Three broadleaf tree species, balsam poplar (Populus balsamifera L.), black cottonwood(Populus trichocarpa Torr. & Gray), and white birch (Betula papyrifera Marsh.) were used for this study. White birch seeds and black cottonwood branch cuttings were collected from trees in the city of Thunder Bay (48˚42'19"N, 89˚26'01"W). Balsam poplar seeds from Kemptville (45˚02"N, 75˚39'W) were obtained from the National Tree Seed Centre in Fredericton, NB, Canada. Balsam poplar and white birth seeds were processed according to The Woody Plant Seed Manual [33] and sown in germination trays in the Lakehead University greenhouse. The cottonwood cuttings were treated with a rooting hormone (Plat Prod Stim Root #3, Plant Products Co. Ltd. Brantford, ON, CA) before being planted. The cuttings were misted continuously in a polyethylene tent during the period of root induction in the greenhouse. The seedlings and rooted cuttings were transplanted into 3.5 L plastic pots filled with peat moss and vermiculite (1:1, v:v). The plants were watered as needed to keep the growing medium moist and fertilized twice a week with 75 mg·L⁻¹ of a fertilizer solution (All-Purpose, 24-8-16 N-P-K fertilizer, Plant Products Co. Ltd. Mississauga, ON, CA). The greenhouse conditions were 23/16˚C day/night temperatures, 16-hour photoperiod and 50% RH. The maximum flux density of photosynthetically active radiation at the canopy level was 500 μmol·m⁻²·s⁻¹ on a sunny day. The environmental conditions were monitored and controlled using an Argus Titan System (Argus Control Systems Ltd., Surrey, BC, Canada).

Following two months of growth, six seedlings were randomly selected from each species. The gas exchange of the 1^st fully expanded leaf on the terminal shoot sample trees was measured using a PP-Systems CIRAS-3 Portable Photosynthesis System equipped with a PLC3 Universal Leaf Cuvette with automatic climate control and a built-in CFM-3 Chlorophyll Fluorescence Module (PP Systems International, Inc. Amesbury, MA, USA). The photosynthetic responses to CO₂, i.e., A/C_i curves, were measured between 9:00 and 16:00. The Laisk script measurements were taken at 200, 150, 100, and 50 μmol·mol⁻¹ CO₂ concentration (C_a) and 300, 150, and 75 μmol·m⁻²·s⁻¹ PAR. Subsequently, full photosynthetic CO₂ response curves (A/C_i curves) were measured at 400, 300, 200, 150, 100, 50, 400, 600, 800, 1000, and 1200 μmol·mol⁻¹ CO₂ and 1000 μmol·m⁻²·s⁻¹ PAR. The apparent quantum yield was derived from measurements taken at 400 μmol·mol⁻¹ CO₂ and 50, 100, 150, 200, 250 μmol·m⁻²·s⁻¹ PAR; the slope of the A-C_i response was derived from light response curves (ΔA/ΔC_i-lrc, Figure 1(c)). All the measurements were made on the same leaf blade.

The point P_v (novel vertical line method in 2.4 section) and P_mp (conventional average midpoint method in 2.5 section) of the Laisk method (Figure 1(c)) were estimated first and then used to calculate R_d and C i * (Figure 1(b)). The variable J method [34] was used to calculate g_m, where the electron transport (J) was calculated from chlorophyll fluorescence according to Momayyezi’s protocol and Γ* was assumed to equal to C i * [35] . The chlorophyll fluorescence measurement was taken using the built-in CFM-3 model in the PP Systems CIRAS-3 system.

The A/C_i data were analyzed using the Plantecophysfitaci function of the R package to produce the maximum rate of ribulose-1,5-bisphosphate (RuBP) carboxylation (V_cmax, μmol·m⁻²·s⁻¹), the maximum rate of photosynthetic electron transport (J_max, μmol·m⁻²·s⁻¹) [10] . The initial slope of A/C_i was recorded as the maximum apparent carboxylation efficiency (ACE = ΔA/ΔC_i).

Figure 1(c) is the Laisk schematic diagram based on actual measurements (Table S1). The three A/C_i lines PAR₃₀₀, PAR₁₅₀, and PAR₇₅ (solid line in Figure 1(c)) are the initial linear part of the Demand Functions (DF) at 300, 150 and 75 μmol·m⁻²·s⁻¹ photosynthetically active radiation (PAR) flux density, respectively; the bold lines are the Supply Functions (SF) at three different ambient CO₂ concentration (C_a) of 100, 150 and 200 μmol·mol⁻¹, the slope of which represents the rate of CO₂ diffusion from leaf surface (C_a) through stomates into the intercellular space (C_i). The three intersecting points of the three supply functions for the same C_a with their corresponding demand functions measured at the three PARs fall on a straight line which we define as the Demand-Supply Function (DSF) (the three dashed lines, DSF₂₀₀, DSF₁₅₀ and DSF₁₀₀ in Figure 1(c)). The X-intercepts of the three DSFs are designated by their corresponding C_a as C_i-dsf100, C_i-dsf150 and C_i-dsf200 (Figure 1(c)). The purpose of this study was to explore the characteristics and physiological significance of the Demand-Supply Functions.

The Laisk method assumes that the initial part of three A/C_i curves measured under three unsaturated PAR flux densities will intersect at a common point ( C i * , −R_d), where C i * represents the CO₂ compensation point at intercellular CO₂ concentration and R_d represents daytime respiration rate in absence of photorespiration. The method is based on FvCB biochemical model of photosynthesis as A = V_c (C_c − Γ*)/C_c − R_d. Where A is the net rate of CO₂ assimilation, V_c represents the Rubisco carboxylation rate, C_c represents CO₂ concentration in the chloroplast, Γ* is chloroplast CO₂ compensation point [13] [36] . When C_c equals Γ*, A equals -R_d. Here we assume that the g_m is infinite as the Laisk method only provides C_i values but not C_c [37] . In reality, however, the three A/C_i lines rarely intersect at a single point. Instead, there are generally three pairwise intersections that form an obtuse triangle (P₁P₂P₃, Figure 1(c)). The conventional protocol uses the average of the three intersecting points (point P_mp in Figure 1(c)) to determine Ci* and −R_d [32] . But the three lines have different weights in terms of slope because of the multiple resistances to CO₂ diffusion, enzyme variables [22] . Furthermore, measurement noises can magnify the separation of the intersecting points when the differences in slope between the three lines are small.

The Demand-Supply Functions (DSF) for different C_a that we derived have similar slopes (DSF₁₀₀, DSF₁₅₀, and DSF₂₀₀ in Figure 1(c), to be explained later) and can be used to estimate C i * and R_d. We proposed to use a line perpendicular to the line with the average slope (DSF_mean in Figure 1(d)) of the three DSFs (i.e., l_v in Figure 1(c), Figure 1(d)) as an auxiliary line for estimating C i * and R_d. This line has a negative slope reciprocal to the average slope of DSFs and an unknown intercept. In theory, line l_v is also the initial section of an A/C_i curve (demand function) at a certain PAR, similar to PAR₃₀₀, PAR₁₅₀, and PAR₇₅ (initial section of A/C_i curves measured at 300, 150, 75 μmol·m⁻²·s⁻¹ PAR, respectively, in Figure 1(c)). l_v is supposed to go through the intersection point ( C i * , −R_d) that all A/C_i curves are theoretically supposed to intersect each other regardless of PAR under which the A/C_i curves are measured.

We use the reduction to absurdity by introducing the vertical line (l_v) from Laisk dataset and developed a novel vertical line method (VL) to calculate the point ( C i * , −R_d). Based on the above description, it’s assumed that that line l_v should pass through point P₂ that the intersection of two A/C_i with the biggest difference in PAR (PAR75 and PAR300, Figure 1(d)). A parallel shift of l_v to the left (l_v1) or right (l_v2) will increase the total length of line segments of P₄P₅, P₄P₆, P₅P₆ which are formed by the intersections of l_v and three A/Ci curves, because the total length of P₄P₆ and P₅P₆ is always greater than the length of P₂P_v150 (Figure 1(d)). According to the FvCB model, these intersections theoretically should converge, when the sum of the distances among the three intersecting points will be minimal and thus the position of line l_v is determined. Therefore, after passing the intersecting point P₂ between PAR₃₀₀ and PAR₇₅, line l_v will intersect PAR₁₅₀ at point P_v150, and the midpoint (P_v) of line P₂P_v150 will be at or near the theoretical converging point ( C i * , −R_d).

In addition to the conventional average midpoint method (MP), Walker and Ort [38] proposed a slope-intercept method (SI) for Laisk data analysis. The slope-intercept method treats the slope and intercept of each A/C_i line (initial part of A/C_i curve) as a point (slope, intercept), and the points from multiple A/C_i lines produce a new regression the slope of which equals to − C i * and the y-intercept of which gives R_d. First, we compared the estimates of C i * and R_d from the Laisk data set using three methods, i.e., conventional average midpoint method, slope-intercept method and our novel vertical line method, and used the estimated C i * and R_d to estimate g_m. Secondly, C i * and R_d estimates using Walker’s data (Table S2) were also produced and compared among the three methods; but we only used the lower three (50, 130, 240 mmol·m⁻²·s⁻¹) of the five PARs because the average midpoint method and the vertical line method can only use for three PARs, and the lower three PARs are more suitable for meeting the sub-saturated light intensity requirement of the Laisk method.

The assay of the carbonic anhydrase activity (CAU) was based on the bromothymol blue colorimetry described by Wilbur [39] . 0.2 g leaf blade was ground in 1 ml of 40 mM potassium phosphate buffer (pH = 8.3) using a mortar and pestle on ice. The homogenate was centrifuged for 10 min at 5000 g and 4˚C, 20 μl of the supernatant was added to 1 ml of the buffer solution containing 20 mg·L⁻¹ bromothymol blue as a pH indicator. 1 ml CO₂-saturated water of 4˚C was then added and the time (as T) that it took for the Ph of the reaction system to change from 8.3 to 6.3 was recorded. 20 μL buffer solution only was used as control and the time of pH change from 8.3 to 6.3 after adding 1 ml CO₂-saturated water was recorded as T0. The carbonic anhydrase activity was calculated as CA (EU) = 10 × (T0 ÷ T − 1).

The differences in the parameters of the supply and demand function (dsf) among different species were tested using one-way ANOVA. The effects of Lasik calculation methods and species on photosynthetic parameters were tested using two-way ANOVA. Tukey-HSD Post-hoc comparisons were conducted when ANOVA showed a significant effect. Pearson correlation analysis and linear regression were performed to examine the relationships between parameter values estimated from the P_v method and those from the other methods. All the analyses were conducted using R 4.0.4. Principal component analysis (PCA) was applied to photosynthetic parameters using the PCA function from the FactoMineR package.

The significant differences in the slope of the Demand-Supply Function (DSF) indicated species specificity (Table S3): the l_v slope was significantly smaller in black cottonwood than in balsam poplar and white birch (Figure 2(a)); however,

the trend for the slope of DSF lines was the opposite, i.e., it was significantly greater in the cottonwood than in balsam poplar and white birch (Figure 2(b)). The DSF slopes of the same species for the ambient CO₂ concentration (C_a, from 100 to 200 μmol·mol⁻¹) were not significantly different from each other, suggesting that the DSF lines were approximately parallel to each other (Figure 2(b)). The coefficient of variation in DSF slope and l_v slope was much greater in white birch than the other two tree species (Table 2). The ratio of C_i-dsf (DSF intercept on the X-axis) to C_a (C_i-dsf/C_a) in the three species was in a narrow range of 0.96 to 0.97 (Figure 2(c)). This suggests that for a given C_a, the X-axis intercept of the fitting line of the Demand-Supply Function was relatively constant.

The slope of the Demand-Supply Function (dsf) was inversely related to stomatal conductance (g_s) and leaf surface CO₂ concentration (C_a) and the relationship varied with species (Figure 3(a) and Figure 3(b)): balsam poplar and white birch had similar trends and had much steeper DSF slopes and smaller stomatal conductance than black cottonwood, and dsf had a strong correlation with and g_s (Figure 3(a)) but a weak correlation with C_a (except white birch) (Figure 3(b)). No significant correlation was observed between C_i/C_a and g_s (Figure 3(c)). C_i/C_a declined with increases in C_a, however, the relationship was not obviously different between the three species (Figure 3(d)).

The multivariate relationships between photosynthetic parameters were summarized using two principal components (PCA, Figure 4). dsf and ΔA/ΔC_i-lrc

clustered on the left side of principal component 1 (PC1) while ACE and AQY grouped on the right side, indicating that there might be negative correlations between the two groups (Figure 4). The differences in these four parameters in the three tree species were similar to those in PCA (Table S4 & Figure 4) and there were similarities within the group (dsf and ΔA/ΔC_i-lrc, ACE and AQY) while opposite variations between the groups (Figure 4 and Figure 5).

An obvious phenomenon was that dsf and ΔA/ΔC_i-lrc almost overlapped in PCA (Figure 4), indicating a close relationship between them. ANOVA analysis verified that dsf and ΔA/ΔC_i-lrc (the values of ΔA/ΔC_i from different PARs) had no significant difference between different C_a but there were differences among species (Table S5), suggesting that dsf and ΔA/ΔC_i-lrc might be species-specific and independent of the measurement conditions (PARs and C_a).

C i * and R_d estimates from our new vertical line method (VL) were not significantly different from those estimated using the conventional average midpoint method (MP) and slope-intercept method (SI) for all three tree species (Table S6 and Figure 6(a), Figure 6(b)). However, white birch had significantly lower R_d than black cottonwood and balsam poplar and the variation and coefficient of variation in R_d were much greater for the combination of the MP method and

birch than the other combinations (Table 2 & Figure 6(a)). There was no significant difference in C i * among the three species, but the variation and coefficient of variation in C i * were much larger for the combination of white birch and the MP method (Table 2 & Figure 6(b)). g_m estimation was influenced by species and the method of C i * and R_d estimation (Table S6). White birch had significantly lower g_m than in the other two species (Figure 6(c)), and the slope-intercept method produced significantly greater g_m estimation than the MP method and vertical line method (Figure 6(d)). The g_m calculated by the MP method was slightly smaller than that calculated by the VL method, but the difference was not statistically significant (Figure 6(d)).

Using the three lower PAR curves of Walker’s data, the C i * and R_d estimates using the three methods were not significantly different from each other and the values from our vertical line method generally fell between those of the other two methods (Table S7). However, the slope-intercept method produced much greater R_d estimates when using 5 PAR curves (0.63) than using 3 curves (0.51) or the other two methods (0.54 and 0.52) (Table S7).

The intersecting points of supply functions and demand functions were estimated using the measurements taken under three different PARs but the same C_a and formed a straight line (DSF). The slope of DSF (dsf, dsf = ΔA_n/ΔC_i, Figure 1(b), Figure S2) represents A response to low PAR at various C_a and its

absolute value represents the apparent quantum year of photosynthesis at various C_a and non-saturating PAR (ΔA_n/ΔC_i-lrc in Figure S1). The apparent carboxylation efficiency (ACE) was derived from the initial slope of A/C_i (ACE = ΔA_n/ΔC_i, using the same algorithm as for dsf and △A_n/△C_i-_lrc). Furthermore, ACE increased with increases in PAR (Figure S3(a)). Similarly, the apparent quantum yield (AQY) derived from the initial slope of a light response curve represents the quantum yield under a certain C_a and AQY increases with increasing C_a (Figure S3(a)) [30] . Many studies have identified similar patterns in ACE and AQY [38] [40] [41] [42] [43] , limiting the application of ACE and AQY because they can only describe the photosynthetic characteristics under a particular measurement condition (PAR or C_a). Under dynamic environmental conditions, plants can rapidly coordinate CO₂ diffusion and biochemical fixation [44] , indicating the existence of internal coordination mechanisms [45] [46] .

The apparent carboxylation efficiency reflects the efficiency of assimilation of the CO₂ in the intercellular space under a certain PAR (usually saturating PAR). It is considered to be the ultimate limiting factor of CO₂ fixation and is related to the amount and activity of Rubisco [47] . Moreover, ACE increased with increases in PAR (Figure S3(a)), indicating that the actual intrinsic carboxylation efficiency (CEi) of a leaf was co-determined by Rubisco activity (related to V_cmax) and RuBP regeneration (related to J_max) [48] . Similarly, the initial slope of LRC (AQY) for a certain C_a increased with increases in C_a (Figure S3(b)). The intrinsic quantum yield (QYi) of a leaf may be affected by both Rubisco activity and RuBP regeneration [31] . V_cmax and J_max (as indicators of photosynthetic capacity) might excert their effects through carboxylation efficiency and quantum yield [25] . These were partially explained by PCA results where ACE and AQY were grouped in the middle region of V_cmax and J_max. However, other studies have shown that V_cmax and J_max are independent of g_s and CO₂ diffusion [48] . To sum up, dsf or ΔA_n/ΔC_i-lrc may provide a link between CO₂ diffusion and biochemical characteristics of photosynthesis because of its close relationships with g_s, ACE and AQY.

The values of Laisk measurements are co-limited by Rubisco and RuBP regeneration. The Rubisco restriction is reflected in the linear or initial portion of an A/C_i curve and RuBP regeneration restriction prevents the lines with different PAR from overlapping [49] . Hence, dsf (or ΔA_n/ΔC_i-lrc) probably acts as an internal coordination mechanism between CO₂ diffusion and fixation in photosynthesis when PAR and C_a both change simultaneously [50] . Rubisco limitation and RuBP regeneration limitation often co-exist in the nature, especially under fluctuating light and drought conditions (may cause C_i decrease) [51] . Therefore, results derived from a normal A/C_i (ACE, V_cmax and J_max) and LRC (AQY) may not be suitable to explain photosynthesis at these conditions.

Another feature of dsf or ΔA_n/ΔC_i-lrc is that it is independent of gas exchange measurement conditions (lower PAR and C_a). The dsf has little dependence on C_a since the Michaelis–Menten constants for the carboxylase (K_c) are relatively large [29] [52] , which may explain why DSF from different C_a were almost parallel to each other and why there was no significant difference between dsf and ΔA_n/ΔC_i-lrc. This property allows the use of dsf or ΔA_n/ΔC_i-lrc to link CO₂ supply and biochemical demand of photosynthesis when Rubisco and RuBP regeneration are co-limiting.

The supply function reflects the diffusion of CO₂ while the demand function reflects the photochemical and biochemical processes of CO₂ assimilation as a function of intercellular CO₂ concentration (C_i) [20] . C_i acts as a CO₂ pool connecting upstream and downstream, which regulates the supply-demand relationship of photosynthesis to a certain extent [53] . Our results showed that C_i/C_a was not closely related to g_s within the range of C_a used in this study although it decreased with decreases in C_a to some extent while no significant differences between in species. Woody plants usually adopt a dynamic leaf gas-exchange strategy and do not maintain a constant value of C_i/C_a [9] . Therefore, C_i/C_a was not suitable to describing coordination between CO₂ diffusion and photosynthetic biochemistry under the Laisk measurement protocol in the three species. However, the slope of the Demand-Supply function was closely related to g_s, and different between the species and thus may better reflect the response of photosynthesis to stomatal conductance.

There are concerns about the validity of Laisk method which is widely used to estimate R_d [23] . Our results demonstrate that the novel vertical line method produced more robust estimates of C i * and R_d than the conventional average midpoint method, especially in white birch. Although the theoretical basis of the Laisk method is the FvCB model, the R_d estimate using the original Laisk method (Laisk 1977) is compromised because decreasing light reduces C i * which in turn affects the estimate of R_d [36] . The Laisk method uses the initial portion of A/C_i curves under several sub-saturation light intensities. However, it is unknown whether photosynthesis is limited by Rubisco activities or by RuBP regeneration under such measurement conditions . Those partial A/C_i curves theoretically should intersect at a single point defined by ( C i * , −R_d), but different curves have different weights on the determination of the intersecting point [22] . The different A/C_i curves measured in the Laisk Method tend to have similar slopes and a great degree of overlapping. Consequently, the determination of the intersection point and thus C i * and R_d estimation are more vulnerable to instrument and operation errors [38] .

The conventional average midpoint method directly averages the coordinate values of the pairwise intersecting points of three A/C_i curves and therefore is also prone to errors if the differences in slopes are small [38] . Mean values of C i * and R_d fluctuate greatly (higher coefficient of variation) when calculated by the average midpoint method in white birch, leading to difficulties for the application of the Laisk method. Our results show that the coefficient of variation of l_v slope (from the average DSF slope) of white birch was significantly larger than those of the other two species, and the DSF showed a slightly increasing trend with increases in C_a and was lower in black cottonwood and balsam poplar, suggesting that the poor performance of the Laisk method (i.e., failure to intersect at one point or the intersection was not in the fourth quadrant) in white birch may be related to the coordination of supply and demand functions. The vertical line method produced more robust estimates of C i * and R_d than the average midpoint method, possibly because the slope of the auxiliary line l_v was much larger than those of the initial A/C_i lines in the conventional Laisk method. Deeper slopes generally produce more stable and reliable determinations of the intersecting point [54] .