;: - 1

((9

for 72 = 1, 2, 3, . . . , N ; and we suppose l>x,>x,>~**>~~>O. Now, it is a standard the necessary described a linear

result in the theory

and sufficient

above are two only: difference

equation

conditions

of difference

for the existence

first, that the sequence of the form

equations

that

of the numbers

rr, Ye, . . . , Y, satisfy

SUMS

OF

EXPONENTIAL

FUSCTIOSS

(E = 1, 2, . . . ) Lv - p), where that

the roots

be distinct

(these

3. DETERhlINATlON

being

OF THE

Let D(s; n) denote

and second,

of order fi,

CrXe- 1 _+ c,,rP - 2 roots

125

TIME

cr, ca, . . , cp are constants;

of the equation ,yiJ -

OF

... I

precisely

INTEGER

(_ 1)PQ = 0,

the

(Xl

x1, xi, . . ., xp of (6)).

$J

the determinant

of s rows and columns

the leading element

is Y, and the element

is Y,, + k _ ?; that

is, let

in which

in the ith row and kth column

I , y,,

Then the necessary to

satisfy

y,,

and sufficient

an equation

y >1 , i

conditions

of the form

y,,

2

--

i-9 -

for the sequence

I

I

rr,

r2, . . . ,7,&

(7) are

I>@; 12) # 0

(9 (ii)

. . .

,- 1

fl(fi + 1 ; ~1)== 0

(I2 = 1, 2,3,.

.),

(?I == 1, 2, 3,

.).

Thus, calculating all the determinants D(s; 12) for ?L = 1, 2, . . . , A - 2s + 2 and for s = 1, 2, 3, . . . , when we reach the stage s = $J at which conditions

(i) and (ii) are satisfied,

the required

At this stage it should be stressed and computational

inaccuracies,

To avoid determining Suppose

that the possible

Then

m) is nonzero,

WC regard

integer

~5 is determined.

owing to both experimental

no determinant

will be precisely

zero.

a false value for fi, a suitable criterion is the following.

true value lies between r,, II@;

that,

error in determining

Ye

is g, (> 0), so that the

g,, and Y,~+ g,,. Suppose that the determinant

while D($ + 1; m) satisfies

D(l, + 1; m) as zero.

For

of D(P + 1; m) by the last row is precisely

the inequalit!

one term

in the expansion

I>. H. PARSONS

126

and

if the

determinant

possible itself,

.4. DETERMINATION

err-or in calculating the latter OF THE

must

this

be regarded

term

is as large

as the

as zero.

COh’STANTS

Suppose that the integer readings being available)

@ be determined

as above

and

that

(N

(This last restriction implies only that at least one determinant D(p + 1; a) can be formed.) Then the equations (7) are a set of N - fi (2 ;b + 1) consistent linear equations in the $ unknowns cr, ca, . . . , cp, any $ consecutive equations being linearly distinct. These equations can therefore be solved uniquely for cr, ca, . , cp. Putting these values in (S), we have an equation of the Pth order in s; and it is easy to show that, in consequence of the condition (i), Section 3, all the roots of this equation are nonzero. Sow, solving (8), and provided that thee roots be distinct, x1, x2, . . , xp, say (we deal later with the possibility of confluent roots), we substitute these values in (B), obtaining thereby a set of N consistent linear equations in the ~5 unknowns B,, B,, . . . , B,; and once more, any p consecutive We can therefore solve these equations equations are linearly distinct. uniquely for B,, B,, . . . , B,. Finally, from (4) and (5) we have

(j-l,:‘,..., The problem 3. THE

is now solved,

CONFLUENT

all the unknowns

/!I). in (2) being

determined.

CASE

It might happen (according to the nature of the biological problem, this may or may not be possible: in the problem treated by Aubert and Milhaud [l] it is not possible) that although the sequence or, ~a, . . ., rn with constant coefficients, of does satisfy a linear difference equation, the form (7), nevertheless the roots of Eq. (8) are not all distinct.

SUMS

OF

EXPONEXTI.ZT,

Ft.SCTIONS

In this event the readings (2), but satisfy

a similar

OF

do not satisfy

relationship

more of the esponential

functions

127

TIhZE

any relationship

of the form

in which the coefficients

are polynomials

of one or

in the time instead

of

constants. To be precise,

let us suppose

that

has a root x1 of multiplicity

WZ~,a root xg of multiplicity

N,, of multiplicity

the

suppose

of (B), we have

lvhere the p quantities constants.

Thus,

an equation

Let

theory

these

equations

as (8)

wz2,. . . , and a

of difference

solve this system

equations

s -= 0, 1, . . , vtk -

values

uniquely

distinct

in (9), which

for the determination

C,,,Y; as before, any p consecutive us now write

equation

that,

of the form

(8) f or the 4 nonzero

solved

X1, X2’ . . . ) sg, we substitute constants

that the resulting

C,?,,7 (Iz = 1, 2, . . . , 4;

having

a system of ,V consistent

\Ve therefore

rl, Ye, . . , r,& exactly

WZ,],where

Then it follows from the standard instead

readings

$ and the constants

But

root

from

the integer

Cl’ cz>. . . , c/, are determined in Sections 3 and 4 above.

equations

then

1) are

numbers constitute

of the /J unknown are linearly distinct.

for the constants

C,<,,.

(3) in the form

and also write

csactly

as in Section

each X, in the right-hand

4.

Then,

substituting

these

side of (9) and rearranging

ials in t,, (whose coefficients

are no\v known),

values for $1 and for the resulting

WV obtain

an expression

of the form

entirely

analogous

to (2), the /, coefficients

D,,,y being

polynom-

known.

128

D.

In this case, we see at once that (I), but is of the rather

where

more

the function

general

H.

PARSONS

r(t) is not of the form

form

mi + m2 $ * f . + my = p.

6. ADVANTAGES

The

great

constants

OF THIS

advantage

are determined

(7) to determine

PROCEDURE

of this

analysis

is that

by simultaneous

the

various

equations:

ci, c2, . . ., co; then a single equation

cletermine xi, x2, . . . , x0; and then simultaneous

first

unknown of all, Eq.

(8) of degree p to

equations

(6) to determine

B,, B,, . . . , B,. Thus there is no point in continuing period.

the experiment

for a prolonged

In fact, the reverse is the case, for since the exponential

and their derivatives earlier

readings

is thus

every

decrease

(in magnitude)

are not only larger, advantage

but changing

in completing

KEFERENCE

,~ialhcmnticnl

Rio.scie?zrr.s 2, 123-

125 (1968)

functions

as the time increases, more rapidly.

the experiment

quickly.

the There