*2.5. Limiting Factor Analysis*

We analyzed global changes in the three regions defined by the climate factor—temperature, precipitation, or radiation—that is most limiting to GPP. Following Nemani et al. [9], we calculated the strength of the limiting factor based on monthly climate data of minimum temperature, precipitation, and cloudy day. Then, we defined the three regions using the highest value among the three data (Figure 1a).

#### *2.6. Estimation of Precipitation Equivalent Water*

Increases in atmospheric CO2 are known to increase the water use efficiency (WUE) of vegetation, estimated as amount of carbon gained per unit amount of water used [11,22,30]. While WUE increases globally with CO2, its changes over precipitation-limited regions are of particular significance for plant growth. To assess the contribution of improved WUE in precipitation-limited regions, we used a slightly different metric called precipitation use efficiency (PUE). PUE is estimated as GPP per unit amount of precipitation. The changes in PUE between the first (2006–2015) and last decades (2090–2099) are then translated to changes in precipitation equivalent, *PEq* (mm), as follows:

$$PE\_{\%} = \left(GPP\_{last} - GPP\_{first}\right) \times \frac{P\_{first}}{GPP\_{first}} - \left(P\_{last} - P\_{first}\right),\tag{4}$$

where *GPPfirst* and *GPPlast* are GPP (kgC/m2) for the first and last decades of the 21st century, and *Pfirst* and *Plast* are precipitation received at each of the grid cells during the first and last decades.

**Figure 1.** Earth system models project easing of temperature, precipitation, and radiation constraints to growth. A map of potential climate limiting factors to plant growth ((**a**) air temperature in blue, precipitation in red, solar radiation in green) was used to guide the spatial analysis of simulated changes in climate and how such changes could impact plant growth around the world. Using outputs from Earth system models of CMIP5 we estimated ensemble mean differences in 2090–2099 minus 2006–2015 monthly air temperature (**b**), solar radiation (**c**), precipitation (**d**), and precipitation use efficiency (expressed as precipitation equivalent; see Methods) (**e**). Changes in constraints by regional mean: temperature-limited, easing 94%, and no significant change 6%; precipitation-limited, easing 23%, tightening 16%, and no significant change 61%; radiation-limited, easing 45%, tightening 19%, and no significant change 36%.
