Worm finance

Hodgkin and Barnes (Proc R Soc Lond B Biol Sci 246(1315): 19-24, 1991) described a weak tra-3 mutant that produces 53% more progeny than the wild-type. This large increase in productivity comes at a price: a 4% increase in the minimum life cycle. The tra-3 mutant produces sperm for 2.6 hours longer than the wild-type, explaining both the large increase in average brood size (from 327 to 499) and the small increase (from 64.4 to 67.0 hours) in life cycle. Once oogenesis begins, progeny are produced at a rate of 5.3 per hour until the sperm supply gives out.

How can the costs and benefits of a trade-off like this be evaluated? The problem is precisely analogous to one that arises in the field of corporate finance. Suppose you have a choice between two investments. Each requires an initial outlay of $1.00. The first (wild-type) will return $5.30 per hour after a delay of 64.4 hours up to a total of $327. The second (tra-3) will return $5.30 per hour after a delay of 67 hours, but will continue up to a total of $499. Although at first glance it appears that tra-3 is the better long-term investment, this conclusion ignores the time value of money: money received early is more valuable because it can be reinvested for a profit. (Similarly, a worm hatched early contributes more to population growth than one hatched later, because in the intervening time the early worm can produce progeny.)

The internal rate of return is used to compare the long-term potential of such investments. If all receipts are immediately reinvested, the value of the total investment rapidly approaches an exponential growth steady-state n = (1 + r)^t. Over the long-term an investment with higher r will outperform an investment with lower r. r is calculated as follows. Each investment is described as a series of cash flows. For the wild-type these are -$1 at time 0 (negative because it is an expenditure), $1 at 64.4 h, $1 at 64.6 h, …, $1 at 125.9 h. r is then the positive root of the equation 0 = S c_i(1 + r)^-ti, where c_i is the i^th cash flow and t_i the time at which it occurs. The doubling time t₂ is ln 2 / ln(1 + r). Financial calculators and spreadsheet programs such as Excel have built-in functions for calculating IRRs given a series of cash flows.

The key assumption in this analysis is that returns can be reinvested. In the biological context, this means that each newly hatched worm has the same opportunities for growth as its parent. In its natural environment C elegans is generally thought to be r-selected; that is, it competes on the basis of rapid reproduction in the presence of abundant resources. Under these circumstances strains with the highest population growth rate will do best.

The table below gives the growth rates calculated from life history parameters measured by Hodgkin and Barnes. This analysis predicts that wild-type will outgrow tra-3: the wild type's 4% time advantage is more important than tra-3's 53% yield advantage. Hodgkin and Barnes measured relative population growth rates in an "eating race" in which the time a population derived from a single worm required to eat a defined large quantity of food was measured. Wild-type finished the race in 2.8 ± 0.6 % less time than tra-3, in excellent agreement with the 3.0% difference in predicted doubling times.

Statistic	wild-type	tra-3	optimum (201 sperm)
IRR (r)	6.99% hourly	6.78% hourly	7.09% hourly
doubling time (t2)	10.25 h	10.56 h	10.13 h
dauers/worm	0.327	0.360	0.306

If there is a trade-off between time and yield, there ought to be an optimum brood size that produces the maximum internal rate of return. To test this prediction, Hodgkin and Barnes measured the brood sizes of 15 independent wild isolates. Finding a tight distribution ranging from 235 to 353, they concluded. "The races appear strikingly similar in self-fertility, despite the diversity of their geographic origins, suggesting that a brood of about 300 self-progeny is a universal optimum for this species." Using Hodgkin and Barnes's measured trade-off rate of 66 sperm/h, I find the optimum brood size to be 201, predicted to grow 1.3% faster than wild-type. 201 is outside the range of normal brood sizes, implying that something is missing from the analysis. It is possible that life history parameters in the wild are subtly different from those measured by Hodgkin and Barnes in the lab. However, there is an alternative explanation: that slower growth provides an advantage when resources become limiting.

Opportunities for extremely rapid growth are brief, both in financial markets and in ecological niches. For instance, at an hourly growth rate of 7% a single worm would grow to a mass larger than Alan Greenspan in less than 3 weeks--it is unlikely that a worm ever encounters a food concentration of that size in the wild. Like a venture capitalist making a rapid growth investment, r-selected organisms need an exit strategy: a way to recoup the gains of rapid growth in a form that can be preserved when a recession strikes. For C elegans, this is the dauer larva. When food runs out, older worms are more valuable than young ones. Eggs and first-stage larvae are likely to die within a week; second -stage larvae can become dauer larvae and survive for months: older larvae can become adults, and adults can support the growth of a new generation by becoming bags of worms, probably to the point where at least some can become dauer larvae. There are more older worms in a slow-growing population than a rapidly growing one. Assuming no death among reproductive or prereproductive age worms, the proportion of the population older than age t is (1 + r)^-t. Thus when food runs out, a slow-growing population has an advantage. To get an idea of the possible magnitude of this effect, assume eggs and first-stage larvae produce no dauers on starvation, second through fourth-stage larvae produce one dauer each on starvation, and adults produce 10 dauers on starvation. The results are in the last row of the table.

This analysis reveals a second trade-off. A fast-growing organism can exploit resources more rapidly than a slow-growing one--this is an advantage in direct head-to-head competition for the same food source. However, a slow-growing population converts growth into lasting benefits more efficiently, an advantage when opportunities for growth are limited, or when the population can obtain exclusive access to a resource pool, either by luck or by erecting barriers to competition. This trade-off results from the greater survivability of older worms. Although the mechanisms that allow older worms to survive appear at first to be specific consequences of C elegans physiology and response to starvation, it is plausible that trade-offs of this type would be common: that mature, stable organisms or investments are better able to survive an unfavorable environment than young, rapidly growing ones.