Effective diffusion rates and cross-correlation analysis of"acid growth"data

We investigated the growth-temperature relationship in plants using a quantitative perspective of a recently derived growth functional. We showed that auxin-induced growth is achieved by the diffusion rate, which is almost constant or slowly ascending in temperature while the diffusion rate of fusicoccin-induced growth increases monotonically with temperature for the entire temperature range, though for some concentrations of IAA"super-diffusion"takes place for unperturbed growth. Furthermore, three kind of experiments were compared: for abraded coleoptiles, coleoptile segments and intact growing seedlings. From cross-correlation analysis it was found that the timing of IAA and FC-induced proton secretion and growth matches well. Unambiguous results, concerning fundamental conditions of the acid growth hypothesis, were obtained by cross- and auto-correlation analysis: (1) For abraded coleoptiles, because of the lowering of the cuticule potential barrier, auxin-induced cell wall pH decreases simultaneously with the change in growth rate; no advancement or retardation of pH (proton efflux rate) or growth rate takes place (2) Exogenous protons are able to substitute for auxin causing wall loosening and growth (3) Although the underlying molecular mechanisms vastly differ, a potent stimulator of proton secretion, the fungal toxin FC, promotes growth similar to auxin, however of much elevated intensity; as for auxin - no advancement or retardation takes place.


Introduction
Plants evolve within the universal constraints imposed by the plant cell wall, dynamically equilibrating the turgor pressure inside the wall (Lintilhac 2014). At the lowest level, the description of cell/plant organ evolution may be expressed in terms of the biophysics and mechanics of the cell wall during growth. Cell extension growth in turgid plant cells/organs is brought about by loosening of the structure of the growth-restraining cell walls, resulting in the relaxation of wall tension and concomitant water uptake (Schopfer 2001). However, the biochemical mechanism of this wall loosening reaction has not yet been fully elucidated. Numerous proteins were recognized as catalysts, in particular cell-wall polysaccharides or expansin family that cause stretch-dependent creep in acidified cell walls by breaking intermolecular non-covalent bonds (Cosgrove 1999(Cosgrove , 2000. Primary wall extension growth ("diffusive growth") is fundamental to plant morphogenesis and the evolution of shape. Permanent volume increase must be accompanied by some kind of stress relaxation, otherwise the enlarged cells would tend to shrink to the original size by elastic interactions. Therefore a viscoelastic stress relaxation response is required (we cite after Lintilhac (2014): Metraux and Taiz, 1978;Dorrington 1980;Taiz 1984; also Haduch-Sendecka et al. 2014, Eqs 6.1 -6.3 and the comment therein). Wei and Lintilhac (2003Lintilhac ( , 2007 have suggested a different approach to model stress relaxation behaviour. While agreeing with the fact that the source of the tensile stress is turgor pressure, they implied that stress relaxation in plant cell walls (at critical pressure) should be treated as a binary switch, a mechanism which may be appropriate in short-term growth processes (Zajdel et al. 2015), caused by low amplitude, high frequency (osmotic) pressure fluctuations, like in pollen tubes.
In recent paper by Pietruszka and Haduch-Sendecka (2015) a solitary frequency f 0 ≈ 0.066 Hz was determined by the detrended Fast Fourier Transform (FFT) of the wall pressure power spectrum, which reveals strict periodicity in turgor pressure of growing lily pollen tubesdata measured in pressure probe experiment by Benkert et al. (1997) and reanalyzed by Zonia and Munnik (2011). A distinct proposal, also leading to quasi-discrete energy levels resembling a binary switch, was put forward independently by Pietruszka (2013a), in the case of periodic growth of pollen tubes, where asynchronous growth dynamics was achieved through an anharmonic potential at constant turgor pressure condition.
In spite of the extensive efforts to explain the effect of temperature response of plants, the subject seems to be insufficiently appreciated and, in this context, the studies of plant cell/organ in the current literature are rarely reported. Usually, focus is put on growth/development/elongation as a function of temperature, but the plots of such temperature dependence are infrequently presented. Only a few papers in which temperature response is treated as a key issue can be mentioned (see next paragraph). It was suggested that cell growth, especially cell elongation, has a high Q 10 factor (which is a measure of the rate of change of a biological or chemical system as a consequence of increasing the temperature by 10 °C), which indicates that this is a chemically rather than physically controlled phenomenon (Went 1953). In the above context, we will show that physical (temperature) constraints act through chemical reactions to direct growth.
Temperature is one of the most important factors that determines plant growth, development and yield (Yan and Hunt 1999). It is intelligible that the accurate indication of plant temperature response is a prerequisite to successful crop management. All biological processes respond to temperature, and several models have been proposed. (a) A linear model which is convenient when the temperature does not approach or exceed the optimum temperatutre (Summerfield and Roberts 1987).
(b) A bilinear model describing separately sub-and supra-optimum temperatures (Olsen et al. 1993) where the derivations may not be always meaningful and the estimates could be inexact and divergent.
(c) A multilinear, multiple parameter (of usually highly empirical origin) model (Coelho and Dale 1980) adopted by crop system simulation packages. (d) Smooth exponential and polynomial models highly inaccurate at both low and high temperature ends (Cross and Zuber 1972;Yan andWallace 1996, 1998), to name a few. The standard density function of beta-distribution was proposed by Yan and Hunt (1999), using only three cardinal numbers. This approach delivered a universal scaling function, although parameters were not of the kind of "fine-tuning" parameters somehow rooted in any accompanying microscopic model. The short term temperature response of coleoptile and hypocotyl elongation growth has also been considered by Lewicka and Pietruszka (2008) based on the Central Limit Theorem for several species (barley, wheat, millet, bean, and pumpkin). For our future use it is important to note that in this model the Gauss function was employed.
Empirical elongation/growth studies usually include the notion of temperature implicitly, while some of them intentionally considered the effect of temperature on elongation growth (Karcz and Burdach 2007), which in this work exhibited a clear maximum at 30 °C in maize, and an upward shift of the maximum in the presence of indole-3-acetic acid (IAA) and fusicoccin (FC). High (supraoptimal) temperatures also promoted auxin-mediated hypocotyl elongation in Arabidopsis (Gray et al. 1998), who showed that Arabidopsis seedlings grown in the light at high temperature of 29 °C exhibit dramatic hypocotyl"s elongation compared with seedlings grown at 20 °C. These results strongly supported the contention that growth at high temperature promoted increase of auxin levels and that endogenous auxin promoted cell elongation in intact plants. For the record, beyond the above Many models were developed in the growth area. We name only few of them. A hormone model of primary root growth where the wall extensibility is determined by the concentration of an unspecified by the authors wall enzyme, whose production and degradation are assumed to be controlled by auxin and cytokinin, was proposed by Chavarria-Krauser et al. (2005). More recently Pietruszka (2012) formulated a biosynthesis/inactivation model for enzymatic wall loosening factors or non-enzymatically mediated cell evolution based on the Lockhart/Ortega type of equation. In this work the physiology and biochemistry of the growth process were related by analytical equations acquiring very high fidelity factors (R 2 ≈ 0.9998, regression P < 0.0001) with the empirical data. Also, in the same context of biosynthesis, biological growth as a resultant effect of three forms of energy (mechanical, thermal and chemical) and their individual couplings, was summarized in the form of an elegant theoretical framework by Barbacci et al. (2013). In their description biological growth was the resulting effect of three forms of energy and their couplings (denoted M/T, M/C and T/C with M for Mechanical, T for Thermal and C for Chemical). For each energy, each couple of intensive and extensive variables was linked by one component of Tisza"s matrix. This derivation, although sophisticated, requires many parameters (13) and externally controlled turgor pressure P and temperature T to retrieve, as an example, the data extracted from the Proseus and Boyer (2008) experiment (see Fig. 5 in Barbacci et al. 2013).
Fast growth of plants requires optimal temperature (e.g. Lewicka and Pietruszka 2006). Below or above this temperature the growth of plant cells and organs is slowed down (ibid., Fig. 3). The latter statement implies that an optimum must exist at the crossover temperature region if one observes slow growth at both high and low temperature ends. At this temperature, it may be presumed, that at least one major factor of the wall-extension governing parameters (couplings) must change intensely. For this approach purposes, we may call it the effective "diffusion rate" k 2 (there may be more such coefficients, forming a multi -Gaussian peak spectrum for each coupling k i in the kparameters hyper-plane). For practical, e.g. agricultural, use this parameter(s) value(s) is/are of major significance. While extending the model, additional relevant coupling strengths (k i , i ≥ 2) must be added, see However, the steps in wall assembly and the specific chemistry controlling rates of enlargement are still lacking the analytic background. Here we partially covered this broad problem by considering the temperature dependence of wall biosynthesis related factors.

Material and methods
Material The manuscript is mostly built upon the experimental data which originate from Karcz and Burdach (2007) paper. Briefly, these experiments were carried out with 10 mm long segments cut from 4-day-old maize coleoptiles of maize (Zea mays L.) 3 mm below the tip, in usual growth conditions.
The experiments were carried out within seven hours each, with measurements taken every 15 min.
These raw individual values were retrieved by us with GetData Graph digitizer and collected in SI Tables 1 -3 for re-analysis. The experimental data obtained by this routine were used in the fitting procedure interrelating the elongation growth data of coleoptile segments and growth functional at Finally, for comparison, we present results of our own (48 hours) measurements, performed with the help of a CCD camera, on intact growing maize seedlings (SI Fig. 4). Seeds of maize were grown in the dark at 27 °C; 4-day-old seedlings of a length of about 2.5 cm were chosen for experiment. The experiment was carried out in both chambers simultaneously for APW (first chamber for control) and for the changing growth factor IAA and FC with concentrations from the interval: 0.5•10 -7 -10 -5 M introduced to the second chamber from the beginning of experiment. The fluid volume in both chambers equaled 30 ml, with 3 seedlings in each chamber. The seedlings of maize were grown in dim green light. A constant temperature was maintained at about 25 °C, and pH, changed by the soaked part of the seedlings and the root system, was measured by two pH-meters (in each chamber independently), type pH/ion meter CPI-501. The images were recorded by Hama Webcam AC-150 every 30 min. From length and time measurements, the relative elongation of marked coleoptile segments (initially 1 cm long fragments indicated by ink spots) was calculated using the formula (l f -l i )/l i , where l i is the initial length and l f is the final. This method allowed for simultaneous measurements of growth and H + efflux. The software OriginPro 8.5.1 (Microcal) was utilized to perform calculations and create graphs in all cases.

Relative elongation growth formula
For mathematical analysis we used the data collected in SI Tables 1 -3 (presented in SI  To prepare temperature-dependent data for further analysis, the additional (intermediate) data points were obtained by linear interpolation (moving average) -SI Table 4 -6.
We also use for analysis the probability density function of the beta distribution (also called the Euler integral of the first kind (Polyanin and Chernoutsan 2011)), for the interval 0 ≤ x ≤ 1, and shape parameters α, β > 0. Beta function is a power function of the variable x and of its reflection (1 − x) and has a normalization constant B = B(α, β). The use of beta distribution was already suggested by Yan and Hunt (1999) for temperature dependent plant growth. The coexistence curve (Fig. 3, ibid.) presented by the authors strongly supports the use of beta function also in this study. The results of the fitting procedure for the growth process amplitude C = C(T), are related to the beta function (2) in the following way

Cross-correlations
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product.
For continuous functions f and g, the cross-correlation is defined by the integral where f* denotes the complex conjugate of f and τ is the time lag. Note, that the cross-correlation is maximum at a lag equal to the time delay (maximum located at lag equal zero means no time delay).
Similarly, for discrete functions (like the usually analysed experimental data points), the crosscorrelation is defined as: which definition is utilized in this work (here: f* = f). Auto-correlation is obtained if f equals g. Crosscorrelation derivative (over time delay τ) is defined as follows (see also Appendix)

Results and discussion
For discussion purposes we omit details of the derivation of Eq. (1), which can be found elsewhere

Effective diffusion rates
First, let us recall (Pietruszka 2012) the first-order differential equation for n(t) which is an actual concentration of WLF (Wall Loosening Factor) which solution was inserted into the truncated form (for constant turgor P) of the Ortega (1985) equation. In Eq. (7) the most interesting for our problem is the diffusion rate k 2 , which is the rate of change of WLF concentration in the cell wall at a given temperature. The coefficient k 1 which originates from the external pool in the model calculations (it may be also interpreted as a "biosynthesis" coefficient), is incorporated into the remaining coefficients of Eq. (1).
With the usual assumption that both cell wall extension and water uptake must occur concomitantly, and the overpressure (by tissue impact) dependence of the yield threshold Y = Y[n(t)] -in the first approximation -may be neglected, we arrived at the solution where n 0 = n(t = 0). We recall, that in general the kinetic coefficient k 2 = k 2 (T) is temperature dependent (Fig. 7 in Pietruszka 2010). Hence, relevant but not mutually interacting coupling strengths (k) must be taken into account while extending the model. In the extended form the equation for the where the completeness relation Next, we recall that the plots presented in Fig. 2 in Pietruszka (2012) exhibit pronounced changes with respect to the "coupling constant" strength (effective diffusion rate k 2 ), in contrast to lesser reactions caused by turgor pressure change. The interpretation from the analytic expression (ibid.) follows that volumetric extension to be effective must be preceded by pressure induced relaxation processes in the cell wall due to WLFs interaction with the wall constituting polymers (Schopfer 2008; Geitmann and Ortega 2009)otherwise growth is less successful since the wall is more "rigid" and not susceptible to the pressure changes.
We also bear in mind that the model involves biosynthesis of WLFs in the cell at a steady rate k 1 and partial inactivation of such created WLFs at a rate k 2 . Note, that k 2 (and/or k i , i ≥ 2 in case of Eq.  Fig. 2A, where a sharp Gaussian peak appears. We need to point out that the obtained diffusion rate k 2 cannot be directly linked to the growth rate, since the volumetric increase in volume can be also built-in into the coefficient C = C(T), which serves as nonlinear, temperaturedependent growth amplitude in Eq. (1). A good example can be drawn by comparison of the results in SI  . 2015), where the diffusion rate k 2 (parameter D) is slightly different for the hypocotyls grown in the dark/light conditions. The over tenfold increase in length is incorporated into parameter C, Fig. 3A -C (ibid.). Note, that the greater k 2 the more substantial decrease of the initial concentration n 0 , and the quicker decrease of the actual "growth factor".
Based on the results from this paper, it was shown that "k 2 factor" decisively influences cell/organ volume. By adding a biochemical substance, which causes similar effect (as WLF), one should observe shifts of the peak in Fig. 2A. Hydroxyl radicals (OH)are capable of unspecifically cleaving cell wall polysaccharides in a site specific reaction (Schopfer 2001). Cell wall loosening underlying the elongation growth of plant organs is controlled by apoplastically produced OH -"attacking load-bearing cell wall matrix polymers" (ibid.).
In the above scenario, molecular factors can be exemplified by: -the dependence of extension on concentrations of ascorbate/H 2 O 2 and Cu 2+ or Fe 2+ (used for generating OHin isolated cell walls of maize coleoptiles (Fig. 3, Schopfer 2001)), -the dependence of extension on pH (Fig. 4, ibid.), -inhibition of auxin-induced elongation growth by Mn-based chemicals (Fig. 9, ibid.).
Implications of our proposal are also supported by a clearly visible shift in Porter and Gawith (1999) study, Fig. 1.
For further discussion we call to mind the results of Lewicka and Pietruszka (2006). Especially, we draw reader"s attention to comparison of Fig. 6 (ibid.) with three characteristic phases (crystalline, semi-liquid and liquid) to the endogenous growth amplitude coefficient C from Eq. (1), presented in Fig. 1A in the present work. From the biomechanical point of view cell membranes are equipped with ionic pumps and channels, water channels and ligand receptors (Berg et al. 2002). The optimal phase of endogenous growth starts to occur in the semi-liquid phase, presumably corresponding to the peak of the Gauss curve located at the phase boundary at about 16 °C in the present work ( Fig. 2A) and a "jump" above 16 °C in Fig. 1A. Then, the endogenous auxin activates H + -ATPase, acidification of cell walls and their loosening. Simultaneously, the K + ions in-flow takes place, through the reduction of the water potential filling up the plant interior, pulling behind water molecules. At this temperature end we encounter a kind of a "phase transition" from a crystalline to semi-liquid to phase. Low temperatures cause membrane depolarization and K + loss, water efflux and growth inhibition. Furthermore, at 0 °C the reservoir of liquid water becomes nearly empty for the sake of crystallisation into the ice phase. At high temperatures, another transition from a semi-liquid to a liquid phase occurs (arrow pointing at a local maximum at 38 °C, Fig. 2a), causing the malfunction of ionic and water channels as well as ionic pumps (and hence the fits for k 2 become inexact, which can be observed at temperatures above 40 °C in Fig. 2a). In effect, we deal with the ionic leakage, a secondary water stress and consequently cessation of growth. Moreover, the auxin receptor proteins change conformation and functionality, causing growth deceleration and termination. The

Cross-correlation analysis
Some crucial arguments against the acid growth theory of auxin action (Kutschera and Schopfer 1985) have been reinvestigated by simultaneous measurements of proton fluxes and growth of Zea mays L. coleoptiles by Lüthen et al. (1990). Among others, it was found that (a) the timing of auxin and fusicoccin-induced (FC) proton secretion and growth matches well and (b) the equilibrium external pH in the presence of IAA and FC are lower than previously recorded and below the so-called "threshold-pH". It was concluded that the acid-growth-theory correctly describes incidents taking place in the early phases of auxin-induced growth. This subject was undertaken next by us and the results are summarized in Table 1 and Figs 3 -10. To save space, the description of our results are partially located in figures captions.
Cross-correlations of pH and elongation growth as a function of time delay τ, Eq. (5), for APW (endogenous growth), exogenous IAA and FC is shown in Fig. 3. A similar plot parameterized by temperature is presented in Fig. 4. The analysis was based on the raw data presented in Figs 1 -5 in Karcz and Burdach (2007). The time retardation (advancement) of the maximum with respect to zero time delay is clearly visible (see also SI Table 7 for the obtained values).
Cross-correlations derivative of elongation growth and pH as a function of time lag τ, Eq. (6), parameterized by temperature, for APW (endogenous growth), exogenous IAA and FC are shown in Figs 5 -6. The discontinuities (representing the relative buffer capacity acidification, approximately proportional to proton efflux rate) in the cross-correlation derivative at τ = 0 correspond to H + ions activity for maize coleoptile segments (see also SI Table 8 for the obtained values).
There are some general problems with the Karcz and Burdach (2007) data, as they did not use abraded coleoptiles like in Lüthen et al. (1990). Instead, they did perfuse the coleoptile cylinders with solution. That appears to work to a certain degree as indicated by the ± nominal pH drops. It may well be that the responses in this system are a bit more sluggish than in abraded coleoptiles. This apparently might affect the meaningfulness of any cross correlation analysis. This seems not quite to be the case. Basically these authors, in order to clarify discrepancies between the earlier Lüthen et al. (1990) and Kutschera and Schopfer (1985) papers, found that the pH of excised coleoptile segments first rises to pH 6.5 (RTreversal time), and then gradually falls to an AE phase (acid equilibrium of about pH 4.8 in maize), which is achieved about 4 hours after excision. When auxin is added, the pH will drop to 4.2 with a time course well matching the growth response. This pattern is also well visible in Fig. 4 in see the auxin effects on a large background of the still ongoing endogenous pH drop. On the contrary, in Peters and Felle (1991) auxin was added at the acid equilibrium, making the pH drop much more clearly visible.
To resolve these doubts we analysed the experimental results presented in Figs 4 and 5 by Lüthen et al. (1990). The apparently unattractive outcomes, presented in Figs 7 -8, we accepted with amazement. In Fig. 7 cross-correlation of growth rate and proton efflux rate is calculated for IAA and FC-induced growth. First, we noted that the timings for each plot matches well and the crosscorrelation intensity for FC is several times stronger than for IAA. No time delay is observed. Second, the triangular shape of both curves and the location of both maxima at zero lag brings to mind the definition of autocorrelation (the cross-correlation of a signal with itself at different points in time, Eq. Finally, for comparison, the results of an "intact seedling" experiment for maize are shown in

Conclusions
We considered the temperature-dependent effective diffusion rate with specific applicability to cell wall loosening factors and application of Euler beta distribution to the amplitude C of the growth functional, Eq. (1). It was shown, that the endogenous/exogenous growth amplitude C = C(T) is realistically reproduced by the Euler beta function as a function of temperature, while the temperaturedependent endogenous diffusion rate D = k 2 (T) is reasonably represented by Gauss distribution (endogenous auxin) or linear function (exogenous auxin of fusicoccin). In the temperature context, the diffusion rate k 2 = k 2 (T) and the amplitude C, describing active (H + ) transport into the wall, provides a measure of thermal energy efficiency of growth. We showed, that the localisation of endogenous growth maximum is essentially determined by temperature (or equivalently by pH, through plasma membrane H + -ATPase), and that the cumulative action of C and k 2 coefficients essentially contribute to growth. It also seems that the localization of the optimum growth is mainly determined by temperature (or pH, Pietruszka 2015) -C coefficient.
The main limitation of our early "temperature" approach to plant cell/organs growth was that it was not accounting for the important role of the biochemical reactions involved in cell wall building processes. Our present study extends our previous proposals into a new territory. The strongly predicative (temperature dependent) semi-empirical equation (1) permit to fine-tune the leading factors in plant cell/organ growth, which implications may be helpful for climatic impact studies onto plant growth.
Our results also suggest that at least for some special experimental conditions (abraded samples like in Lüthen et al. 1990), the timings of growth and proton efflux match, while the interaction expressed by cross-correlations is much stronger for fusicoccin than for auxin.
The molecular mechanism of fusicoccin and auxin differ: (1) 14-3-3 protein interaction with the ATPase in the case of FC and (2) TIR1 and/or ABP1 binding and an unknown signaling pathway in the auxin case. However, the H + efflux mechanism by which auxin and fusicoccin cause the promotion of growth may be effectively similar on the level of primary wall tissues. At this lowest (molecular) level, auto-correlation analysis of Lüthen et al. 1990 data led us to the conclusion that the primary wall growth rate and H + -efflux rate coincide It seems that investigating "acid growth hypothesis", and resolving mounting controversies, is today impossible using solely biological experiments. It could be explored further using modelling as shown in this work. In the case of this work the experimentum crucis for a biological problem belongs paradoxicallyto physics and mathematics.
In conclusion, we believe that model equation (1)  The cross-correlation of continuous functions f and g is defined in Eq. (4). By assuming f ≡ pH(t) and g ≡ u(t) [μm] the cross-correlation derivative (over time delay τ) can be calculated explicitly as where u" is a growth rate.    Table 4. Fit parameters: χ 2 = 0.09056 and determination coefficient R 2 = 0.95.
Excluded area at temperatures exceeding 40 °C: see dense (red) pattern due to high error values (SI Table 4). Arrow pointing at the upper local maximum at T 2 .  Table 4 Table 6).      (1990). Note that the timings for each plot almost coincide and the cross-correlation intensity for FC is about 4 times stronger than for IAA.

Figure 8
Auto-correlations, f = g in Eq. (5), calculated for the kinetics of (a) IAA-induced growth rate (A) and proton efflux rate (B) and (b) action of FC on growth rate (C) and proton secretion rate (D) for the simultaneously measured both parameters, presented in Figs 4 -5 by Lüthen et al. (1990). Note, that auto-correlations deliver almost identical results (vertical scale neglected), both for fusicoccin (FC) and auxin (IAA) action. This result can be treated as convincing argument for the "acid growth hypothesis", applicable not only for FC but IAA as well. Growth rate and proton efflux rate coincide for both FC and IAA, and can be used interchangeably.