The Application of the Logistic Function to ... - ACS Publications

T H E XPPLICAITIOK O F T H E LOGISTIC F C K C T I O S TO EXPERILIESTAI, DATA*

Downloaded by TEXAS A&M INTL UNIV on August 27, 2015 | http://pubs.acs.org Publication Date: January 1, 1928 | doi: 10.1021/j150299a014

BY LOTVELL 6 . R E E D AS-D JOSEPH B E R K S O S

An extensive use has been made of this function for the elucidation of research problems in many fields. Yet in almost all cases, the mathematical phases of the treatment have been faulty, with consequent cost to the precision and validity of the conclusions drawn. Indeed, in numerous instances that have come under our notice, erroneous inferences were drawn as a result of a misconception of certain mathematical characters of the function. 1T-e deem it important, for these reasons, that a reviejv of the basic properties of the equation be made, and some niethods for its analysis indicated, towards which end this paper is contributed. The curve has been frequently referred to as that of an autocatalytic reaction, froin the fact that Ostwald applied it to a c l a s of chemical reactions subsumed by him under that term. *As will be illustrated belowv,however, it is descriptive of processes, even in thc field of chemistry, in situations where t,he concept of autocatalysis has no place. We would recommend, therefore, that the more general term “logistic,” which does not commit it to any specific mechanism, be retained in its stead. The equation expressing the function is as follows:

Logarithmically expressed it id In

K - (y - d) = In C y - d

+ rt

The differential form is

3 = $ [y - d] [I(- (y - d)] dt

y is the dependent variable t is the independent variable K , d, T , and C are parameters In signifies the logarithm to the base e K is the distance between two asymptotic values of y, and so demarks the range of the limiting values of y. The quantity d is a constant which must be algebraically subtracted from y ; it is this difference which varies logistically with t. I n an experimental

* This paper is issued jointly froin the Department of Biometry and Vital Statistics and the Institute for Biological Research of the Johns Hopkins University. I t constitutes KO.148 in the numbered series of publications of the Department of Biometry and Vital Statistics.


A P P L I C l T I O S O F THE LOGISTIC F U S C T I O X

761

situation, one may think of d as a n element incorporated gratuitously with each observational determination of IJ, itself not subject to the changes taking place during the course of the experiment. I n many esperirnents, there is no such element, and (I = 0. T is a parameter associated with the intrinsic rate of the reaction: it is in fact I< times the rate a t which y is changing, expressed as a fraction of the product of ( y - d i by the difference between Ii and (y - d)." c' is a constant of integration, depending matheniatically on the choice of origin for the a s k , and varying in any experiment with the value of the initial observation. I t will simplify the presentation that follows, t o assume its value as unity, and to discuss the significance of a departure from this value when vie consider the experimental applications.

Description of the Function The appearance of the function xi11 change as the parameters assume different values. We shall consider the various possibilities seriatim.

Case 1 * *

Figure

c

d = o

I.

= +I

FIG.I Logistic for Case

I

The following characters are notable: ( I ) There are two asymptotes parallel to T axis, a lower at y = 0, and a p upper a t y = R. (2)

At the point where t

=

0, y = y o = K

2;

(7)

* Vide equation ! 3 ) . * * K T?-ill he assumed positive in all the cases to be considered. K h e r e it is negative, the equation may be transformed by suitable rotation of the axis of reference t o one of the forms with K positive, and is omitted from the discussion here.

LOWELL J . R E E D A S D JOSEPH BERGSON

762

At this point the curve has a point of inflection and the slope d y dt is a maximurn.


( 3 ) The form th) is the form (a) rotated about the I? axis. .It any point t , the distance f r o m thp upper asymptote t o (a)>is equal to the distance fr.om

the IoLYer asymptotc to i b ) . (j) The slope of the curve ii a t any point proportional tothe product of the distances of the point from the lower and upper asymptotes. (6) The curve has a point of symmetry a t t = 0, y = K 2 ; if one-half of the curve is rotated about this point through an angle of 180°, it Trill coincide with the other half.

Case 2.

Figure

2.

c=-

d = +

I

fal

d

r = +

11

Y

Y

Ib I

I

I

d

T

0

0

T

PIC;. 2

Logistic for Case z

d] [I< Here n e note, That foinis (roa) and (Iob) result respectively fiom ( l a ) and (4b)> (I) nhen the T axis IS iiioved in the I' direction, a distance equal t o minus d . Hence, they are exactly these equations with the quantity (y - d ) substituted for y. (2)

(3)

Khere t = 0, y = yo = For forin (a) v - ~ = +d, For form (b) u+- = K

K

+

-

2 d y-, = R d, y-, = (1

+d

(13)

(14) (15,


APPLICATIOX OF THE LOGISTIC FUNCTION

763

(4) There are two asymptotes K distance apart as before, but the lower one is a t y = cl, not zero. ( j ) The movement of the T axis has been represented as in the minus direction, and this corresponds experimentally to the addition of a constant element d to each observation. An analogous situation exist,s, if in a logistic process, each observation excludes a fixed quantity d. This corresponds geonietrically to the inovenlent of the Taxis in a plus direction, and the relations given in this section would then hold with the sign of d changed from minus t o plus. Case 3.

Figure 3. d = o

c = -1

r = k

L (IY I

K

-t.

y

.r7 K

0

FIG.

0

.

T

3

Logistic for Case 3

K e note here t h a t : !I) The two fornis (16a) and (16b), though very different in geometric appearance from (4a) and (4b)>bear a close resemblance t o them analytically. They result from the latter, in fact,, if the sign in the denominator be changed t o minus. (2) There are three asymptotes, two in a direction parallpl to the abscissa, and one t o the ordinate. The abscissa1 asymptotes are a t a distance K apart, the lower at a position y = 0. The vertical a s p i p t o t c i p approached by both branches of the curve, one in the plus and the other in the minus direction.

LOWELL J. R E E D .4ND JOSEPH B E R K S O S

764


(3) (4)

*

When t = 0,y = y o = 03 For form (a) yAL?;= 0 , y--3 = +E; For form (b) y+= = +IA,, the mass law gives

where C1 is a constant of proportionality, depending on temperature, etc. From ( j 6 ) we may write Downloaded by TEXAS A&M INTL UNIV on August 27, 2015 | http://pubs.acs.org Publication Date: January 1, 1928 | doi: 10.1021/j150299a014

dy, d t

=

C1 @--Ao) [y-A, - cn,-A,j]

(Si)

( 5 7 ) is seen t o be the logistic of forni ( 2 5 ) in which

K

=

Bo-&

d = A0

(58)

r = C1(Bo-Ao)

(59)

We take the following esarnple.1 Methyl bromide and sodium thiosulphatc were placed together for reaction and the following gives observations niade of the quantities of the decreasing sodium thiosulphate. A, = concentration of CH3Br at, the beginning of the experiment = 14.; units Bo = concentration of S a 2 S 2 0 3a t the beginning of the esperinient = 24.3 units y = observed concentration of SasSZOa a t time t K = Bo - A, = 9.6 from (58) Equation ( 2 2 ) applies and r is to be determined. TI-e may utilize Method I with convenience since K is known. (See Table 11,Fig. 8). Following the same procedure as that used for obtaining (51)we have )'=

- 0 I

08455

c = ,604; The equation now becomes y

= I

from 1\37) froin 138) 96--

- ,boJie-

(60)

Oosl*at

The calculated and ohserred values ale compared in Table I1 and Fig. 8.

TABLE I1 t

y (observed)

log (1C-y) y

y calculated from equation 160)

'1.3 22 . g j

,7817

20.j

,725:

20.54

18.8 Ii.35

,6897 , 6joo ,5979

'7,45

15 .9 14.35

9.6 Slator: J. Chem. Soc., 85, 1295 (1904)

,

,

7647

j1ya

. 0000

25.29 22

.84

18.85 :j . 9 0

14.55 9.60

BPPLIC.4TION OF THE LOGISTIC FUSCTION

773


3. Bzniolecitlar Reaction wzth Equal I n i t i a l Concentration of Reactants. From ( j 6 ) , if Bo 7 A , we have dy/dt = ci(Xo-y)? (6 I where c1 is a constant of proportionality, from which by integration

FIG.8 Graphical Presentation, Example 2 Legend: y = Concentration S a & 0 3 t = time A , = Concentration CHaBr,original Bo = Concentration Sa2SP08,original .

.

where c ? is a constant of integration that depends on the initiallvalue of y. Form (62) is seen t o be the hyperbolic form 133) in which d = A, b = CI C? = CL (63)

LOWELL d. REED A S D JOSEPH BERKSOS

ii4

SVe take the following example.’ Equivalent quantities of acetamide and hydrochloric acid in a water colution reacted t o form ammonium chloride, and the progress of the reaction followed by observing a t succesive times t the amount of nitrogen produced a s ammoniuni chloride. A, = 26.80 (observed)


We have now to determine 2, and CZand mayconvenientlyuse the method outlined under IT. (See Table I11 and Fig. 9).

The equation now becomes y = 26.80 -

(64)

I

.03 j

+ .ooo56ot

The calculated and observed values are compared in Table I11 and Fig. 9. t

Y

y calculated from (64)

I5

4.42

,044

30 45

7.53 I O .16

.Oj2

60

I20

12.13 13.68 14.97 16.97

Ijo

18.40

I 80

19.53

13

90

,060 ,068 ,076 ,085 . IO2

,119 , I38

4.

--

3.54 ”

( , > I

I O . 13 12.31 13 .SI I j .03

17

.oo

18.40

19.45

Autocutulytic Reaction, S e g a f i c e Autocatalysis. If, in a catalytic reaction, one of the products acts as a “negative” catalyst, Le., inhibits the main reaction, we have analogous t o (50) the following

From which we may write

This is seen t o be form (24) in which

An illustration is the following.?

* K. Ostwald: 2

J. prakt. Chem., 135, 14 (1883). J. Berkson and L. B. Flexner: J. Gen. Physiol., 9, j,433 (1928).

773

.iPPLICATIOS OF THE LOGISTIC F U S C T I O S

The proteolysis of gelatin by pancreatin was folloxed by measurements of the \-iscosity of the gelatin mixture. The time required for the flow of the mixture through a capillary was found t o be a logistic function of the duration of the reaction, a5 folio\\ b .


./5

c

*

JP

60

30

t

/20

153

o obseried

FIG.9 Graphical Presentation. Example 3 Legend: g = S as S I l a C l t = time :I9= total S as SH4C:I yi - y, - 1 T-o - L vi c-rt in n.hich yo y is the time of flow at tinic t yo is the tiine of flow a t t = 0 y, is the time of flow of watcr yf is the tiriie 6f flow !Then protcolysi.5 is coriiplcte y-y,-

>-\,

/eo

2 4

LOWELL J. REED .4SD JOSEPH BERKSOX

776

This is seen to be the logistic of form

Vi = 35.5


It remains t o determine Table IT7 and Fig. I O ) .

y. = T.

in which

(22)

1.0 = j 9 . s

2g.j

This is accomplished by use of Method I. ( S e e

r = -0.006j81 = 6 C' = 0.800

from ( 3 7 ) from ( 6 9 )

K

The equation now bpcomes

The calculated and obqerved values are compared in Table IT- and Fig.

IO.

TABLE IT' y fobserved)

t

log

y - ti -I< y -d ~

-

y calculated from

equation 1701

1,9031

.io 5

7.8901

s6 1

i.8732

53 4 j I 5

1 8602 ~

1.8459

39 8

i.8,312 i,8202

38 4

8035

i.i8j3 -

1 jj62

47

1

36

I

45 0 31 I

6. OZidCltiCJ71 U 7 i d RPdliCfiCJll PCJtoltl'OlS. The equation governing the electrode potentials in an oxidation-reduction reaction is given as follows:' E h =

E,

RT S, -In nF

S:

in which

Ei, = observed electrode potential nwasurrd against a hydrogen electrode E,' = n constant characteristic of the particular system R,T,F, = constants n = the number of molecules involved S, = concentration of the reductant P, = concentration of the oxidant K. 3lansfield Clark: 1.. 8. I'ub. Health Rep., 38, 943 (1923).

APPLICATIOS O F THE LOGISTIC FL-XCTIOS

ii7

+

If we denote by K = So S,,the total amount of substance involved in the reaction and by c the constant RT ,IF, we may rewrite (;I) as follom: K - s, In -- = - EL c I I C Et, (i2)


s,

+

' \I 1

0

Observed

-CalculQtea

FIG. I O

Graphical Presentation, Esample 4 Legend: !i = viscosity. time of flwv t = time y,'. = -d = viscosity for I l i O y, = visrosity, final

This is .wen to be the logistic of the form (E;) in which y = s, t = Eh r = I/c In c = - E L ' C (7.1) 7T-e take t,he following example.' I-napthol-2-sulphonate indophenol was reduced by addition of titanous chloride and the electrode potentials observed. The number of molecules involved, n, was independently determined as 2 . I t is desired to determine E;. Since r = nF RT is known, Method I1 can be conveniently used. I t will be necessary to assume some single observation as yo, and me choose one in the middle of the series a t y = 16. 1

IY,Ilansfield Clark: V, S. Pub. Health Rep., 38, 946

(1923).

778

LOWELL J. REED A S D JOSEPH B E R K S O S T =

76.57889 - constant

yo - yert

vs. y is plotted (see Table 1-ar.d Fig. I - ert and C are determined as follows:


40

and from the slope m, K

11)

r

1

:serve;

-0-

,

,

20

IS

Ca/Lv/afed

Os o

/O

5

15

) 30

8, f c c ) FIG II

Grapliical Presentation, Erample j T2egenti:

I/ =

t

Sr

=

= E h -

Ti +--cc. ,1224 = Potential against 11 electrode -

1224

779

A P P L I C l T I O S O F THE LOGISTIC F C S C T I O S

from (42) from (40)

K = myo = 32.68 C = m-1 = 1 . 0 4 2 2 j E: = - c I n C = -.oooj = -,GOO;

+

,1224 =

from ( 7 3 )

.1219inoriginalunits.

The equation noiv becomes sr


E h = ,1219

- .0130j84In 32.68- S,

(74)

The calculated and observed values are compared in Table T and Fig.

I I.

TABLE

Ti+Jcc.

Eh

Observed

t E h --.1224

yo -yert 1-ert

Eh

Calculated from (74)

0

,287

,1646

2

,1581 . 1479

,0357

1.0272

.1575

,0246

I ,8489

6 8

. I41 j

0191

,1368

,0144

2.985 4.02;

10

,1327

12

. I292

0103 ,0068

6.14j

,1476 ,1414 ,1367 ,1326 ,1290

14

. 1 2 j6

,003 2

6 . 795

. I ? jj

16 18

. I224

. 00

,1192

,0032

20

,1159

,006j

22

,1124 ,108 j

.OIOO

4

24

,0139 ,0183

26 28

,1041

30

,090j

,0239 ,0319

,036

,0864

32

,0985

.8

j, 0 0 2

00

,1224

8.797 9.801 I O . 786 II , 7 7 1

,1193 '1159

12.730

,1042

1 3 , io7

,098j

14,667 16.116

,0904

,1125

. io86

The Application of the Logistic Function to ... - ACS Publications

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