The authors first measured base metabolic rates of subjects at rest as a baseline. Then they walked or ran on a treadmill at various degrees of inclination, extending over a range of grades from ‒45% to +45%. Metabolic rate was measured in each case and found to be proportional to the speed, but with nontrivial function of slope.
In cycling, it's relatively simple: at a given speed, there is a power component proportional to linearized slope (grade / √1 + grade²). But running is different. In cycling, you can coast downhill, whereas when running down hill, your muscles are doing work no matter what the grade. I was going to propose an analytic form for this relationship purely on conjecture, but thanks to Andrew Coggan's tip, I found Minetti's experimental data.
Here's a plot of Minetti's result, the energy per unit distance consumed on the various grades, along with equations I fit to his data. Minetti provided his own fit, a sixth-order polynomial, but I didn't like that much as polynomials tend to have catastrophic extrapolation characteristics. Plus, my formula has only four fitting parameters in contrast to Minetti's six, and fewer fitting parameters is generally better. Skiba uses Minetti's polynomial; perhaps I can convince him to switch over to my formula.
The result is that for downslopes, the steeper the slope beyond a certain level, it becomes more, not less, difficult to run a certain distance. This certainly speaks to the relative difficulty of hilly trail runs: they add difficulty both going up and coming down.
Another interesting, but completely predictable result of this plot is that walking is always easier than running. That means at a given speed if you can walk, do it. This was advice given to me by Don at Zombie Runner in Palo Alto, and I've taken it to heart. On steep climbs I try to walk and find, doing so, I'm able to stay in the top 10% on climbs against people who are doing extra work bouncing around but not going any faster.
Now there's a tricky issue with this, which is that it's for metabolic energy, not work done. Cycling power meters measure work done. If the human body can produce work at a given fixed efficiency, then the two are interchangeable, with an efficiency typically reported to be around 22%. However, the human machine isn't so simple. It would be a mistake to use the same efficiency to convert these running and walking metabolic costs to work.
Fortunately, the approximation of fixed efficiency may be relatively better for cycling, which is an optimized interface between the body and the road. For example when riding a geared bike on different slopes, we can shift gears to keep the force on the pedals and the rate of pedal rotation similar to what we encounter on a flat road. So for the purpose of comparing cycling to running, these data might be suitable, as long as we convert the cycling work done (as reported by power meters) to metabolic energy.