There are some uninviting-looking graphs on this page, but don't be alarmed - the story they tell is a very simple one and everything should be intelligible if we proceed a step at a time.

Let's think about a newly-planted field of some crop or other.  It sits there for a while and then seedlings begin to emerge from the earth, leaves grow and eventually the soil isn't visible any more because the foliage has formed a closed canopy.  After a further time leaves begin to go yellow and eventually the golden colour of the ripened crop is the sign for harvest to begin.

Let's call the period from planting to harvest the season and say it lasts for S days.

Agronomists have a term for the area of foliage covering a given area of ground - it's called the leaf area index or L, and it's an important measure of how effective the crop is at intercepting incoming sunlight.

If we add up L for the whole season, S, we get a quantity called the leaf area duration or D.  For many crops, there is a very close correlation between the value of D and the final yield.

So one approach to increasing crop yield is to increase D.  Many of the inputs used by modern agriculture - fertilizers, pesticides, irrigation - preserve or increase D.  And plant breeding has also been successful in easing genetic constraints on D.

How much further can this strategy take us?  Let's try to imagine the best possible pattern of seasonal growth.

Ideally, the crop would leap out of the ground on the day it was planted and immediately establish maximum (or, to avoid problems of mutual shading etc, better say optimum) L.  It would hold this for the whole of the season, then on day S it would senesce (ripen) instantaneously, reverting to an L of zero just like that.

Of course, unless you had magic beans, your crop couldn't possibly behave this way...

In this graph of L against Time the grey rectangle is the Jack and the Beanstalk case.

The other lines represent slightly more realistic trends for leaf area index over the growing season.  The S-shaped (sigmoidal) pattern of crop growth on the left and its reverse during senescence on the right combine to make some kind of bell curve.

It's an interesting exercise to work out how close you can get to the Jack and the Beanstalk ideal while still retaining the necessary biological property of sigmoidal behaviour.

The little graph in the top right-hand corner of the L/time figure shows this (if you're interested in how it's worked out, look here).  The conclusion is that the best you can hope for is something like about 0.7 (in other words 70%) of the absolute Jack and the Beanstalk maximum leaf area duration.

My friend John Sheehy at the International Rice Research Institute thinks we're already at about that limit for rice, and there's evidence that the same is true for maize.  Sorghum (with the development of advanced staygeen varieties) is coming up fast.

The message is that the canopy architecture of the major food crops is already near or at its productivity limit.  So where do we go from here if we want to squeeze even more yield out of these hard-working species?

Sheehy and colleagues at IRRI think we have no alternative but to revisit the strategy that has engaged, frustrated and ultimately defeated past generations of crop scientists - increasing photosynthetic carbon fixation.  The next graph explains.

You can work out the net amount of carbon an individual leaf contributes to plant yield by subtracting carbon expenditure (carbon lost through respiration plus structural carbon) from carbon income via photosynthesis.

The graph shows the pattern for a bean leaf from birth to death.  Net carbon gained is indicated as the interesting green-shaded shape.  I've indulged in mildly amusing alliteration by calling this the carbon credit contour.

To improve yield we want to increase the size of the CCC.  As we discussed at the level of crops and canopies, faster growth and later senescence would do the trick (moving the edges of the CCC in the directions of arrows 1 and 2).

Alternatively, we could move the boundaries in the directions of arrows 3 and 4.  This means increasing photosynthesis and/or decreasing respiration.

This is essentially John Sheehy's proposal for the next Green Revolution in rice.  You can read the details here:

Mitchell PL, Sheehy JE (2006) Supercharging rice photosynthesis to increase yield. New Phytologist 171: 688–693.

Will crop scientists be able finally to crack this most difficult of problems?  The rice genome has been sequenced; if, like the Human Genome Project, this really is the scientific revolution it appears, increasing the CCC in all dimensions is a worthy Grand Challenge for applying the new tools and knowledge.