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Genome-wide association mapping of growth dynamics detects time-specific and general quantitative trait loci.

Bac-Molenaar JA, Vreugdenhil D, Granier C, Keurentjes JJ - J. Exp. Bot. (2015)

Bottom Line: Genome-wide association (GWA) mapping of the temporal growth data resulted in the detection of time-specific quantitative trait loci (QTLs), whereas mapping of model parameters resulted in another set of QTLs related to the whole growth curve.The positive correlation between projected leaf area (PLA) at different time points during the course of the experiment suggested the existence of general growth factors with a function in multiple developmental stages or with prolonged downstream effects.In addition, the detection of QTLs without obvious candidate genes implies the annotation of novel functions for underlying genes.

View Article: PubMed Central - PubMed

Affiliation: Laboratory of Plant Physiology, Wageningen University, Droevendaalsesteeg 1, 6708 PB Wageningen, The Netherlands Laboratory of Genetics, Wageningen University, Droevendaalsesteeg 1, 6708 PB Wageningen, The Netherlands.

No MeSH data available.


Related in: MedlinePlus

Modelling of growth using three mathematical functions, Expo1, Expo2, and Gom (see Table 2 for details). (A) S-curve as a model for determinate growth consisting of three phases: the acceleration phase, the linear phase, and the saturation phase. The S-curve has three characteristic points: the inflection point ‘K’ where the growth changes from increasing to decreasing, the point of maximal acceleration ‘s1’, and the point of maximal deceleration ‘s2’. (B) Data and curve fits for line CS76226 (Se-0), representative for the 11% of the plants for which the growth curve reached ‘K’. The data were fitted to the three models. It is shown that even if the inflection point ‘K’ is reached, model Expo1 and Expo2, which both assume indeterminate growth, resulted in good fits. (C) Data and curve fits for line CS76308 (ZdrI2-25), representative for 56% of the plants for which the growth curve did not reach ‘s1’. The data were fitted to the three models. (D) Magnification of (C) allowing better comparison of the various models for the time window of the experiment.
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Figure 3: Modelling of growth using three mathematical functions, Expo1, Expo2, and Gom (see Table 2 for details). (A) S-curve as a model for determinate growth consisting of three phases: the acceleration phase, the linear phase, and the saturation phase. The S-curve has three characteristic points: the inflection point ‘K’ where the growth changes from increasing to decreasing, the point of maximal acceleration ‘s1’, and the point of maximal deceleration ‘s2’. (B) Data and curve fits for line CS76226 (Se-0), representative for the 11% of the plants for which the growth curve reached ‘K’. The data were fitted to the three models. It is shown that even if the inflection point ‘K’ is reached, model Expo1 and Expo2, which both assume indeterminate growth, resulted in good fits. (C) Data and curve fits for line CS76308 (ZdrI2-25), representative for 56% of the plants for which the growth curve did not reach ‘s1’. The data were fitted to the three models. (D) Magnification of (C) allowing better comparison of the various models for the time window of the experiment.

Mentions: To be able to quantify the dynamics in rosette growth over time, growth was modelled using different mathematical functions (Table 2; Supplementary Table S2 at JXB online). Determinate growth (i.e. growth that terminates before the end of the life cycle of an organism) can in many cases be described by a sigmoid function (S-curve). Rosette growth of Arabidopsis is known to be determinate, following such an S-curve (Boyes et al., 2001). S-curves are characterized by an accelerating phase, a linear phase, and a saturation phase (Fig. 3A). Within the linear phase, which is not really linear, but can be approached by a linear function, the inflection point ‘K’ is located. At ‘K’, the curve changes from increasing growth to decreasing growth. Near the end of the acceleration phase, which can be approached by an exponential function, the point of maximal acceleration ‘s1’ is located. Near the beginning of the saturation phase, the point of maximal deceleration ‘s2’ is reached and, thereafter, the growth gradually stops and the final rosette size ‘Amax’ is reached. Determinate growth was modelled using the Gompertz function (Gom), which results in an S-curve (Gompertz, 1825; Winsor, 1932) (Table 2). This function is a slightly modified form of the basic logistic function, which was first described by Pierre Verhulst in 1838 (Verhulst, 1838). The modifications of the basic logistic function change this basic symmetric growth curve into an asymmetric one. The Gompertz function used here contains three parameters: ‘Amax’, the final rosette size; ‘b’ that determines the position of the curve along the time axis; and ‘r’ that determines the growth rate at the inflection point ‘K’ (Table 2). The combination of these three parameters determines on which day ‘s1’, ‘s2’, and ‘K’ are reached. As the growth curves fitted with Gom were investigated, none of the plants in this experiment reached ‘Amax’ and only 4% reached ‘s2’within the window of the experiment. Even ‘K’ and ‘s1’ were not reached by the majority of the plants: for 89% of the plants ‘K’ was not reached before day 28 and for 57% of the plants ‘s1’ was not reached before day 28. This means that for most plants the collected data points are located in the accelerating phase and the beginning of the linear phase of the growth curve, implying that the plants had not yet entered the saturation phase. This was expected for the plants that had not yet bolted, but for the 30% of the plants that were bolting on the last day it was expected that they would have reached at least the saturation phase, because earlier studies reported that Arabidopsis rosettes reach the final size when they start to flower (Boyes et al., 2001). Because most plants were in the acceleration phase even on the last day of the experiment, the growth was modelled not only with the Gompertz curve that describes determinate growth, but also with models that describe indeterminate growth (e.g. exponential growth). The simplest indeterminate growth model used was based on an exponential function (Expo1) with one parameter ‘r’, which represents the growth rate (Table 2). This model assumes that the growth rate is equal during the whole growth period and that the initial size (A0) is 1 (Table 2). Exponential growth was also modelled with a function (Expo2) with two parameters, growth rate ‘r’ and the initial size (A0) (Table 2; Supplementary Table S2). A0 not only represents the starting value, but it is also a magnification factor. This means that for two plants with equal ‘r’ and a factor two difference in A0, plant size is also a factor of two different during the whole experiment. To illustrate the use of the three models, data from two plants that showed determinate and indeterminate growth were used for curve fitting (Fig. 3B–D). The plant with determinate growth is representative for 11% of the plants in which growth reached ‘K’ within the course of the experiment (Fig. 3B). The plant with indeterminate growth is representative for the 56% of the plants for which growth did not reach ‘s1’ within the course of the experiment (Fig. 3C, D).


Genome-wide association mapping of growth dynamics detects time-specific and general quantitative trait loci.

Bac-Molenaar JA, Vreugdenhil D, Granier C, Keurentjes JJ - J. Exp. Bot. (2015)

Modelling of growth using three mathematical functions, Expo1, Expo2, and Gom (see Table 2 for details). (A) S-curve as a model for determinate growth consisting of three phases: the acceleration phase, the linear phase, and the saturation phase. The S-curve has three characteristic points: the inflection point ‘K’ where the growth changes from increasing to decreasing, the point of maximal acceleration ‘s1’, and the point of maximal deceleration ‘s2’. (B) Data and curve fits for line CS76226 (Se-0), representative for the 11% of the plants for which the growth curve reached ‘K’. The data were fitted to the three models. It is shown that even if the inflection point ‘K’ is reached, model Expo1 and Expo2, which both assume indeterminate growth, resulted in good fits. (C) Data and curve fits for line CS76308 (ZdrI2-25), representative for 56% of the plants for which the growth curve did not reach ‘s1’. The data were fitted to the three models. (D) Magnification of (C) allowing better comparison of the various models for the time window of the experiment.
© Copyright Policy - creative-commons
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4585414&req=5

Figure 3: Modelling of growth using three mathematical functions, Expo1, Expo2, and Gom (see Table 2 for details). (A) S-curve as a model for determinate growth consisting of three phases: the acceleration phase, the linear phase, and the saturation phase. The S-curve has three characteristic points: the inflection point ‘K’ where the growth changes from increasing to decreasing, the point of maximal acceleration ‘s1’, and the point of maximal deceleration ‘s2’. (B) Data and curve fits for line CS76226 (Se-0), representative for the 11% of the plants for which the growth curve reached ‘K’. The data were fitted to the three models. It is shown that even if the inflection point ‘K’ is reached, model Expo1 and Expo2, which both assume indeterminate growth, resulted in good fits. (C) Data and curve fits for line CS76308 (ZdrI2-25), representative for 56% of the plants for which the growth curve did not reach ‘s1’. The data were fitted to the three models. (D) Magnification of (C) allowing better comparison of the various models for the time window of the experiment.
Mentions: To be able to quantify the dynamics in rosette growth over time, growth was modelled using different mathematical functions (Table 2; Supplementary Table S2 at JXB online). Determinate growth (i.e. growth that terminates before the end of the life cycle of an organism) can in many cases be described by a sigmoid function (S-curve). Rosette growth of Arabidopsis is known to be determinate, following such an S-curve (Boyes et al., 2001). S-curves are characterized by an accelerating phase, a linear phase, and a saturation phase (Fig. 3A). Within the linear phase, which is not really linear, but can be approached by a linear function, the inflection point ‘K’ is located. At ‘K’, the curve changes from increasing growth to decreasing growth. Near the end of the acceleration phase, which can be approached by an exponential function, the point of maximal acceleration ‘s1’ is located. Near the beginning of the saturation phase, the point of maximal deceleration ‘s2’ is reached and, thereafter, the growth gradually stops and the final rosette size ‘Amax’ is reached. Determinate growth was modelled using the Gompertz function (Gom), which results in an S-curve (Gompertz, 1825; Winsor, 1932) (Table 2). This function is a slightly modified form of the basic logistic function, which was first described by Pierre Verhulst in 1838 (Verhulst, 1838). The modifications of the basic logistic function change this basic symmetric growth curve into an asymmetric one. The Gompertz function used here contains three parameters: ‘Amax’, the final rosette size; ‘b’ that determines the position of the curve along the time axis; and ‘r’ that determines the growth rate at the inflection point ‘K’ (Table 2). The combination of these three parameters determines on which day ‘s1’, ‘s2’, and ‘K’ are reached. As the growth curves fitted with Gom were investigated, none of the plants in this experiment reached ‘Amax’ and only 4% reached ‘s2’within the window of the experiment. Even ‘K’ and ‘s1’ were not reached by the majority of the plants: for 89% of the plants ‘K’ was not reached before day 28 and for 57% of the plants ‘s1’ was not reached before day 28. This means that for most plants the collected data points are located in the accelerating phase and the beginning of the linear phase of the growth curve, implying that the plants had not yet entered the saturation phase. This was expected for the plants that had not yet bolted, but for the 30% of the plants that were bolting on the last day it was expected that they would have reached at least the saturation phase, because earlier studies reported that Arabidopsis rosettes reach the final size when they start to flower (Boyes et al., 2001). Because most plants were in the acceleration phase even on the last day of the experiment, the growth was modelled not only with the Gompertz curve that describes determinate growth, but also with models that describe indeterminate growth (e.g. exponential growth). The simplest indeterminate growth model used was based on an exponential function (Expo1) with one parameter ‘r’, which represents the growth rate (Table 2). This model assumes that the growth rate is equal during the whole growth period and that the initial size (A0) is 1 (Table 2). Exponential growth was also modelled with a function (Expo2) with two parameters, growth rate ‘r’ and the initial size (A0) (Table 2; Supplementary Table S2). A0 not only represents the starting value, but it is also a magnification factor. This means that for two plants with equal ‘r’ and a factor two difference in A0, plant size is also a factor of two different during the whole experiment. To illustrate the use of the three models, data from two plants that showed determinate and indeterminate growth were used for curve fitting (Fig. 3B–D). The plant with determinate growth is representative for 11% of the plants in which growth reached ‘K’ within the course of the experiment (Fig. 3B). The plant with indeterminate growth is representative for the 56% of the plants for which growth did not reach ‘s1’ within the course of the experiment (Fig. 3C, D).

Bottom Line: Genome-wide association (GWA) mapping of the temporal growth data resulted in the detection of time-specific quantitative trait loci (QTLs), whereas mapping of model parameters resulted in another set of QTLs related to the whole growth curve.The positive correlation between projected leaf area (PLA) at different time points during the course of the experiment suggested the existence of general growth factors with a function in multiple developmental stages or with prolonged downstream effects.In addition, the detection of QTLs without obvious candidate genes implies the annotation of novel functions for underlying genes.

View Article: PubMed Central - PubMed

Affiliation: Laboratory of Plant Physiology, Wageningen University, Droevendaalsesteeg 1, 6708 PB Wageningen, The Netherlands Laboratory of Genetics, Wageningen University, Droevendaalsesteeg 1, 6708 PB Wageningen, The Netherlands.

No MeSH data available.


Related in: MedlinePlus