A new method for examining the complexity and relationships of “timers” in developing systems

DEVELOPMENTAL

BIOLOGY

95,

73-91

(1983)

A New Method for Examining the Complexity and Relationships of “Timers” in Developing Systems R. SOLL'

DAVID Department

of Zoology, University

Received February

of Iowa, Iowa City, Iowu 52242

16, 1982; accepted in revised form

August 27, 1982

Simple methods are developed for analyzing the rate-limiting pathways, or “developmental timers,” for consecutive stages in a developing system. Two conditions are first defined for short and long timing to a developmental stage. Shifts are then performed at time intervals from short to long and long to short conditions. The total time to the stage (time under first condition plus time under second condition) is scored and plotted as a function of the time of shift, resulting in two plots, one for shifts from the short to long condition, and the other for shifts from the long to short condition. Elach plot is then analyzed for the number of components, slopes of components, absolute times of origins and termini of components, and discontinuities between components. This information is then used (1) to distinguish betwe,en single- and multiple-component timers, (2) to assess the sensitivity of each timer component to the change in the environmental condition employed in the method, including reversibility, (3) to test for the addition of a new timer component under long conditions, and (4) to test for an identity change of a timer component between short and long co:nditions. These interpretations in turn provide a minimum estimate of the complexity of the ratelimiting pathway to a developmental stage, temporally define major transition points between timer components, and provide some insight into the nature of timer components. By characterizing the rate-limiting pathway from the origin of a developmenta. program for each consecutive stage in that program, distinctions can also be made between single, parallel, sequential, and branching timer relationships. From these interpretations, a detailed temporal “map” of the rate-limiting program can be generated for any developmental system in which consecutive stages can be reproducibly monitored with time.

mental timers (Soll, 1979). These methods depended upon the differential sensitivities of timers to small changes in a single environmental parameter. By timing the stages of a developing system at different temperatures within a small range, and by measuring the intervals between two stages after a shift from low to high and from high to low temperature at the time the first stage is formed, distinctions could be made between single, sequential, and parallel timer models. When these methods were applied to a model developing system, morphogenesis in the cellular slime mold Dictyostelium discoideum, unexpected results were obtained. A number of independent timers were distinguished for consecutive developmental stages, and several of these timers appeared to function in parallel. However, as pointed out in this report, the original methods used to interpret timer relationships were based upon the underlying assumption that timers were not complex (i.e., were not themselves composed of multiple, consecutive processes) and therefore were uniformly affected along their entire lengths by the environmental change employed. Thus, interpretations of timer relationships employing this method were tentative. In the present report, we present a simple and direct method for testing this assumption. By very simple shift experiments, minimum timer complexity can

INTRODUCTION

There are a number of questions which remain unanswered concerning the regulation of timing in most developing systems. These include the number, complexity, identity, and relationships of those essential pathways which are rate limiting and thus serve as developmental “timers.” There are several possible reasons for this lack of information. First, it has been assumed that because the stages of a developing system are in sequence, the rate-limiting pathways for consecutive stages are also in sequence, an assumption which has recently been questioned in at least one system (Sol1 and Waddell, 1975; Soll, 1979). Second, it has been tacitly assumed that all essential processes for the genesis of a particular developmental stage are rate limiting, an unlikely possibility for developmental changes involving large numbers of gene functions. Finally, few methods have been available for first distinguishing and then analyzing those essential processes which are also rate limiting for consecutive stages in a developing system. In a recent report, a formal set of methods were developed for investigating the relationships of develop* To whom all correspondence should be addressed: Department Zoology, University of Iowa, Iowa City, Iowa 52242.

of

73 0012-1606/83/0100’73-19$03.00/O Copyright All rights

0 1933 by Academic Press, Inc. of reproduction in any form reserved.

74

DEVELOPMENTALBIOLOGY

be assessed. This information in turn can be used to determine more definitively the relationships of timers for consecutive developmental stages, and affords insight, in some cases, into the characteristics of timers under different environmental conditions. In an accompanying article (Varnum et al., 1983), the method is applied to the rate-limiting pathways for the consecutive stages of slime mold morphogenesis. A RIGID DEFINITION

OF A DEVELOPMENTAL

TIMER

A “developmental timer” will be rigidly defined as the rate-limiting pathway to a speciJic developmental stage under a single set of environmental conditions. This definition does not address timer complexity (i.e., the number of components in sequence along a single timer). It also does not preclude the possibility that the identity of the rate-limiting pathway for a stage may change when an environmental variable is changed (see later section on “identity change model”). Finally, it does not enumerate the number of stages regulated by a single timer (i.e., it does not distinguish between single and multiple timer models). A distinction must also be made between those processes which are essential for the genesis of a developmental stage and those processes comprising a timer (Soll, 1979). In a complex developmental system, large numbers of gene functions may be essential for the genesis of a stage and many may function in parallel, but it is only the slowest sequence of essential functions, or the slowest necessary “pathway,” which is rate limiting under a particular set of environmental conditions and which therefore represents the timer for the stage. It should be noted that this interpretation does not imply a special regulatory function or cuing role for a timer. This possibility must be tested. A METHOD FOR DETERMINING MINIMAL

TIMER COMPLEXITY

The “complexity” of a timer is defined as the number of individual processes (timer “components”) functioning in sequence along the rate-limiting pathway for a developmental stage under a single set of environmental conditions. The following method (diagramed in Fig. 1) distinguishes the minimum number, sequence, and lengths of components along a timer for a developmental stage (D.S.): two conditions are defined in which a single environmental parameter is varied and in which the time to the single developmental stage is altered. Under one condition, the time to the developmental stage is shorter than that under the second condition. The former condition is referred to as the “short condition” (S), and the timer under the short condition is

VOLUME95, 1983

referred to as the “short timer.” The latter condition is referred to as the “long condition” (L), and the timer under the long condition is referred to as the “long timer.” In describing this method, temperature will be employed as the single varied environmental parameter, but it should be kept in mind that a small change in any environmental parameter causing a change in the time to a stage can be employed (e.g., Mercer and Sol1 (1980) employed ionic strength to examine the complexity of the aggregation timer in D. discoideum). Development is initiated in a number of parallel cultures under short and long conditions (Fig. 1). Then, at short time intervals prior to the genesis of the developmental stage, cultures under the short condition are shifted to the long condition (S - L shift), and cultures under the long condition are shifted to the short condition (L - S shift). Total time to the developmental stage is measured for each shifted culture. Total time includes the time prior to the shift (under the first condition) plus the time following the shift (under the second condition). Although there are several ways to analyze the data obtained, this analysis will be limited to a single graphical method (Fig. 1). The total time to the developmental stage is plotted as a function of the time of the shift on separate plots for shifts from the short to long condition and for shifts from the long to short condition. In the former case, the abscissa of the plot represents the length of the short timer, and in the latter case it represents the length of the long timer. The minimum number of components comprising the plot, the sign and magnitude of the slopes of the components, the transitions or discontinuities between components, and the developmental times at which each component begins and ends can then be interpreted in terms of timer complexity. In the following sections, the possible relationships between short and long timers will be examined for a single developmental stage, and the hypothetical data which one obtains from the reciprocal shift experiment for each relationship will be presented. The complexity of a timer will be discussed in terms of timer components. If a timer is uniformly sensitive to a change in an environmental parameter along its entire length, then it will be considered a single-component timer-i.e., by the reciprocal shift experiment, only one sensitivity and therefore only one component is discernable along the entire length of the timer. If a timer exhibits two sensitivities along its length, then it will be considered a two-component timer-i.e., by the reciprocal shift experiment, two different sensitivities and therefore two components in sequence along the length of the timer are discernable. We will limit our detailed discussion of complex timers to two-component timers. Our discussion will not be complete, but

Timers

R. SOLL

DAVID

75

in Development

RECIPROCAL SHIFT METHOD FOR ANALYSING TIMER COMPLEXITY I.

Two conditions are developed timing to a developmental

SHORT CONDITION YEi

oi

: 1

: 3

: 4

' 4

' i,

: .: 7 8

sy

10

from The condition scored.

short total plus

Sy-(?I-*

L!-(?)-b

D.S.

D.S.

so+

S~-;?)-~ Lt-'?\-.

3.

l D.S

: 9

LONG TO SHORT SHIFTS (L+S\

SHORT TO LONG SHIFTS (S+L)

2.

long

IL1 : 2

Shifts are performed at time intervals to long and long to short conditions. time to the stage (time under first time under second condition) is then

1.

and

Ds

LONG1 TIMER0

D.S.

D.S

LO4

S++?+

soy L”?\

III.

short (D.S.).

I.51

LONG CONDITION

II.

for stage

D.S.

D.S.

Total developmental of time of shift conditions (S+L)

for and

time is plotted as a function shifts from short to long long to short conditions(L-S).

1,,,,,,,,,,)

z9 -0 0l-7 Ub z I=5 0C-JTE-I

0

SHORT TIMER

1 2

3 4 5 LONG

6 7 8 9 10

TIMER

TIME OF SHIFT IV.

Each plot is then analysed for number of components, slopes of components, absolute times of oriqins and termini of components, and discontinuities between components. Timer interpretations are then made according to the combinations of these characteristics for the two plots.

FIG. 1. Scheme of the reciprocal

shift experiment

it should develop in the reader an awareness of the data sets which may be obt,ained by reciprocal shift experiments and the most probable interpretations of these data sets in terms of timer complexity.

for analyzing

timer complexity.

A. A Single-Component

Timer

In the original method for analyzing of timers (Soll, 1979), the simplifying

the relationships assumption was

76

DEVELOPMENTAL BIOLOGY 012345

VOLUME 95, 1983

B. Multiple-Component

Timer

If a timer is composed of two components with different sensitivities to the environmental condition which is being varied, and if the identities of the components remain unchanged under both the short and the long condition, then one obtains a plot composed of two straight lines in sequence and with different slopes. The transition point at which a change in slope occurs represents the discrete transition between components. There is a variety of possible characteristics for the two components: TIME OF SHIFT FIG. 2. Model and hypothetical data for a single-component timer. The equivalency map in the upper portion of the figure represents an irreversible timer uniformly expanding in the short (S) to the long (L) direction. The numbers along the timer represent arbitrary time units. The lower graphs represent hypothetical data for shifts from and the long to the short the short to the long condition (S -L) condition (L-S). The axes are graduated in arbitrary time units equal to those along the timers in the equivalency map. The vertical axis represents total time to the developmental stage (time under first condition plus time under second condition). See Fig. 1 for the details of the shift experiment.

made that each timer was uniformly affected along its entire length by the change in temperature employed, and that the identity of the timer did not change when the temperature was changed. If a timer is uniformly affected and does not change its identity when the temperature is varied over a short range, then one obtains a straight plot with a single negative slope when shifts are made from the short to the long condition, and a straight plot with a single positive slope when shifts are made from the long to the short condition. In Fig. 2, we have presented data for a hypothetical timer which extends 5 time units under the short condition and 10 time units under the long condition, and which therefore exhibits a uniform short to long equivalence of 0.5 (“equivalence” in this case is calculated by dividing the length of the short timer by the length of the long timer). Since timer components are discriminated by differences in their sensitivities to a change in an environmental parameter, one must keep in mind that a timer which in fact is composed of two components which are in sequence but which exhibit identical sensitivities to the particular environmental change employed will be interpreted as a single-component timer in this analysis. It is therefore worthwhile to repeat the interval shift experiment, varying a second environmental parameter when a uniform effect is initially obtained in order to verify a single-component timer interpretation.

1. One insensitive, one sensitive. In Fig. 3A, the short timer is composed of two components each 2.5 time units long. The first is insensitive to the change in the environmental parameter and is therefore 2.5 time units long under the long condition as well; the second component is sensitive and exhibits a short to long equivalence of 0.33. Therefore, the last 7.5 time units of the long timer represents a uniform expansion of the last 2.5 time units of the short timer. Both the plot for shifts from the short to long condition and the plot for shifts from the long to short condition are composed of two straight, connecting lines with different slopes. In the former case the first slope is zero and the second slope is negative; in the latter case, the first slope is zero and the second positive. Very clear transition points occur at 2.5 time units in each plot. If a timer is composed of two components with reversed characteristics (i.e., the first component is sensitive and the second insensitive), one again obtains plots composed of two straight lines with clear transition points. For shifts from the long to short condition, the first slope is negative and the second zero; for shifts from the long to short condition, the first slope is positive and the second zero. 2. Both components sensitive. If a timer is composed of two components both sensitive to the change in the environmental parameter and sensitive in the same direction (i.e., both expand under the long condition and therefore both exhibit short to long equivalences of less than l.O), then one obtains plots for shifts from the short to long condition and from the long to short condition composed of two straight, continuous lines, both possessing the same slope sign but different slope values. In the hypothetical situation depicted in Fig. 3B, the first component exhibits a short to long equivalence of 0.67, and the second component exhibits an equivalence of 0.4. The signs of the slopes of the lines composing the plot for shifts from the short to the long condition are both negative, and the signs of the lines composing the plot for shifts from the long to the short condition are both positive.

DAVID

R. SOLL

Timers

77

in Development

A. 012345 SC:: I ! I

LI

0

.

1

:+r, I ; ‘\ -:;y--, ,. I 1. . . . 2

I

3

4

t,

0~0123456709lO SHORT TIMER

5

.

;

;

6

7

0

1

I,,,,,,* LONG

:I,

9

10

TIMER

TIME OF SHIFT

C.

S--L

0

1

:2

3

4

5

0

1

:2

3

4

5

6

7

8

2

3 4 LONG

>‘., 9 10

L-S

0

1

5 6 7 TIMER

8

9

10

TIME OF SHIFT

01

2345678910 LONG

TIMER

TIME OF SHIFT FIG. 3. Models and hypothetical data for two component timers. The equivalency maps of the two components are presented in the upper portion of each panel. The en,ds of the arrows in these maps represent the ends of the single components. See Fig. 1 for the details of the shift experiment. The “direction of sensitivity” in panels B and C refers to the direction of expansion (i.e., in the short to the long or the long to the short direction).

In the situation depicted in Fig. 3B, the first component is less sensitive to the change in the environmental parameter than the second component. Reversing the level of sensitivity so that the first component is more sensitive than the second component does not alter the relationships of the signs of the slopes: for shifts from short to long, both slopes are again negative, and for shifts from long to short, both are again positive. Note that if both components are equally sensitive, one would obtain th.e results described in Fig. 2 for a single-component timer. If a timer is composed of two components, both sensitive to a change in the environmental parameter, but in opposite directions, then one obtains plots for shift experiments which are composed of two straight, continuous lines, the slopes ‘of which exhibit opposite signs. In the example depicted in Fig. 3C, the first component

expands under the long condition, and the second component contracts. The first component exhibits a short to long equivalence of 0.25, but the second component exhibits a short to long equivalence of 1.5. The signs of the slopes of the lines comprising the plot for short to long shifts are negative and positive, in that order, and the signs of the slopes of the lines comprising the plot for long to short shifts are positive and negative, in that order. If the sensitivities of the components were reversed, then all characteristics of the two components of the plots would be reversed. C. Timer Reversibility A shift in a single environmental parameter could result in the reversal, or “erasure,” of progress along a timer component (Mercer and Soll, 1980; Sol1 and

SHORT TIMER

1

2

3 4 5 6 7 LONG TMER

8

9

10

f8

8

13

14

15

gm u.49

5

6 ‘i:5

TIME OF SHIFT

7 6

7

9

: \:

:\

:y:

0;

\

SHORT TIMER

S--L

O123456789~lGL0123456789W

g11

$2

TIMER

012345 1 -1 -. :-r-\ ‘\ -. .,-=--,---,

L:

s;

C.

ll-

0

I1 I,,,,,,* 012345678910 LONG

TIME OF SHIFT

5-

L-S

10

12-

13-

SHORT TIMER

?6-

8-

9-

lo-

ll-

12-

13-

14-

15-

?-

;

8-

:

012345 St : : : : p=.Z~x\\‘.--~--==~-~~ ___ . 1 : : : -: 7-7y--r-:-,

:?

14-

15-

t

: L-s

:‘:-r-:

TIME OF SHIFT

LONG

\;wN==--

0123456789lO

5- -/

lllo6897-

1312-

: *

f

012345 SW:

TIMER

:

:

-‘--- ------r-

*

.

r.

I

:

I

1

FIG. 4. Models and hypothetical data for single-component, reversible timers. The origins of the dashed arrows in the models in the upper portion of each panel represent the shift points along the timer under the initial condition; the termini of the dashed arrows represent the continuation points under the second condition. The vertical dashed lines in the graphs in the lower portions of the panels represent discontinuities in the plots. See Fig. 1 for details of the shift experiment.

0m

012345

=i=

012345 SI : /’ ! __-! : * : ./--L-Lp r : f : ! : : O12345678910LO12345676910

A.

DAVID

R. SOLL

Timers

Waddell, 1975; Waddell and Soll, 1977). Reversibility may occur for shifts in both the short to the long and the long to the short direction, or it may be limited to a shift in one direction only. These three possibilities are graphically described in Figs. 4A-C, respectively, for a single-component Itimer. In the situation in which the timer is reversed in both directions (Fig. 4A), shifts from short to long conditions result in a plot composed of a single component with a slope of f1.0, beginning at 10 hr and ending at 15 hr total development time. Note that the times for shifted cultures are greater than those for cultures maintained continuously under the long condition (i.e., without shifts). In this case (Fig. 4A), shifts from the long to the short condition again result in a plot compos’ed of a single component with a slope of +l.O, but beginning at 5 hr and ending at 15 hr. Note again that the times for cultures shifted during the latter half of progress of the long timer are greater than the times for cultures maintained continuously under the long condition. For situations in which reversibility occurs in onl:y one direction (Figs. 4B and C), only the plots for the shifts in the direction causing a reversal result in developmental times in excess of that for cultures maintained continuously under the long condition. In these cases, shifts in the opposite direction, which do not cause reversals, result in the standard plots for single-component timers (compare L to S plot in Fig. 4B and S to L plot in Fig. 4C with plots in Fig. 2). By comparing the sets of plots for reciprocal shifts in Figs. 4A-C with the plots in Fig. 2, it becomes obvious that reversible and irreversible timer components are readily distinguishable. If a timer is composed of two components and one or both components are reversible, there are 123 different combinations dependent upon which component, or components, are revers#ible, the direction or directions of the shifts which cause reversibility, and the sensitivities of the two components (Table 1). For this reason, we will limit our discussion to the few unique features of reversible, complex timers, and describe in detail two specific cases (Figs. 5A and B). For the case in which the first component of a two-component timer is reversible, and the second irreversible, a discontinuity exists in the plot for shifts in the direction of reversibility. In the example depicted in Fig. 5A, the first component is reversible in both directions and expands in the short to the long direction; the second component is irreversible and insensitive. For shifts from the short to the long condition, one obtains a plot composed of two discontinuous lines (the discontinuity in the plot is depicted by a vertical, dashed line), the first with a positive slope of one representing the reversible component and terminating, in this specific case, at 12.5 time units, 2.5 units longer than the time

in Development

79

for a system maintained continuously under the long condition. The second line is horizontal at 5 hr. Shifts from the long to the short condition give a similar plot, with an obvious discontinuity (Fig. 5A). Reversibility in the first of two components is therefore an easy timer feature to identify because of the radical discontinuity in the shift plots and has been observed for the first component of the aggregation timer of the slime mold D. discoideum for shifts from the short to the long condition employing ionic strength (Mercer and Soll, 1980) or temperature (Varnum et al., 1983) as the environmental variable. One does not obtain a discontinuity in the reciprocal shift plots for timers composed of two components in which the second component is reversible. In the case depicted in Fig. 5B, the first component is irreversible and insensitive, and the second component is reversible in both directions and expands for shifts in the short to the long direction. For shifts from the short to the long condition and from the long to the short condition, one obtains plots with similar characteristics, an initial straight line with a slope of zero followed by a straight line with a positive slope. Again, plots of the reversible components are always characterized by late values which exceed the developmental time of cultures maintained continuously under the long condition. It should be noted that in this section, we have treated reversibility as an all-or-none event. However, it is quite possible that reversibility may be incomplete for a given component, leading to an overestimate of complexity. If shifts at different times lead to variability in the reinitiation point, especially for shifts in one direction only, it would indicate that a single component is incompletely reversed by the shift. This possibility will not be dealt with graphically in this paper, but it must be kept in mind when developing models for shift data. D. Addition

Models

It is also possible that the increase in the time to a developmental stage under the long condition is due to the addition of a new component at the origin or terminus of the short timer, or to the insertion of a new component along the length of the short component. Again, a number of possible models exist in each category depending upon the direction of sensitivity and the reversibility of the individual components, just as in the case of the reversibility models (see Table 1). Therefore, we have limited our detailed discussion to simple examples of each case. 1. Addition at the origin. In the case depicted in Fig. 6A, the short timer is composed of a single insensitive component which is equivalent to the last 5 time units

80

DEVELOPMENTALBIOLOGY

VOLUME95, 1983

TABLE 1 THE NUMBEROF POSSIBLEREVERSIBILITYMODELS One-component timer Reversibility S-LandL-S S - L only L - S only 3

3

Two-component timer First component reversible, second irreversible Reversibility of first component S-LandL-S S - L only L - S only

Sensitivity of first component insensitive sensitive, + sensitive, 3

x

3

x

Sensitivity of second component insensitive sensitive, + sensitive, 3

-(2 X 2)” = 23

x

Sensitivity of second component insensitive sensitive, + sensitive, 3

-(2 X 2)” = 23

First component irreversible, second reversible Reversibility of second component S-LandL-S S - L only L - S only 3

Sensitivity of first component insensitive sensitive, + sensitive, x

3

Both components reversible Sensitivity of first component insensitive sensitive, + sensitive, -

Sensitivity of second component insensitive sensitive, + sensitive, -

Reversibility of first component S+LandL-S S - L only L - S only 3

x

3

x

Sensitivity of second component insensitive sensitive, + sensitive, 3

x

3

-(2 x 2y = 77

Total = 126 Note. The symbols S - L and L - S represent the direction of shift, from the short to the long condition in the former case and from the long to the short condition in the latter case. A positive (+) sensitivity represents expansion in the short to the long direction, and a negative (-) sensitivity represents contraction in the short to the long direction. a Insensitivity cannot occur simultaneously in both directions. In addition, both components cannot contract simultaneously in the short to the long condition. Therefore, four possibilities (2 X 2) must be subtracted.

of the long timer. The first 5 time units of the long timer represent a component which must be progressed through only under the long condition. Two alternate plots are possible for shifts from the short to the long condition, depending upon the reversibility of the short timer component. If the component is irreversible, shifts from the short to the long condition any time after the onset of the developmental program result in a uniform total developmental time of 5 units (Fig. 6A, filled circles in S - L plot). Alternatively, if progress along the short timer component is reversible, shifts from short to long conditions will result in a plot beginning at 10 time units and ending at 15 time units

with a positive slope of 1.0 (Fig. 6A, open circles in S L plot). It should be noted that total developmental time for a shifted culture is longer than that for a culture continuously maintained under the long condition and that the results for shifts in the short to the long direction are indistinguishable from a simple reversibility model (compare results of short to long shifts in Figs. 4A and 6A). In the case depicted in Fig. 6A, two alternative plots are also possible for shifts from the long to the short condition, again depending upon the reversibility of the component being traversed. First, if the initial component of the long timer is irreversible, shifts from the

DAVID R. SOLL

Timers

in Deuelopment

81

82

DEVELOPMENTAL BIOLOGY

VOLUME 95, 1983

A. 012345 s I .._. :.!-I ,::<:z=::., L,

:

B I -.--.-----. ‘.:-.:.::>.:. ) r:“.

:

:-:-r-

B

012345676910

ITI,

OmO12345676910 SHORT TIMER

I

II

III*

LONG TIMER

TIME OF SHIFT

C.

B. 01 !jI I

: I

2345 i : I

I 1 I LL:::::::::

i I1

I

‘:v? s O1?9?5

B

i

012345B7*810B

L0l-l

. . . . . . . . . . . . . . . . . . .D++++* ....................

L @,+.

-.

.................

L @,+-H-b..

-

\=pj~

0

1 2 3 4 SHORT TIMER

5

I

I

!

1

0

1

2

3L&s,,;ER7

I

I

I

I1

g~(~

I

8

9

10

TIME OF SHIFT I

01234501234507B910 SHORT TIMER

I

s,

I

LONG

I

I

I,

I-

TIMER

TIME OF SHIFT FIG. 6. Models and hypothetical data for timers increased by the addition of a new component under the long condition. (A) Addition of the new component at the origin. Dashed lines represent equivalency of short timer with second component of long timer. For S - L, the short timer may be irreversible (0) or reversible (0). For L - S, the new component may be irreversible (A) or reversible (A). (B) Addition of new component at the terminus. Dashed lines in model represent equivalency of short timer and first component of long timer. Vertical dashed lines in the graphs represent discontinuities in the plots. For L - S, new component may be irreversible (a) or reversible (A). (C) Insertion of a new component along the length of the short timer. Dotted portion of models represent inserted component. The three long timers represent insertions at different points, indicated by circled numbers. Vertical dashed line in S - L graph represents a discontinuity in the plot, For L - S, the new component may be irreversible (A) or reversible (A).

long to the short condition will result in a plot composed of a single horizontal line at 10 timer units (Fig. 6A, open triangles in L - S plot). Alternatively, if the first component of the long timer is reversible, shifts from the long to the short condition will result in a plot composed of two continuous lines, the first with a positive slope of 1.0 and the second with a slope of zero. If the component added to the origin of the timer under the long condition is sensitive to an increase in temperature, one obtains the same results for short to

long shifts as those for the case in which the added component is insensitive. For shifts from long to short conditions, however, one obtains the results observed for a two-component timer with the first component sensitive, if the added component is irreversible. It is obvious that there are a number of combinations for the origin-addition model, 16 in fact, depending upon the direction of sensitivity and reversibility of the timer components. In some cases, the addition model results in data similar to the more simple two-component mod-

DAVID R. SOLL

els for long to short shifts; however, in all cases, some aspect of the data resulting from shifts in the short to long direction is different. Therefore, distinctions can be made between these two models so long as shifts are performed in both the short to long and long to short direction. If the short timer component is sensitive to a temperature change, expanding in the short to the long direction, and is irreversible, one obtains a plot composed of a single line with a negative slope for shifts from the short to long condition. It should be noted that the plot begins at a time point which is less than that for cultures maintained continuously under the long condition. If sensitivity is in the opposite direction, expanding in the long to the short direction, one obtains a plot which is composed of a single line with a positive slope for shifts from the short to the long condition and which terminates at a time point which is less than the total time for cultures maintained continuously under the long condition. If the short timer component is sensitive and reversible, one obtains a plot composed of a single line with a positive slope regardless of the direction of sensitivity. For shifts from long to short conditions, one obtains plots composed of two continuous lines with slopes dependent upon the direction of sensitivity and reversibility of the short component. In the case depicted in 2. Addition at the terminus. Fig. 6B, the short timer is composed of a single, insensitive component which is equivalent to the first 5 time units of the long timer. The last 5 time units of the long timer represent a component which must be progressed through after the short timer but only under the long condition. For shifts from short to long conditions, one obtains a plot composed of a single horizontal line at 10 time units. If the short timer is sensitive, one would still obtain a plot composed of a single line, but the slope would be positive or negative depending upon the direction of sensitivity. For shifts from long to short conditions, two alternative plots are possible, depending upon the reversibility of the second component. First, it is possible that once the terminal component of the long timer has been initiated, it is irreversible. In this case, shifts from the long to short condition result in a plot composed of two horizontal and, therefore, discontinuous lines, both with zero slope, the first at 5 units and the second at 10 units (Fig. 6B, open triangles). The discontinuity between the two lines is drawn as a dashed vertical line with a slope of infinity. Alternatively, it is possible that when shifted from long to short conditions during the progress of the second component of the long timer, further progress terminates and the developmental stage is immediately generated. In this situation, one obtains a plot composed of two adjoining lines: the first is horizontal at 5 time

Timers

83

in Development

units, and the second has a positive slope of one (Fig. 6B, filled triangles). If the short timer is sensitive to the change in the environmental parameter, the slope of the initial component would not be zero, but would rather be positive or negative, depending upon the direction of sensitivity. If the added terminal component is sensitive to a shift from long to short conditions and is irreversible, one would obtain a plot composed of two discontinuous lines, but the second would possess either a positive or a negative slope, depending upon the direction of sensitivity. 3. Insertion model. In the case depicted in Fig. 6C, a shift from the short to the long condition results in the cessation of progress along the short timer component, initiation and completion of a new component, and, finally, renewal and completion of the unfinished short timer component. In this case, progress along the short timer component is conserved under long conditions until the completion of the inserted component. Shifts from the short to the long condition would lead to a plot in which all points would fall on a horizontal line at 10 time units, except for the very last one, which would be at 5 time units. This is schematized in Fig. 6C: the time of shift from the short to long condition is denoted by the circled numbers along the short timer, and the corresponding timer models after the shift to the long condition are presented below. When initiated under the long condition, the insertion component would begin at the origin. The possible plots for shifts from the long to short condition, in this case, would again depend upon the reversibility and sensitivity of the components, just as in the case of the origin-addition model (see previous section for a discussion of possible plots for shifts from the long to short condition for the origin-addition model). If an insertion does occur, it is more probable that it will be located between the terminus of one component and the origin of the subsequent component. This would result in a two-component plot for short to long shifts and a three-component plot for long to short shifts. It also accentuates a prevalent characteristic of addition models: in many cases, the plot for shifts from the long to the short condition contains one more component than the plot for shifts from the short to the long condition. E. Timer Identity

Change

Since a developmental timer represents the rate-limiting pathway to a developmental stage under a single set of environmental conditions, it is possible that a change to a second set of conditions results in a change in the identity of the rate-limiting pathway. This

84

BIOLOGY DEVELOPMENTAL

012345 SI : :,: /x- ! 4 _ /‘--<: : : : LyT : O12345676910L012345676910

;

:

:

012345 SI : : ppy~w , ; ‘:

:

: __ \z=-=-----; .: -pyy-r

_

;-,

S-L

i

5J O;lbl; SHORT

0

1

TIMER

2

I I I r I I I*

3

4

LONG

5

6

7

6

9

10

TIMER

TIME OF SHIFT

FIG. 7. Model and hypothetical data for an identity change after shifts in either direction for a reversible timer. The origins of the dashed arrows in the models represent the shift points along the original timer; termini of the dashed arrows represent continuation points under the second condition. In this particular case, the timer arrows (solid and horizontal) in the models represent different ratelimiting processes under the short and the long conditions and hence have no equivalencies. Vertical dashed lines in the graphs represent discontinuities in the plots.

change could conceivably occur in two ways. First, a shift from either the short to the long condition or the long to the short condition could cause a complete reversal of the pathway in progress and initiation of a different rate-limiting pathway under the second condition (Fig. 7). The results of reciprocal shifts in this case are indistinguishable from the simpler case of reversibility without a change in identity (compare Figs. 7 and 4A). In fact, all cases of identity change for oneand two-component timers involving reversibility in both directions cannot be distinguished from cases of reversibility in both directions not accompanied by identity changes. Alternative and more likely cases of identity change are described in Figs. 8A and B. In these cases, parallel pathways “a” and “b” are both necessary for the genesis of the developmental stage, but pathway “a” is the last to be completed and therefore the rate-limiting pathway under the short condition, whereas pathway “b” is the last to be completed and therefore the rate-limiting pathway under the long condition. In the situation depicted in Fig. 8A, pathway a is insensitive and pathway b is sensitive to a change in temperature. For shifts from the short to the long condition, one obtains a plot composed of two adjoining lines, the first with a negative slope and the second with a slope of zero. For

VOLUME95,1983 shifts from the long to the short condition, one again obtains a plot composed of two adjoining lines, the first with a slope of zero and the second with a positive slope. In the situation depicted in Fig. 8B, both pathways a and b are sensitive, but b far more than a. For shifts from the short to the long condition, one obtains a plot composed of two adjoining lines, both with negative slopes. For shifts from the long to the short condition, one obtains a plot composed of two adjoining lines, both with positive slopes. A very interesting situation occurs if both pathway a and pathway b are necessary for the genesis of the developmental stage, both are sensitive to a temperature change in the same direction, pathway a is rate limiting under both short and long conditions for cultures continuously maintained under either of these conditions, but pathway b is reversible when shifts are performed in either direction (Fig. 8C). In this situation, pathway b is rate limiting for shifts midway along both the short and the long timer. For shifts from the short to the long condition and from the long to the short condition, pathway a is rate limiting for early and late shifts, but not for intermediate shifts. This situation results in plots composed of three lines. The first and third lines exhibit the same slope, but the second line exhibits a different slope. For shifts in both directions, one obtains a discontinuity between the second and the third line. DISTINGUISHING

BETWEEN MODELS

OF TIMER COMPLEXITY The methods which we have developed in this article do not afford one with an unequivocable interpretation of timer complexity for a particular developmental stage. Rather, they afford one with the best interpretation or interpretations in the absence of alternative methods for doing so. To distinguish between most of the possible models described in the preceding sections, shifts must be performed in both directions, and careful analysis must be made of the resultant plots for (1) the number and length of components, (2) the developmental times at which each component begins and ends, (3) the signs and magnitudes of the slopes of the components, and (4) continuity between components along the plots. For one- or two-component timers, the combinations obtained for shifts from the short to the long condition and from the long to the short condition allow one to minimize the number of interpretations to one or, at most, two. In some situations, subtle differences in the data sets allow distinctions to be made between model interpretations. For instance, both a two-component system in which both components are sensitive in the same direction (Fig. 3B) and an identity change

DAVID R. SOLL

Timers

85

in Development

A. EOUIVALENCE

s

AL % iTiT 4.0

012345 Oc--e-e-c-c, bOT 012345 a-

Lb,

:

:

:

:

:

:

; ;

i *

I

I

012345670910

\,I

0;01234537897

I

I

I

I

I-

LONG TIMER

SHORT TIMER

TIME OF SHIFT

C.

8. EOUIVALENCE

14 is

23 b

a”m

S bOT

EGUIVALENCE b % &I 115

012345

S

01234567

La,:::::::-, bl,,

.

0123456789IO

r

u 1 r

a-

bOm

, a $9

0 a)

1 :

2 :

3 :

4 :

5 :

6 :

7 ;

0 :

9 ;

10 .

LbOm

L-s

S-L

0123450123456789lO SHORT TIMER

I 1 1 I I I I I

I *

$4J//~

LONG TIMER

TIME OF SHIFT 0~012345378970 SHORT TIMER

I,,,I,I

,,I* LONG TIMER

TIME OF SHIFT FIG. 8. Models and hypothetical data for an identity change after shifts for irreversible timers. Pathways a and b in the models represent two parallel pathways essential for the genesis of the developmental stage. The longer of the two under a single set of environmental conditions is rate limiting. The equivalencies presented in the upper right-hand corner of each panel represent the ratios of the lengths of each pathway under long and short conditions. In panel C, pathway b is not rate limiting under either short or long conditions when no shifts are performed, but becomes rate limiting after shifts at intermediate times because of its reversibility. Vertical dashed lines represent discontinuities in the plot.

in which the short timer is sensitive (Fig. 8B) result in experimental plots which are roughly similar: in both, shifts from the short to long condition result in a plot composed of two adjoining lines both with negative slopes, and shifts from the long to the short condition result in a plot composed of two connecting lines both

with positive slopes. However, a subtle difference does exist between the results of the two situations which allows one to distinguish between them. If one compares the relative magnitudes of the slopes of the components in each plot, one finds that for the two-component systern, the absolute slope of the first portion of the plot

86

DEVELOPMENTALBIOLOGY

is less than that of the second portion for shifts from the short to the long condition and for shifts from the long to the short condition (Fig. 3B). However, for the identity change in Fig. 8B, the absolute slope of the first portion of the plot is greater than that of the second portion for shifts from the short to the long condition, but the absolute slope of the first portion of the plot is less than that of the second portion for shifts from the long to the short condition. Subtle differences also may exist in the values of the origins and termini of data obtained from the reciprocal shift experiment which allow discrimination between models. For instance, an irreversible, sensitive, singlecomponent timer can be distinguished from a sensitive single-component timer which is reversible in one direction by the value of the terminus of the plot for shifts in the direction of reversibility (compare Fig. 2 with either Fig. 4B or 4C). In this case, the slopes of the plots for shifts in the direction of reversibility will also differ from those for an irreversible component. Many other subtle differences allow distinctions to be made between interpretations, but limited space does not allow us to discuss each situation. Our brief discussion should alert the reader to the types of differences which must be considered when interpreting the results of shift experiments. In previous sections, we have considered the few cases in which distinctions could not be made between models. In particular, an identity change accompanying reversibility (Fig. 7) cannot be distinguished from a simple reversible timer which does not undergo an identity change (Fig. 4A). In such cases, differing sensitivities of the rate-limiting pathways under short and long conditions to various inhibitors and environmental conditions other than the one used to affect the short and long conditions can be examined for an assessment of identity. However, it should be stressed that in most cases of one- and two-component timers, relatively clearcut model interpretations can be made. A careful comparison of the hypothetical data sets in Figs. 2 through 8 and an analysis of real data for Dictyostelium in an adjoining article (Varnum et al., 1983) should reinforce this assertion. A UNIFORM TIMER COMPONENT NEED NOT BE LINEAR We have argued that a timer component consists of a single reaction or process, uniformly affected along its length by a particular environmental perturbation, such as a temperature shift. This does not in turn require that the process or reaction be linear. If the process is logarithmic or exponential, but is uniformly sensitive to the environmental perturbation, then its expansion in the short to the long direction will be

VOLUME95, 1983

proportional along its entire length, resulting in a uniform expansion as is depicted in Fig. 2 for a singlecomponent timer. CASES OF HIGHLY

COMPLEX TIMERS

The preceding analysis allows one to investigate timers of relatively low complexity (i.e., timers composed of a minimum number of components). However, it is conceivable that the timer for a particular stage may be composed of a large number of rapidly completed components. If these numerous components exhibit diverse sensitivities to changes in temperature, then the shift experiments would generate very complex plots. In some cases, pseudoexponential or pseudologarithmic plots would be obtained. In other cases, plots composed of a squiggly line would be obtained. Fortunately, in the one developmental system to which this method has been applied, the timers have been found to be of low complexity and thus amenable to interpretation (Mercer and Soll, 1980; Varnum et al., 1983). EMPLOYING THE METHOD To ASSESS TIMER RELATIONSHIPS

Possible Timer Relationships Besides affording information on the complexity of developmental timers, the method which we have developed also allows an assessment of the relationships between timers of consecutive stages in a developing system. There are basically three simple models of timer relationships: a single timer model, in which the times of consecutive stages are regulated by a single rate-limiting pathway (Fig. 9A); a sequential timer model, in which an initial pathway regulates the time of the first stage which in turn cues the onset of a second rate-limiting pathway for the second stage (Fig. 9B); and a parallel timer model in which the rate-limiting pathway for the late stage is independent of and in parallel to the rate-limiting pathway for the early stage (Fig. 9C). Combinations of these three simple relationships are of course possible and the most important of these is the branching timer model, in which the initial component is common and the final components differ in the rate-limiting pathways to consecutive stages B and C (Fig. 9D). It should be noted that only a fine distinction exists between single and sequential timers if timers are complex, and it depends upon whether the timer for the second stage begins at exactly the same time that the first stage is generated. This distinction is not that important since in both cases the rate-limiting pathway for stage B is shared by stage C. A far more funda-

Timers

DAVID R. SOLL

87

in Development

B.

A.

‘B-C I

I

I I TIME -

I

I

C.

I

I

I TIME-

I

I

I I TIME -

I

I

D.

-B I

DC -cl I I TIME -

I

I

I

I

FIG. 9. Basic models for the regulation of the timing of consecutive stages B and C in the developing system. The horizontal (thick) axis of each arrow renresents m-ogress of the timer. Vertical axis (thin) does not represent temporal progress. More complicated models can be developed by various combinations of these basic models.

mental distinction exists between these two models and the parallel timer model. In the latter case, stage B and stage C do not share a rate-limiting pathway. As will be evident in the following discussion, the major strength of the methods which we have outlined in this report lies in the assessment of parallelism. It should also be noted that one can obtain a diverging model of branching such as the one presented in Fig. 9D, but one cannot obtain a converging model. In the latter case, the rate-limiting pathway, by definition, would follow the longest path, resulting in a single timer composed of more than one component. Finally, it must be emphasized that a description of timer relationships does not include the many pathways which are necessary but which are not rate limiting for the genesis of the developmental stages under the two sets of conditions used for the shift experiment. This point is dealt with in more detail in a previous report (Soll, 1979) and in the accompanying article (Varnum et al, 1983). The Method of Assessment To assess the relationships of the timers for the consecutive developmental stages B and C, one simply characterizes the rate-limiting pathway from the origin of the developmental program for each stage by the reciprocal shift experiment and then compares the characteristics of parallel portions of the pathways of the different stages. From this comparison, distinctions can be made between models (Figs. lOA-I). We have limited

our discussion to the timers of two consecutive developmental stages, B and C, and to a maximum timer complexity of two components. In the diagrams presented in Fig. 10, the complexity and the similarity or dissimilarity of the timer components for stages B and C are presented in the upper portion of each panel, and the applicable model, or models, as well as the excluded model, or models, are listed in the lower portion of each panel. 1. Both pathways single component. If the rate-limiting pathways for consecutive stages B and C are each composed of a single component, and both exhibit identical sensitivity to the environmental change employed in the shift experiment (Fig. lOA), then the best interpretation is that of a single timer model, although the remaining models (see Fig. 9) cannot be excluded. In our previous method for analyzing timer relationships (Soll, 1979), we characterized the sensitivity to a change in temperature by the parameter %At, the percentage change in the time to a stage between high and low temperature. In the case depicted in Fig. lOA, the % At’s for the pathways to the consecutive stages are equal (Soil, 1979). If the rate-limiting pathways for consecutive stages B and C are each composed of a single component, but they exhibit dissimilar sensitivities to the environmental change employed (Fig. lOB), then only the parallel timer model is applicable and the single, sequential, and branched models are excluded. In the case depicted in Fig. lOB, the % At’s of the timers for stages B and C would be unequal (Soll, 1979).

88

DEVELOPMENTAL BIOLOGY A

Each similar

timer one component, characIerisIics.

VOLUME 95, 1983 B

Each timer one component. dissimilar characlerrslics

CC

-C Best inlerp. single Excluded: nothing.

C.

Best interp. parallel Excluded: single, sequential. branched.

The B Iimer one component. the C timer Iwo components. The B timer and first component of C timer similar.

Q.

Best interp.: sequential Excluded: single.

E.

Best interp.: parallel Excluded: single. sequential, branched.

The B timer Iwo components, the C timer one component. The first component of B timer similar IO C Iimer. second component dissimilar.

F.

a

*

-C

-C

Best interp.: parallel Excluded: single, sequenIial. branched.

Best interp.: branched Excluded: single, sequential

The B timer and C timer both Iwo components; first and second components similar.

H.

The B timer and two componenIs; second dissimilar.

C timer both firs1 similar,

--B

--B \\

a

-C

Best interp.: single. Excluded: nothing.

I.

The B timer two components. the C timer one component. The first component of B timer dissimilar IO C timer, second component similar or di$similar

--B

--B

G.

The B timer one component. the C timer Iwo components. The B timer and first component of C timer dissimilar.

Best interp.: branched Excluded: single, sequential

The B timer and C timer both Iwo components; first dissimilar, second similar or dissimilar

Best interp.. parallel Excluded: single, sequential, branched

FIG. 10. Method for assessing timer relationships for consecutive stages B and C in a developing system. Each horizontal arrow represents a single timer component. Angled equivalence symbol (Q) indicates that the parallel components for stages B and C exhibit the same sensitivities to environmental change. Dashed equivalence symbol (s) indicates that the parallel components for stages B and C exhibit dissimilar sensitivities.

DAVID R. SOLL

Timers

2. One pathway single component, one pathway two components. If the rate-limiting pathway for stage B is composed of a single component and the pathway for stage C is composed of two components, and if the pathway for stage B exhibits the same sensitivity as the first component of the pathway for stage C, the best interpretation is the sequential timer model (Fig. 1OC). The single timer model can be excluded in this case, but the parallel timer model cannot, since the first component of the C pathway and the B pathway may represent different rate-limiting processes with similar sensitivities to the environmental change. If the rate-limiting pa.thway for stage B is composed of one component and the pathway for stage C is composed of two components, and if the first component of the pathway for stage C exhibits a different sensitivity than the pathway for stage B (Fig. lOD), the best interpretation is the parallel timer model; all other models are excluded. If the rate-limiting pathway for stage B is composed of two components and the pathway for stage C is composed of a single component, and if the first component of the pathway for B exhibits the same sensitivity as the pathway for C, the best interpretation is a branched model (Fig. 10E). In this, case, the branch would consist of the second component of the pathway for stage B which would emanate from the initially common pathway to C. Both the single and the sequential timer models are excluded, but the parallel timer model cannot be since the first colmponent of the B pathway and the C pathway may represent different rate-limiting processes for the respective stages with similar sensitivities. If the rate-limiting pathway for stage B is composed of two components and the pathway for stage C is composed of a single component, and if the first component of the pathway for B exhibits a different sensitivity than the pathway for stage C, only the parallel timer model is applicable; the single, sequential, and branched models are excluded in this case (Fig. 10F). 3. Both pathways two components. If the rate-limiting pathways for both stages B and C are composed of two components, and if the first and the second components exhibit similar sensitivities, respectively, the best interpretation is the sequential timer model, but one cannot exclude any of the major models (Fig. 10G). If the rate-limiting pathways to stages B and C are both composed of two components and if the first but not the second components exhibit similar sensitivities, the best interpretation is a branched timer model with the branch point at the transition between the first and the second components (Fig. 10H). Both the single and the sequential timer moldels can be excluded in this case, but the parallel timer lmodel cannot be since the first components of the two Ipathways may represent differ-

in Development

89

ent rate-limiting processes for the respective stages but with similar sensitivities. If the rate-limiting pathways for stages B and C are both composed of two components, and the first components exhibit dissimilar sensitivities, then regardless of whether the second components exhibit similar or dissimilar sensitivities, only the parallel timer model is applicable; all other models are excluded (Fig. 101). It should be emphasized that for all cases in which the initial parallel components of the rate-limiting pathways for stages B and C exhibit different sensitivities, only the parallel timer model is applicable and all other timer models are excluded. It should therefore be obvious that the strongest aspect of this method is the assessment of parallelism. GENERAL APPLICATION OF THE PROPOSED METHODS A number of biochemical and genetic methods have been developed for the purpose of dissecting the programs of viral assembly (e.g., Jarwick and Botstein, 1973; Wood et ah, 1968), the cell cycle (e.g., Hartwell, 1974; Pringle, 1978), and both simple and complex developing systems (e.g., Coote and Mandelstam, 1973; Frankel et ab, 1976; Loomis et al., 1977; King and Mohler, 1975). These methods are useful in defining the temporal order and dependencies, or independencies, of gene functions essential for the particular developmental program, but they do not discriminate between those gene functions which are rate limiting and those which are not. Of course, if all gene functions essential to the genesis of a developmental stage were ordered in a single dependent sequence, then by definition, they would all be rate limiting. At least in the cases of viral development and the yeast cell cycle, parallel pathways of dependent events have been established (e.g., Wood et ab, 1968; Nurse 1976; Hartwell et al, 1974), indicating that even in simple systems, the developmental program may not consist of a single dependent sequence of gene functions. In addition, the large number of gene functions estimated to be involved in several more complex developing systems (e.g., Mohler, 1977; Loomis, 1978) indicates that it is unlikely that essential gene functions in these systems are ordered in a single sequence. We therefore have concentrated our efforts in developing a set of simple conditional methods for analyzing only those events which are rate limiting in a developing system. In a previous report, we developed methods which were based upon the then untested assumption that the rate-limiting pathway to a developmental stage was composed of a single component and was therefore uniformly affected by an environmental change (Soil, 1979). In the present report, we have de-

90

DEVELOPMENTAL BIOLOGY

veloped a more relined set of methods which are not based on any major underlying assumption and which test the previous assumption of uniformity. These methods first provide a minimum estimate of the complexity of the rate-limiting pathway to a developmental stage, temporally define the transition points between components along the rate-limiting pathway, in select cases indicate changes in the identity of the rate-limiting events, and afford some insight into the characteristics of the rate-limiting events by virtue of their sensitivity to the environmental change employed in the shift experiments. The methods should prove to be useful in the dissection of developmental periods during which few biochemical or ultrastructural markers have been found, especially of so-called “latency” periods which follow the initiation of developmental induction and precede either the first major phenotypic change or the point of phenotypic commitment. Examples of the types of latency periods which may be amenable to these methods include the precommitment period during bacterial sporulation (Vinter, 1969), the preaggregation period in slime mold morphogenesis (Mercer and Soll, 1980), the pre-evagination period for synchronous bud or mycelium formation after release from stationary phase in Cundida albicans (Mitchell and Soll, 1979), the period preceding flagellar outgrowth during ameboflagellate transformation in Naegleria gruberi (Fulton and Walsh, 1980), G1 of the yeast cell cycle (Hartwell, 1974), the induction period of erythroid differentiation for murine erythroleukemia cells (Levinson and Housman, 1979, 1981) and developmental hormone induction periods (Tata, 1980). In addition to providing a detailed description of the rate-limiting pathway preceding each stage in a developmental sequence, the methods developed in this report can be employed to assess the relationships between the rate-limiting pathways for these consecutive stages. In particular, the methods allow one to distinguish in many cases between the single, sequential, parallel, and branching timer models (Fig. 9). As I have argued, the particular strength of the methods lies in identifying parallel timers for consecutive stages. This method should be especially useful in reexamining systems in which it has been proposed that a single regulatory pathway (i.e., a single sequence of gene functions) is basic to the developmental program (e.g., Loomis et ah, 1977) or that a single clock mechanism dictates the length of the cell cycle, especially in systems in which limiting cycle oscillators have been proposed (e.g., Klevecz, 1978; Kauffman and Wille, 1975). In the latter cases, the reciprocal shift experiment could potentially test whether the hypothesized increment periods composing the rate-limiting program of the cell cycle exhibit common characteristics.

VOLUME 95, 1983

In a more general context, investigations of timer relationships in complex developmental programs will be especially interesting in testing the previously proposed possibility that particular developmental pathways have evolved as cues for the genesis of consecutive stages in developing systems (Soll, 1979). These pathways may serve to ensure that all preparatory events for the genesis of a particular developmental stage have taken place, and together they may be viewed as the potential underlying regulatory program. This hypothesis leads in turn to several potential characteristics of temporal regulation which should be considered. First, the formation of a developmental stage may not take place as a continuum of observable events, but rather as a relatively discrete, observable event at the end of a latency period. Second, if particular pathways have evolved as rate-limiting pathways, or developmental timers, then one might expect these pathways to be rate limiting under most conditions (i.e., under both short and long conditions, thus excluding the possibility of identity changes). In this context, one would not expect the observable formation of a developmental stage to begin prior to completion of this rate-limiting pathway. Finally, one might expect timer pathways to be relatively uncomplex (i.e., to be composed of only one or a limited number of components) and perhaps to share common characteristics. In an accompanying article (Varnum et al., 1983), we have applied the methods developed in this report to the program of multicellular morphogenesis in the discoideum. It is demonslime mold Dictyostelium strated that (1) in contrast to our previous assumption (Soll, 1979), the rate-limiting pathways in this system are composed of more than one component (i.e., they are not uniformly affected by a change in temperature), but they are still relatively simple, no timer exhibiting more than four identifiable components along its entire length; (2) parallel rate-limiting pathways do exist for consecutive stages, although not as abundant as originally suggested (Soll, 1979); (3) a branch point is clearly distinguished midway in the temporal program; and (4) at least two examples of an identity change and one example of a reversible timer component are demonstrated. Using this information, a far more detailed and accurate map of the rate-limiting pathways for the consecutive stages of slime mold morphogenesis is developed (Varnum et al., 1983). It is hoped that these new methods will also be applied to other developing systems in order to assess whether general rules exist for the regulation of timing in developing systems. The author is deeply indebted to Ms. B. Varnum for important suggestions which helped in developing both the methods and the various timer interpretations, to Mr. A. Gaylon for his work on the

DAVID R. SOLL

Timers

figures, and to Mr. L. Mitchell and Mr. J. Mercer for their involvement in portions of this project. This investigation was supported in part by Grant PCM78-15763 from the National Science Foundation and by Grants GM25832 and AI15743 from the National Institutes of Health.

REFERENCES

COOTE,J. G., and MANDELSTAM, J. (1973). Use of constructed

double mutants for determining the temporal order of expression of sporulation genes in Bacillus subtilis. J Bacterial. 114, 1254. FRANKEL, J., JENKINS, L. M., and DEBAULT, L. E. (1976). Causal relations among cell cycle processes in Tetruhyemena pyrifwmis: An analysis employing temperature sensitive mutants. J. Cell Biol. 71, 242. FULTON, C., and WALSH, C. (1980). Cell differentiation and flagellar elongation in Naegleria gr&eeri. Dependence on transcription and translation. J. Cell Biol. 85, 346. HARTWELL, L. H. (1974). Sacch,aromyces cerevisiae cell cycle. Bacterial. Rev. 38, 164. HARTWELL, L. H., CULOTTI, J., PRINGLE, J. R., and REID, B. J. (1974). Genetic control of the cell division cycle in yeast. Science 183, 46. JARWICK, J., and BOTSTEIN, D. (1973). A genetic method for determining the order of events in a biological pathway. Proc. Nut. Acad. Sci. USA 70, 2046. KAUFFMAN, S., and WILLE, J. J. (19’75). The mitotic oscillator in Physarum polycephalum. J. Theor. BioL 55, 47. KING, R. C., and MOHLER, J. D. (1975). The genetic analysis of oogenesis in Drosophila melanogaster. In “Handbook of Genetics” (R. C. King, ed.), p. 757. Plenum, New York. KLEVECZ, R. R. (1978). The clock in animal cells is a limit cycle oscillator. Cell Reprod. 12, 139. LEVINSON, R., and HOUSMAN, D. (1979). Memory of MEL cells to a previous exposure to inducer. Cell 17, 485. LEVINSON, R., and HOUSMAN, D. (1981). Commitment: how do cells make the decision to differentiate? Cell 25, 5.

in Development

91

LOOMIS, W. F. (1978). The number of developmental genes in Dictyostelium. Birth Defects 14, 497. LOOMIS, W. F., DIMOND, R., FREE, S., and WHITE, S. (1977). Independent and dependent sequences in development of Dictyostelium. In “Eukaryotic Microbes as Model Development Systems” (D. O’Day and P. L. Horgen, eds.), p. 177. Dekker, New York. MERCER, J. A., and SOLL, D. R. (1980). The complexity and reversibility of the timer for the onset of aggregation in Dictyostelium. Differentiation 16, 117. MITCHELL, L., and SOLL, D. R. (1979). Commitment to germ tube or bud formation during release from stationary phase in Candida albicans. Exp. Cell Res. 120, 167. MOHLER, J. D. (1977). Developmental genetics of the Drosophila egg. I. Identification of 59 sex-linked cistrons with maternal effects on embryonic development. Genetics 85, 259. NURSE, P., THURIAUX, P., and NASMYTH, K. (1976). Genetic control of the cell division cycle in the fission yeast Schizosaccharomyces pombe. Mol. Gen. Genet. 146, 167. PRINGLE, J. (1978). The use of conditional lethal cell cycle mutants for temporal and functional sequence mapping of cell cycle events. J. Cell PhysioL 95, 393. SOLL,D. R. (1979). Timers in developing systems. Science 203, 841. SOLL,D. R., and WADDELL, D. R. (1975). Morphogenesis in the slime mold Dictyostelium discoideum. I. The accumulation and erasure of “morphogenetic information.” Dev. Biol. 47, 292. TATA, J. R. (1980). The action of growth and developmental hormones. Biol. Rev. 55, 285. VARNUM, B., MITCHELL, L., and SOLL, D. R. (1983). An analysis of developmental timing in Dictyostelium discoideum. Deu. Biol. 95, 92-107. VINTER, V. (1969). Physiology and biochemistry of sporulation. In “The Bacterial Spore” (G. W. Gould and A. Hurst, eds.), p. 73. Academic Press, New York. WADDELL, D. R., and SOLL, D. R. (1977). A characterization of the erasure phenomenon in Dictyostelium. Dev. BioL 60, 83. WOOD, W. B., EDGAR, R. S., KING, J., LIELAUSIS, I., and HENNINGER, M. (1968). Bacteriophage assembly. Fed. Proc. 27, 1160.