A theoretical analysis of how plant growth is limited by carbon allocation strategies and respiration

. Improving crop yield is essential to meet increasing global food demands. Boosting crop yield requires the coordination of carbon acquisition by leaves and carbon utilization by roots and seeds. Simple modelling approaches may be used to explain how this coordination is achieved within plant growth. Here, the limits to allocation strategies and the inluence of maintenance costs are explored by analysing the sensitivity of a simple root–shoot carbon allocation model for vegetative and reproductive growth. The model is formulated based on fundamental constraints on plant growth and therefore can be applied to all plants. This general but quantitative approach shows that the relative costs of root and leaf respiration alter the relationship between carbon allocation and inal plant size, enabling a range of allocation strategies to produce a similar total amount of plant material during vegetative growth. This plasticity is enhanced by increasing assimilation rate within the model. Results show that high leaf allocation during vegetative growth promotes early reproduction with respect to yield. Having higher respiration in leaves than roots delays the optimal age to reproduce for plants with high leaf allocation during vegetative growth and increases the restrictions on lowering time for plants with high root allocation during vegetative growth. It is shown that, when leaf respiration is higher than root respiration, reallocating carbon towards the roots can increase the total amount of plant material. This analysis indicates that crop improvement strategies should consider the effects of maintenance costs on growth, a previously under-appreciated mechanism for yield enhancement.


Introduction
Improving crop yield is considered vital for meeting increasing global food demands (Fischer et al. 2009;Beddington et al. 2011), and new approaches are needed to break through existing yield barriers. Modern crops have been selectively bred to have increased proportions of biomass in harvestable material in order to maximize yield (Génard et al. 2008). Additionally, light interception has been maximized to allow of 80-90 % of all visible light during growing seasons (Long et al. 2015). Out of all plant processes, photosynthesis is the best understood at the mechanistic level, which facilitates the current focus on investigating the eficiency of photosynthesis in crop research. However, the beneits gained from such increases in photosynthetic rates are not only limited by nutrients, temperature and water, but also the energy requirements for growth of nonphotosynthetic plant parts (e.g. roots and seeds) and their maintenance.
Crop yields depend on the accumulation of biomass via growth and its allocation (partitioning of resources) to harvestable plant parts. Environmental trade-offs and metabolic costs can impact the allocation of carbon between leaves and roots. For example, a lack of water or nutrients within the soil can lead to a plant allocating more carbon into roots (Bongarten and Teskey 1987;White et al. 2016). Conversely, if there is a lack of light or a loss of leaves, more carbon will be partitioned towards the leaves (Kozlowski 1949;Barney 1951;Nelson 1964;Murthy 1990). A plant's size can impose limitations on carbon acquisition when considering light limitation. Growing surplus leaves per unit ground area (increasing leaf area index [LAI]) leads to self-shading and reduces potential light absorption by the lower leaf layers. Additionally, a large plant requires more energy for respiration and tissue turnover than a small plant. For example, in the case of forest stands, a decline in net primary productivity with stand age and size is thought to arise from high levels of respiration compared with photosynthesis, restrictions on water transport and nutrient limitations on photosynthesis (Hunt et al. 1999).
The costs of maintenance respiration may differ between leaves and roots. Johnson (1983) suggests, from a theoretical analysis, that the cost of respiration in the roots is higher than that in the leaves, which is a consequence of nitrogen and anion uptake and assimilation being very expensive within the roots. Conversely, Tjoelker et al. (2005) show empirically that leaf respiration varies between 1 and 2.25 times higher per unit mass than root respiration for 39 grassland and savannah species. These examples show that the cost of maintaining a leaf may not always be equal to that required to maintain a root. Amthor et al. (2019) argue that respiration should be a focus in optimizing crop productivity. Knowing that there are species where respiration in the leaves is more than twice the cost of respiration in the roots (Tjoelker et al. 2005), it becomes important to investigate the limitations on carbon allocation with different ratios of leaf and root maintenance.
Whole plant processes which affect plant growth are well deined in isolation (e.g. photosynthesis (Farquhar et al. 1980) and respiration (Amthor 1986)), but there is a lack of knowledge about how such processes deining growth work together to determine the allocation of growth to above-and below-ground biomass. Here, modelling plant growth can be beneicial. Modelling can provide a way of assessing how different mechanisms interact, which can be limited by experiments alone due to a lack of ine control over each process. It also helps to reine the potential range of questions that are addressed with experiments, and allows more targeted hypothesis testing. In particular, it formalizes quantitative predictions based on the current understanding that can be tested via experimentation. A modelling approach is therefore adopted for the speciic research questions in this paper.
It remains unclear what internal mechanisms are responsible for biomass partitioning, since plant growth models use a variety of different assumptions for allocation (Ewel and Gholz 1991;Luxmoore 1991;Weinstein et al. 1991;Dewar et al. 1994). Some more recent models assume allocation based on a functional balance of resources to leaves and roots and is calibrated to maximize plant relative growth rate (Zerihun et al. 2000;Buckley and Roberts 2006;Feller et al. 2015). Most carbon allocation or plant growth models investigate the effects of environmental conditions, herbivory, senescence and/or pruning on plant growth (Hogsett et al. 1985;Ford and Keister 1990;Luxmoore 1991;Weinstein et al. 1991), and ignore the effects of different potential allocation strategies between leaves and roots on growth and how cellular processes such as photosynthesis and respiration alter this. However, any strategy for increasing productivity must balance the allocation of carbon to the growth of source and sink tissues. Here, a carbon source is deined as a net exporter of carbon to the rest of the plant, where carbon is acquired from the environment (i.e. mature leaves via photosynthesis), while sinks are net importers of carbon from internal sources (Doehlert 1993), and include young leaves, seeds and roots.
Understanding the developmental stage at which a plant should reproduce can bring insights into optimal strategies that a plant should adopt when faced with environmental or biotic hazards and has a limited amount of time to reproduce before it is killed by an external factor. Equally, when considering crop production in a seasonal climate, development must be completed during a growing season of limited duration. Many have investigated the effects of lowering time on yield (Cohen 1971;King and Roughgarden 1982;Guilbaud et al. 2015) and speciically focused on environmental heterogeneity (Paltridge and Denholm 1974;Ledder et al. 2004), multiple reproductive phases (Chiariello and Roughgarden 1984) and photosynthetic rates (Schaffer et al. 1982). Yet, the effects on yield when varying allocation between leaves and roots during vegetative growth were not addressed. This paper explores the following questions: (i) How does varying the allocation of carbon for growth between leaves and roots alter total plant biomass during vegetative growth? (ii) How do the processes of respiration and photosynthesis constrain possible strategies of allocation between leaves and roots? (iii) Does increasing allocation towards the roots always lead to a decrease in overall plant size? (iv) How does varying allocation between leaves and roots during vegetative growth and lowering time alter reproductive output (which is equivalent to seed yield in the case of crops)? These are addressed by analysing the sensitivity of a simple carbon allocation model for vegetative and reproductive growth. All values within this paper are dimensionless in order to look at the general behaviour of the model. Our model of plant growth dynamics is based on the balance between carbon assimilation rate (photosynthesis) and carbon usage rate (growth and respiration). While the absolute values of these processes vary significantly in different plants, we aim to capture the generic features of the dynamic balance between them. In order to do so, we study our model in dimensionless form. Plant growth depends on many different factors and therefore plant growth models can become extensive and include many parameters which do not necessarily exert strong control over the results. Importantly, the solution of the model may not necessarily depend on all of its parameters independently. Non-dimensionalization allows a model to be simpliied by removing those extra parameters that can be combined, and removing the units. This approach has the advantage of identifying the speciic combinations of model parameters that control the scaling of model solutions. It results in a model that contains a minimal set of effective parameters, whose solutions depend on the relative magnitudes of different processes. Importantly, the solutions of any dimensional form of the model are obtained from the non-dimensional solutions by scaling. This approach allows the sensitivity of plant growth to changes in allocation, photosynthesis and respiration to be explored in their broadest sense. The model results are described as changes in ratios or relationships between parameters; the inclusion of units does not change these relationships (see Appendix 1 for further detail).

Model description
The model considers only carbon sources and sinks, with no soil or other environmental interactions. Sink or source strength can be deined as the combination of source or sink size and activity, which relates to the uptake or export rate of a particular resource. By combining two alternative, but complementary, perspectives, the model simultaneously accounts for net carbon export rate from the leaves (source strength), internal carbon allocation to leaves for growth (growth of source size) and roots or seeds for growth (growth of sink size), and the development of leaves, roots and seeds (White et al. 2016).
The physiological perspective deines growth by the acquisition and loss of carbon via photosynthesis and respiration, respectively (Lambers et al. 1990). This expresses carbon gain as the difference between carbon acquired via photosynthesis (source strength) and the loss of carbon through respiration: where Al represents the rate of carbon assimilation (A is assimilation rate per unit leaf tissue and l is the amount of leaf tissue) and R 1 l and R 2 r is the rate of use of carbon for maintaining plant material (R 1 and R 2 are leaf and root respiration and r is the amount of root tissue [dimensionless]). This formulation accounts for the fact that not all carbon assimilated is used for the growth of new plant tissue; some is used for the maintenance of existing tissue and other metabolic processes. The second perspective is the development of source and sink tissues. Here, plants are conceptualized as modular structures, where a module is one of the repeating units from which a plant is constructed (e.g. leaves, roots or cells) and growth is assumed to be a function of module initiation rates (the rate at which each module is constructed). Growth in this case is deined as the development of individual leaves and roots (Pritchard et al. 1999), which can be expressed as the combination of organ initiation rates: where µ 1 and µ 2 are the module initiation rates of leaves and roots, and M l and M r are the sizes of leaf and root modules, respectively. Individually, these perspectives bring limited insights. The physiological approach is resource driven and the developmental perspective is driven by the carbon requirements for the growth of new organs. By combining these perspectives, the model simulates plant growth based on fundamental constraints (White et al. 2016), allowing the exploration of how physiological rates, allocation and developmental rates interact to control growth.

Assumptions
Simplifying assumptions mean that the model is best interpreted as a simulation of monocarpic species growing in warm, fertile conditions, with strong competition for light within a closed leaf canopy. This is closest to the situation for annual crops. Although the developmental pattern of these plants means that the limits simulated by the model are never reached in nature, these limits set boundaries that development cannot exceed. Roots are modelled as carbon sinks, which account for all nonphotosynthetic plant parts. The model does not simulate soil nutrient limitations or capture other environmental limitations such as temperature or drought. This allows the effects of carbon allocation on growth to be directly analysed.

Photosynthesis
The rate of carbon assimilation by photosynthesis is assumed to be the sole mechanism of carbon acquisition and is modelled as the average canopy photosynthetic rate, which is a decreasing function of the total leaf area. As LAI increases, the average unit rate of photosynthesis (per unit of leaf area) decreases due to self-shading (Pury and Farquhar 1997). Assimilation rate is chosen to be where θ is the shading coeficient, A 0 is the unit rate of photosynthesis for an unshaded leaf and l is the amount of leaf tissue. When l = θ, the rate of photosynthesis per unit leaf area is half of its maximum (unshaded) value. Increasing the shading coeficient (θ) reduces the effect of shading on the assimilation rate.
Determinate growth Acquired carbon can either be allocated for growth and its associated cost (growth respiration) or used for maintenance respiration, where the cost of maintenance is linearly dependent upon the size of the plant (Ryan 1991) (Eq. (9)). The carbon allocated towards growth encompasses the cost of growth respiration since this is a dimensionless system. A necessary condition for growth is that the assimilation rate must always be greater than the respiration rate. This provides a natural constraint on appropriate parameter values.
In the model, plants grow until canopy photosynthesis is exactly counterbalanced by total plant respiration, which sets a inal plant size. Once this size is reached, no further growth occurs. Although a maximum size set by source-sink balance has been hypothesized for forest trees (Hunt et al. 1999;Day et al. 2001), this idea has been superseded by hypotheses relating to water and nutrient limitation of growth in tall trees and reductions in photosynthetic eficiency (Gower et al. 1996;Murty et al. 1996). However, despite being inappropriate for trees, this approach may be suitable for plants which do not grow indeinitely (determinate growth) such as monocarpic, herbaceous species, including annual crops, where plants reproduce once before dying and can be assumed to maximize reproductive output (i.e. seeds) using available resources. For many crops, the harvestable plant parts are the roots; therefore, within this framework, seeds are used to describe any reproductive organ that stores carbon. For annual crops, the cessation of vegetative growth is brought about through a pre-determined developmental pattern. However, a useful approximation for this is when resource uptake and use become balanced (due to selfshading), given that there is no senescence of leaves or roots in the model. It can be argued that the decline in plant relative growth rates is caused by plants becoming less eficient as they grow due to self-shading and tissue ageing (Evans 1972;Rees et al. 2010), while others suggest it is due to increased allocation of growth towards non-photosynthetic plant compartments (roots and stems) and reductions in soil nutrients (Paine et al. 2012;Philipson et al. 2012). Simulating steady-state plant growth is commonly used for plant growth models (Thornley 1972;Charles-Edwards 1976;Reynolds and Thornley 1982;Hirose 1986;Yin and Schapendonk 2004) and it can be argued that they can be applied to nonsteady-state conditions (Hirose 1986;Agren and Ingestad 1987;Hirose et al. 1988: Van Der Werf et al. 1993.

Reproductive growth
Roots are the only sink organ modelled during vegetative development, and roots and leaves both stop growing after a plant transitions from vegetative to reproductive growth. During reproductive development, the only sinks are seed growth and the respiration required to maintain existing roots and leaves. This transition to reproductive growth is imposed at numerous stages of a plant's life. These assumptions approximately replicate what happens during the development of annual crops such as wheat and rice (Atwell et al. 1999).
Senescence There is no turnover or senescence of plant tissues in the model, and source strength is maintained during reproductive development. This is a simpliication that ignores the decline in source activity that occurs in annual crops during seed growth, which is linked to the remobilization of nitrogen from leaves to seeds (Masclaux et al. 2001).

Resource allocation
The balance of carbon allocation between sources and sinks is assumed to be ixed such that the model simulates one allocation strategy throughout determinate growth. This ensures easy analytical comparisons between allocation strategies, but ignores ontogenic drift in allocation. Allocation strategy is dependent upon module initiation rates of leaves and roots, where the root module initiation rate is a ixed proportion of leaf initiation rate: where α is the allocation coeficient. For example, when α = 2, twice as much carbon is allocated to roots than leaves and when α = 1/2, twice as much carbon is allocated to leaves than roots.

Vegetative growth
Leaf and root growth can be individually expressed as and where m l and m r are leaf and root module size, respectively. Using Eq. (4), root and leaf growth are related by where m = m r /m l . Therefore, growth trajectories are given by As a consequence of assuming a constant allocation strategy, the growth trajectories are straight lines with gradient mα. Each growth trajectory represents the duration of plant growth from a seedling to the point, in which plant material reaches steady state as a function of leaf and root tissue and varies dependent upon initial seedling size. The physiological perspective (Eq. (1)) deines the growth rate as the difference between the rates of photosynthesis and respiration. Carbon assimilated by the plant via photosynthesis is distributed between maintenance respiration and growth (including growth respiration): Substituting from Eq. (7): and therefore Using Eq. (7) again gives The model is given by Eqs (11) and (12). Steady states of the model satisfy the equation which corresponds to a continuous curve in the (l, r) phase space. The function A(l), which represents the effect of self-shading on photosynthesis, is a decreasing function of l. Using the functional form given in Eq. (3), the general form of the steady-state curve is illustrated in Fig. 1. The curve has a peak at l = θ Therefore, the qualitative shape of the blue curve is independent of R 2 , which determines only the height of the curve. The location of the peak of the curve is determined by A 0 /R 1 ; therefore, optimizing the ratio between the unshaded rate of photosynthesis and leaf respiration increases inal plant size within this framework, which is to be expected. The steady-state curve (black line) begins with a steep increase, which is the ratio of root tissue to leaf tissue increasing because a much higher quantity of carbon is required for root respiration to balance out assimilation from the leaves. As leaf tissue increases, the level of self-shading increases. Once assimilation rate reaches its maximum, additional leaves are costly and less root tissue is required for respiration to balance the canopy assimilation rate. This explains the decline in the steadystate curve. For the purposes of behaviour illustration, the following default parameters: A 0 = 10, θ = 10, m = 1, α = 1, R 1 = 2, R 2 = 2, with initial conditions of l = 0.01 and r = 0 will be used. The orange line in Fig. 1 is the vegetative growth trajectory, which represents the growth of a plant from seedling to a steady state at which the rates of photosynthesis and respiration are equal. The growth  showing the local direction of growth trajectories. An example trajectory is also shown (orange line). Using A 0 = 10, θ = 10, R 1 = R 2 = 2, αm = 1 with an initial seedling size of l = 0.01, r = 0. Assumes vegetative growth only (no reproduction). All model parameters are dimensionless.
Downloaded from https://academic.oup.com/insilicoplants/article-abstract/1/1/diz004/5509959 by University of Sheffield user on 05 July 2019 trajectories are straight lines in the l, r plane with gradient αm. Allocation is a combination of balance between allocation and module size. The straight line signiies a constant ratio of root to leaf tissue allocation as the plant grows. The root to leaf ratio at steady state depends on αm, initial leaf and root tissue, and the black steady-state curve. Fig. 1 also shows the vector ield in blue [(dl/dt), (dr/dt)]. This illustrates the fact that only the portion of the steady-state curve with a negative gradient (solid line) corresponds to steady states that are stable. Biologically, this represents plants in which the roots account for the majority of total plant material with very few leaves. These situations are unrealistic given the assumption that roots are only considered as carbon sinks. The maximum stable allocation strategy is deined as Fig. 2a shows several growth trajectories for different allocation strategies (α) when the cost of leaf and root maintenance respiration is equal (R 1 = R 2 ). The choice of allocation strategy determines the point at which the growth trajectory intersects the steady-state curve. This determines the ratio of root and leaf material and total plant size at steady state. These inal plant sizes can be related to yield as a larger plant produces a greater yield than a smaller one. For these parameters, steady-state total plant size is a decreasing function of allocation strategy. In other words, strategies which favour leaf growth result in larger steady-state total plant size than those favouring root growth, when roots are Root mass Total plant mass only considered as sinks. The same effects also apply to growth rates (Fig. 2b). The constant allocation ratio implies that the intersection between the vegetative growth line and the steady-state curve (inal plant size) is dependent upon the choice of initial seedling size and the steepness of the growth trajectory (allocation strategy). Varying the allocation strategy can alter the point at which the growth trajectory reaches the steady-state curve. The effects of shading are a decreasing function of the shading coeficient θ; thus, steady-state total plant size is a decreasing function of θ, since both the peak of the steady-state curve and maximum leaf tissue depend on shading. Increasing A 0 has a positive impact on inal plant size as expected (Fig. 2c). As the shading coeficient tends towards ininity, the assimilation rate becomes unconstrained and plant material continuously increases. Conversely, imposing a very small shading coeficient highly constrains assimilation rate and leads to a very small inal plant material.
These behaviours imply that the model represents the growth of a plant in a reasonable manner reproducing well-known phenomena.

Reproductive growth
Formulated in terms of the current model, a plant undergoes vegetative growth from an initial seedling size using Eqs (11) and (12) with a pre-determined allocation strategy. The plant can then decide to reproduce at any developmental stage. Once the plant has made the life history decision to reproduce, no further vegetative growth occurs, total leaf and root mass become ixed at this point in time and the reproductive sink increases in size until the steady state is reached. Since the growth of leaves and roots is set to zero, all carbon made available from the plant minus respiratory costs at its chosen developmental stage is invested into seed production. Leaf and root biomass from the chosen developmental stage are the new initial conditions for the reproductive stage of growth. Within this framework, the seeds themselves have the same respiratory cost as roots and simply represent an additional sink. The model then becomes where s is the amount of seed material, A is assimilation rate, l is leaf tissue, r is root tissue, and R 1 and R 2 are maintenance respiration for leaves and roots and seeds, respectively. Within this framework, there are two decisions that a plant can make which alter potential seed yield: (i) The allocation strategy during vegetative growth; (ii) The developmental stage at which a plant reproduces.

How do maintenance costs limit carbon allocation?
Within the model, growth depends on the allocation of carbon to plant compartments, the assimilation of carbon via photosynthesis and loss of carbon via maintenance respiration. How these processes interact together can provide insight into the limitations on growth. In particular, how do maintenance costs limit the range of carbon allocation strategies which alter inal plant size?
When leaf and root maintenance costs are equal, unshaded assimilation rate reduces limitations to carbon allocation In the circumstances when the costs of leaf and root maintenance respiration are equal, increasing allocation towards the leaves increases inal plant size. Fig. 2d shows the effect of varying allocation strategy on inal plant size. With default parameters, inal plant size decreases as α increases, until α = 4, then the plant does not have enough carbon to grow when allocation favours the roots. The maximum allocation strategy for stable steady-state solutions is α max = 1.2361. This implies that carbon allocation is limited by an upper bound on root strategies; therefore, only plants which allocate up to 1.2361 times more carbon to root growth than leaves are realistic given the parameter set. A sensitivity analysis shows how this upper bound is changed by alternative model parameterizations (see Appendix 2). There are two parameters within the model that vary the effect of assimilation rate; the unshaded assimilation rate and the shading coeficient. Increases in the unshaded assimilation rate cause both the maximum inal plant size and the upper bound of root allocation to increase. For example, when increasing unshaded assimilation to A 0 = 20, the limit to root allocation is reached approximately when α max = 2.1623. Fig. 3 shows the relationship between allocation strategy and inal plant size with multiple unshaded assimilation rates. The gradients of these lines are not altered when varying A 0 . Therefore, the model behaves reasonably, conirming that, having a higher unshaded assimilation rate ensures that the plant has more carbon available to allocate towards new modules, enabling a plant to allocate more towards the roots.
In contrast, increasing the effect of shading on growth has no effect on the limits to carbon allocation. Increasing the shading coeficient (i.e. reducing the effect of shading) increases the maximum inal plant size but the upper bound on root allocation remains the same. This is because α max is independent of θ (Eq. (14)). The shading coeficient (θ) simply delays the effect of shading on assimilation rate; therefore, the cost is not applied until the total leaf canopy size is high (small α). This means that canopy architecture does not constrain the allocation of growth to roots within this framework. Further to this, using different initial conditions would alter the limits to carbon allocation since they alter the rates of photosynthesis and respiration.

When leaf respiration costs more than root respiration, increasing carbon allocation to the roots can increase inal plant size
When leaf and root respiration are equal, increasing allocation towards the leaves increases inal plant size and increasing allocation to the roots decreases inal plant size. Only the negative slope of the steady-state curve has stable solutions during vegetative growth (Fig. 1b). The gradient of this curve deines the relationship between the allocation strategy and inal plant size. If the gradient of the curve is less than −1, then when α is large (i.e. allocation favours roots over leaves), the trajectory intersects the steady-state curve at a higher point (Fig. 4a). At this point, a larger plant is produced compared with the point of intersection for a smaller α. This is the opposite effect to when the gradient of the curve is greater than −1, when intersecting a higher point on the curve produces a smaller plant when compared with having a smaller α (Fig. 4b).
When the cost of leaf respiration is higher than the cost of root respiration, decreasing leaf allocation reduces the amount of carbon lost via respiration, enabling a larger plant with a smaller leaf allocation strategy. When increasing allocation to a much larger root production strategy, there would be insuficient carbon assimilated to produce a larger plant. This means that, in the instance where the gradient of the steady-state curve is less than −1, increasing allocation to the roots increases inal plant size. But are there any realistic scenarios where this would happen? In order for this effect to occur within the model, the gradient of the steady-state curve (differentiated Eq. (13)) must be less than −1, which is equivalent to therefore, (a) A necessary condition for growth is that A 0 > R 1 ; therefore, the cost of leaf respiration must be greater than the cost of root respiration. The relationship between the allocation strategy and inal plant size changes substantially when making leaf respiration larger than root respiration. Fig. 5b shows multiple trajectories intersecting the steady-state curve with different allocation strategies. Higher allocation strategies (large α) intersect a higher point on the steadystate curve and lower allocation strategies (small α) intersect a lower point on the curve. Towards the top of the steady-state curve, the gradient is less steep than −1. Thus, as α increases, the point of intersection moves further along the steady-state curve, declining in plant size until the gradient reaches zero. This can be seen in Fig. 5a. It shows the increase in inal plant size as allocation tends towards favouring the roots. Once the α reaches 1.2, the inal plant size decreases. Therefore, there is only a small region along the steady-state curve, where increasing allocation strategy increases the inal plant size. This graph implies that allocating more carbon towards the roots when α<1.2 would improve yield. It is also important to note the small range of inal plant size in Fig. 5a-i.e. when varying allocation between two times more carbon to the leaves and two times more carbon to the roots, there is only a change of 4% in the inal plant size when A 0 = 10, θ = 0.1 and m = 1 (Fig. 5a). This implies that steady-state plant size can be plastic with respect to allocation strategy, such that over a range of α, any allocation strategy can be implemented and achieve the same inal plant size. This identiies two key questions: what range of parameter values deining carbon uptake (shading coeficient θ and unshaded assimilation rate per leaf A 0 ) allows this plasticity of inal plant size to occur? Can altering these values limit or extend this plasticity within the model?
Varying the shading coefficient has no effect on the range of plasticity within the model The scenario in which leaf respiration costs twice the amount of root respiration is fairly common across different plant species (Hansen and Jensen 1977;Reich et al. 1998;Loveys et al. 2003;Tjoelker et al. 2005). This implies that there is a large range of parameter values which allow plasticity to occur with minimal change in inal plant size. In the following sensitivity analysis, all parameters are varied within the model, while leaf and root respiration remain constant.
Increasing the shading coeficient has no effect on the region of allocation strategies which do not alter inal plant size. For plasticity to occur, Eq. (16) must be satisied. Since this equation is not dependent upon θ, it cannot have any effect on the range of plasticity. When the shading coeficient is 0.5 (high shading), a plant can have any strategy of α between 0.5 and 1.5 (Fig. 6a). Within this framework, the range of allocation strategies which produce similar inal plant biomass is not altered by shading as increasing shading only reduces the availability of carbon for allocation.
As assimilation increases, there is more available energy to grow; therefore, much more carbon needs to be allocated to the roots when growth stops in the model. This means that the range over which inal plant size increases and then decreases (parabola-like curve shown  in Fig. 5a) is much larger. Though this is the case, there is a much larger change in inal plant size under these scenarios. Fig. 6b shows the relationship between inal plant size and a range of allocation strategies which produce a stable steady-state when varying A 0 . When unshaded assimilation rate is A 0 = 10, plasticity occurs when α is between 0.5 and 2.47 (Fig. 6b), while as assimilation increases, this range shifts to root-favoured strategies. When A 0 = 70, plasticity occurs when α is between 4 and 9.83 (Fig. 6b). Therefore, increasing assimilation rate increases the parameter space which allows plasticity of inal plant size to occur and promotes a rootfavoured allocation strategy.

How does the timing of reproduction inluence yield?
The steady-state condition when photosynthetic carbon gain is exactly balanced by respiratory carbon loss is a special case that may not be commonly achieved in nature. One situation when this may occur is in mature forest stands, as size becomes a limiting factor on growth. This balance of the rate of photosynthesis and respiration rate is one mechanism proposed to explain why forest biomass does not increase indeinitely, but tends to approach a plateau (Hunt et al. 1999;Day et al. 2001). The general problem with such a steady state is that, once this point is reached, the system is essentially 'stuck' unless tissues turn over. One biological situation where this is not a problem is at the end of life in a monocarpic plant. Monocarpic species reproduce once at the end of their lives (Harper and White 1974), and thus it might be assumed that they invest all remaining photosynthetic carbon into seed production at this point, so that growth approaches a steady-state situation. If this argument is accepted, this is an interesting case to consider because many crop species are annuals (i.e. monocarpic), including all cereal and grain legume crops.

Seed production depends on allocation strategy and developmental stage
Vegetative growth is simulated with a variety of allocation strategies. This produces multiple growth trajectories with different gradients. An initial root tissue of 0.1 is used for each trajectory, and initial leaf tissue is determined by r = αml. This sets the initial seedling size for each vegetative growth trajectory. Reproductive growth is then simulated by taking a point along the growth trajectory with a distinct amount of leaf and root tissue as the initial conditions to solve Eq (17) (Fig. 7a). The time taken to produce seed can be solved analytically using Eq. (17) for a range of allocation strategies and developmental stages. The amount of seed produced is calculated and compared. The dependence of seed production on the developmental stage of reproduction is shown in Fig. 7b for a range of allocation strategies when leaf and root respiration are equal. Each developmental stage represents the total amount of leaf tissue as a percentage of the total amount of leaf tissue reached at steady state during vegetative growth (percentage of total age) or in other words, various stages along a plant's lifespan if it did not reproduce. As the amount of leaf tissue increases during growth, the amount of energy available to produce seed increases and so does the level of self-shading. Once the plant produces a certain amount of leaf tissue, assimilation per unit leaf tissue begins to decrease (due to shading costs) while the cost of maintaining the plant is still increasing. This leads to a decline in the amount of seed tissue. A plant with an allocation strategy favouring leaf growth should reproduce at 25-31 % of its total potential vegetative age to produce the maximum seed. A plant with an allocation strategy favouring root growth, should reproduce at 35-49 % of its potential vegetative age. A plant favouring leaf growth can reproduce at an earlier ontogenic stage as it has already invested more energy into leaf growth whereas a plant favouring root growth would need more time to develop enough leaves to create the most seed. High leaf allocation therefore promotes early reproduction. Although the timing of reproduction is affected by factors such as day length, temperature and stress, the balance of carbon between leaves and roots during vegetative growth imposes limits that all potential reproductive scenarios sensitive to these factors must stay within. The optimal strategy for maximizing reproductive output is deined by the maximum vertical distance between the vegetative growth trajectory and the steadystate curve. This difference is the potential energy available to synthesize seed material. This potential energy (∆E) can be expressed as where A is assimilation rate, l is the amount of leaf tissue, r is the amount of root tissue, and R 1 is maintenance respiration for leaves and R 2 is maintenance respiration for roots and seeds. Equation (18) states that the potential energy for reproduction available at a given time is equivalent to the difference between the energy produced via photosynthesis and the energy required to maintain the leaves and roots. With the largest potential energy producing the maximum yield, allocation strategies with the least steep trajectories have the largest potential energy. This implies that allocation strategies favouring leaves will have the largest potential energies for reproduction. Fig. 7b conirms this, where it is clear that allocation strategies which favour leaves produce the largest amount of seed tissue. The strategy which allocates the largest amount of carbon to the leaves produces the most seed and as allocation towards the roots increases, yield decreases.

During vegetative growth, leaf favoured allocation strategies promote early reproduction
The time taken to produce seed can be calculated by integrating Eq. (17) and rearranging to yield: Seed mass produced The range of root to leaf allocation patterns when growth stops (blue line) with a vegetative growth trajectory (orange line) starting from an initial leaf and root tissue of l = 0.1, r = 0.1 and a reproductive growth trajectory (green line) starting from half of its total possible vegetative size (l = 20, r = 20), when A 0 = 10, m = 1, α = 1, θ = 10 and leaf respiration is equal to root respiration (R 1 = 1 = R 2 = 1). Sink size includes the combined tissues of roots and seeds, while source size is the amount of leaf tissue. (B) The relationship between transition to reproduction at different percentages of total age (percentage of the amount of total plant material at vegetative steady state) and seed production, comparing multiple allocation strategies with an amount of initial root tissue of 0.1, and when A 0 = 10, m = 1, θ = 10, R 1 = 1 and R 2 = 1. The data points (circles) represent the maximum amount of seed tissue produced for each allocation strategy. Developmental stage is a percentage of the amount of total plant material at vegetative steady state. All model parameters are dimensionless.
Downloaded from https://academic.oup.com/insilicoplants/article-abstract/1/1/diz004/5509959 by University of Sheffield user on 05 July 2019 where t is the time taken to produce seed, r ss is the total sink size (seed + root tissue) at steady state and r 0 is initial root tissue (amount of root tissue at the end of vegetative growth). The time taken to produce seed decreases with the developmental stage for plants favouring leaf allocation strategies during vegetative growth, implying that a plant reproducing later along the vegetative trajectory takes less time to reproduce. Fig. 8a shows that there is a positive linear relationship between the amount of seed tissue and the time taken to produce seed. The time taken to reproduce decreases as allocation towards the roots increases. This is because less carbon has been invested in leaf growth, so there is less energy available from photosynthesis to produce seed and it is exhausted more rapidly. For plants with a root favoured strategy during vegetative growth, the relationship between developmental stage and time taken to produce seed is much weaker (Fig. 8b). This is due to root allocation strategies (α>1) creating a much steeper vegetative trajectory, reducing the variation in potential seed production for the range of developmental stages along the trajectory. This suggests that there is a trade-off between the time taken to reproduce and yield. For a plant in a hazardous environment creating a reduced growing season, it may be beneicial to reproduce earlier and not obtain the maximum possible yield. For plants without constraint on the length of the growing season, the optimal age to reproduce can be chosen based on the total seed produced as shown in Fig. 7b.

Higher leaf maintenance costs delay reproduction for plants allocating more carbon to the leaves
When increasing the cost of leaf maintenance, there is a reduction in seed production and therefore also in the time taken to produce seed. The maximum sink size (at the peak of the steady-state curve) is equivalent to Therefore, as the cost of respiration tends towards the amount of carbon assimilated via photosynthesis, the maximum sink size decreases. Not only does the maintenance cost affect yield, but it also affects the optimal age to reproduce in order to maximize yield. When maintenance costs are equal and A 0 = 20, for high leaf allocation strategies, it is beneicial to reproduce between 20 and 23 % of the potential vegetative age if it did not reproduce. In contrast, for high root allocation strategies, it is beneicial to reproduce within 27-41 % of the potential vegetative age for default parameters. However, when maintenance costs are unequal (R 1 = 2R 2 ), for high leaf allocation strategies it is beneicial to reproduce between 25 and 28 % of the potential vegetative age and for high root allocation strategies, it is beneicial to reproduce within 30-38 % of the potential vegetative age. This change in maintenance costs shifts the optimal region for leaf allocation strategies by 5 % towards later reproduction and reduces the optimal region for root allocation strategies by 6 %. This implies that having a higher maintenance cost for leaves than the roots, whilst allocating more carbon to the leaves, Time to reproduce

Fig. 8. (A)
The relationship between the time taken to produce seed during reproductive growth and the amount of seed tissue for multiple allocation strategies when A 0 = 10, m = 1, θ = 10 and R 1 = R 2 = 1. When α<1, more carbon is allocated to the leaves and when α>1, more carbon is allocated to the roots. Each data point represents a different developmental stage. (B) The relationship between the choice of reproduction at different percentages of total age and the time taken to produce seed for multiple allocation strategies. All model parameters are dimensionless.
Downloaded from https://academic.oup.com/insilicoplants/article-abstract/1/1/diz004/5509959 by University of Sheffield user on 05 July 2019 delays the optimal age to reproduce and further limits the optimal age to reproduce, when the plant is allocating more carbon towards the roots. This is because having a higher cost of leaf material reduces the amount of energy available to produce seed. A plant must therefore be more established in order to have the maximum energy available to reproduce (data not shown).

Discussion and conclusions
The aims of this paper were to understand the consequences of different allocation strategies and costs of maintenance respiration for plant growth in a general but quantitative manner, and to investigate any possible limitations to carbon allocation. These were addressed by using a simple root-shoot carbon allocation model. The effects of varying the source-sink allocation ratio on plant growth are not explicitly explored in the work of most plant growth models. Typically, these incorporate an allocation assumption and investigate the effects of environmental conditions or perturbations on growth (Hogsett et al. 1985;Ford and Keister, 1990;Luxmoore 1991;Weinstein et al. 1991). Ignoring the effects of environmental conditions made it possible to determine which underlying processes have the greatest inluences on plant growth. In particular, the model suggested that maintenance respiration plays a signiicant role in the effect of allocation strategies on growth. When maintenance costs are equal between leaf and root tissue, increasing allocation to the leaves increases the growth rate and therefore the inal plant size. Many propose that the most eficient means for a plant to maximize its growth rate is to allocate just enough resources to the roots for nutrient assimilation and allocate the majority of resources to the leaves (Mooney 1972;Wareing and Patrick 1975;Russell et al. 1977;Reynolds and Thornley 1982). In the case when maintenance costs are equal, there are limitations on carbon allocation to the roots and high assimilation rates alleviate these limitations. Thus, high rates of photosynthesis enable both large plant size and high root allocation.
However, it is not necessarily the case that leaf and root maintenance respiration are equal, and published data show that leaf respiration can be up to 2.25 times larger than root respiration (Hansen and Jensen 1977;Reich et al. 1998;Loveys et al. 2003;Tjoelker et al. 2005). In the circumstance where leaf respiration is higher than root respiration, increasing root allocation can in theory lead to a larger plant. This effect is especially pronounced when canopy self-shading is high (i.e. the shading coeficient in the model is low), and contradicts the widely held assumption that carbon should be preferentially allocated to the leaves to optimize growth.
Instead, it shows that maintaining a large number of shaded leaves within a canopy can be detrimental for production when those leaves carry a high maintenance cost. When leaves become too ineficient and costly to maintain, plants senesce their leaves.
Long-standing theory in vegetation modelling predicts that plants should add leaf layers until the lowest layers fail to make a positive net contribution to canopy carbon gain (Woodward et al. 1995). However, modern crops such as soybean seem to violate this prediction, producing very dense leaf canopies, especially when supplied with high atmospheric CO 2 , and the dense shading of lower leaves means that they contribute little to canopy carbon assimilation (Drewry et al. 2010a, b). Recent modelling and experimental manipulation of soybean crop canopies shows that these plants overinvest in leaves, and that leaf removal can actually improve yields (Srinivasan et al. 2017). Under certain circumstances, crop plants, therefore, produce more leaves than is optimal for growth and yield, and the model shows how this behaviour can arise from the high cost of maintaining leaves and a declining rate of photosynthesis with shading. It has been hypothesized that the overproduction of leaves evolves in wild plants from the beneits of shading out competitors in dense, competitive plant communities (Anten 2005). This may increase the itness of individual plants but, in the case of crops, breeders and farmers aim to maximize the yield from the whole population of plants within the ield, and it is advantageous to reduce competition among individuals (Denison 2012;Anten and Vermeulen 2016). Respiration accounts for a large proportion of carbon loss within the plant (Gifford et al. 1984;Amthor 1989Amthor , 2000Cannell and Thornley 2000), providing a key mechanism which can be manipulated to boost crop productivity. Advances in respiration research identify genes responsible for substrates, enzymes and transporters that are essential for protein turnover and transport, which can be targeted to optimize respiration (Amthor et al. 2019).
In the model, circumstances when leaf maintenance costs are higher than root maintenance costs also introduce a certain level of plasticity with respect to inal plant size, such that a range of allocation strategies can lead to similar inal plant size (this occurs when the gradient of the steady-state curve is less than −1). When maintenance costs are equal, leaves are much more beneicial for growth than roots, but when leaves cost more to maintain than roots, the beneits of growing leaves and roots become more equal. This allows for a range of root:shoot ratios to produce the same inal plant size. Work has been done to investigate the effect of environmental conditions (Ericsson 1995) or competition (Waite and Hutchings 1982) on the plasticity of allocation but not necessarily how maintenance costs alter the plasticity of allocation. Increasing the shading coeficient has no effect on the range of allocation strategies where this plasticity occurs. However, increasing assimilation rate increases the range of allocation strategies where inal plant size does not change.
When looking at reproductive growth, the optimal developmental stage for the plant to reproduce is deined by the carbon available to produce seeds or other reproductive organs such as tubers. Therefore, allocation strategies favouring leaf growth will have the largest potential energies for reproduction, implying that high leaf allocation during vegetative growth promotes early reproduction. This corroborates the work of Guilbaud et al. (2015) who suggest that high growth rates correlate with early lowering, since high leaf allocation leads to a higher growth rate. Their work extends that of Cohen (1971) by investigating how nitrogen dynamics alter the decision to lower. Cohen (1971) paved the way for reproductive growth models by investigating the effect of transition to lowering on yield. He determined that one transition which allocates all resources from vegetative growth to reproductive growth is the most beneicial for yield, and that lowering time is dependent upon the length of the growth season. Other models build onto this by including loss terms (King and Roughgarden 1982) or environmental conditions and hazard rates (Paltridge and Denholm 1974;Ledder et al. 2004). Others investigate the effects of multiple reproductive phases (Chiariello and Roughgarden 1984) and the effect of photosynthetic rate on reproduction (Schaffer et al. 1982). However, there has been little previous work investigating the effect of varying vegetative allocation strategy on yield. Having higher leaf respiration than roots delays the optimal age to reproduce for leaf allocation strategies and increases the restrictions on lowering time for root allocation strategies.
The model results hinge upon the assumption that growth stops when carbon sources and sinks are balanced. The extent to which this situation arises in natural or crop systems is unclear, and three factors would tend to act so that steady state is not reached. First, new growth is always required to replace tissues as they turnover. Secondly, the requirement for roots (in terms of anchorage, nutrient and water uptake) may be less than the limit imposed by maintenance costs. A model incorporating tissue turnover and functional roots would be required to evaluate the magnitude and consequences of these effects. Finally, the development of short-lived plants, in particular, ensures that the limits imposed by carbon balance are not reached. Nonetheless, these limits set boundaries beyond which development cannot stray.
The model simulates carbon allocation by using a constant allocation ratio throughout the ontogeny of a plant and, therefore, has only one pre-determined allocation strategy throughout. However, allocation is usually allometric, such that it depends on size, and the proportion of carbon allocated to leaves or roots varies with time (Weiner 2004). The choice of a simpliied assumption of allocation within the model provides a framework to test distinct strategies of allocation and make predictions on how certain behaviours can inluence plant growth. Further analysis with this model could consider an allocation strategy that is variable and dependent upon plant size.
Having the roots solely as sinks within the model underpins the inding that minimal allocation to roots maximizes plant growth. However, resources obtained via the roots inluence the allocation of carbon to sources and sinks and also overall plant growth (Running and Gower 1991). Environmental conditions control luctuations in carbon and nitrogen availability, causing crosstalk between signalling pathways of carbon and nitrogen (White et al. 2016). This crosstalk determines allocation to sources and sinks. When there is ample nitrogen, cytokinins are produced, which increases sink strength (Kuiper 1993;Ghanem et al. 2011;Thomas 2013), and this also increases carbon acquisition. When there is high carbon availability, nitrogen sources are up-regulated (Stitt and Krapp 1999) and sink activity is increased (Klein et al. 2000;Kaiser et al. 2002;Reda 2015). There is a need for a new type of allocation model which includes the interaction of carbon and nitrogen feedback and feedforward mechanisms deining allocation to further understand how allocation can be used to improve growth.
Overall, this quantitative modelling approach has revealed that the maintenance costs of leaves and roots alter the limitations on allocation by increasing the plasticity of inal plant size in vegetative growth and restricting the range of optimal developmental stages for reproduction. It has shown that when leaf respiration is higher than root respiration, reallocating carbon away from leaves and towards the roots can improve plant growth. This analysis indicates that crop improvement and management strategies should consider the effects of canopy maintenance costs for improving growth.

Sources of Funding
The authors thank the Leverhulme Centre for Advanced Biological Modelling (X/009780-11-22) for inancial support.   This demonstrates that the only role that the parameter η plays in the behaviour of the model is to regulate the rate of change of plant mass. Neither the relative amounts of leaf or root tissue nor the inal plant mass depends on the value of η. This is illustrated in Fig. A1.

Appendix 2.
The black lines represent parameter space where final plant material remains the same and the colours represent different final plant materials (yellow is high plant material and blue is virtually no plant material). For low shading coefficient values, the black lines of constant final plant material are horizontal for the majority of allocation strategies (Fig. A2). As allocation strategy continues to increase after α = 2.7 , these lines become almost vertical. As the shading coefficient increases, the region where the black line is approximately horizontal for a given shading value decreases in size and is much smaller using θ>7. This implies that the shading coefficient reduces the level of plasticity within the model by increasing light penetration into the leaf canopy.