Derivation of a macroscopic model for nutrient uptake by hairy-roots

Summary• Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes.• We developed a mathematical model of nutrient transport and uptake in the root hair zone of single roots growing in soil or solution culture. Accounting for root hair geometry explicitly, we derived effective equations for the cumulative effect of root hair surfaces on uptake using the method of homogenization.• Analysis of the model shows that, depending on the morphological and physiological properties of the root hairs, one of three different effective models applies. They describe situations where: (1) a concentration gradient dynamically develops within the root hair zone; (2) the effect of root hair uptake is negligibly small; or (3) phosphate in the root hair zone is taken up instantaneously. Furthermore, we show that the influence of root hairs on rates of phosphate uptake is one order of magnitude greater in soil than solution culture.• The model provides a basis for quantifying the importance of root hair morphological and physiological properties in overall uptake, in order to design and interpret experiments in different circumstances.

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“…Furthermore, we consider Cartesian geometry rather than the more computationally complex cylindrical geometry (Ptashnyk, in press) (Fig. 1b).…”

Section: Methodsmentioning

confidence: 99%

“…In the Mathematical Notes S1 we derived the effective equations using multiscale expansion. A proof that these equations are actually the unique limit of this homogenization problem when ε → 0 can be found in Ptashnyk (in press).…”

mentioning

confidence: 94%

A dynamic model of nutrient uptake by root hairs

et al. 2009

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“…Thus, we have that the operator K : L ∞ (0, T ; W(Ω)) → L ∞ (0, T ; W(Ω)), defined by K( u ε ) = u ε , where u ε is a weak solution of (1), is a contraction for sufficiently smallT , where T depend on the coefficients in the equations and is independent of (p ε , n ε , b ε , u ε ). Hence, using the Banach fixed point theorem and iterating over time intervals, we obtain the existence of a unique weak solution of the microscopic problem (1) Due to the fact that viscous term is defined only in the cell wall matrix and is zero for cell wall microfibrils, to conduct the multiscale analysis of the viscoelastic problem (1) we first consider a perturbed problem by adding the inertial term ϑ∂ 2 t u ε , where ϑ > 0 is a small perturbation parameter: (25) ϑχ…”

Section: Resultsmentioning

confidence: 99%

“…Theorem 4.3. A sequence of solutions (u ε , p ε , n ε , b ε ), of the microscopic problem (2), (3), (25), converges to a solution (u ϑ , p ϑ , n ϑ , b ϑ ) of the macroscopic perturbed equations…”

Section: Resultsmentioning

confidence: 99%

Homogenization of a viscoelastic model for plant cell wall biomechanics

Ptashnyk

Seguin

2017

ESAIM: COCV

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Abstract.The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin-Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations governing the viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for the degenerate viscoelastic equations is conducted using a perturbation argument.

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“…By applying the theory of invariant regions [51, Theorem 2] and [55,Theorem 14.7], we obtain the non-negativity of n ε j , j = 1, 2, and b ε in the same way as in the proof of Theorem 3.3. Taking n ε and b ε as test functions in (12) and using the non-negativity of n ε j and b ε and the boundedness of p ε , along with the assumptions on F n , Q n , Q b , and J n , see Assumption 1, yield (49) n ε (τ ) 2…”

Section: 1mentioning

confidence: 99%

Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

Ptashnyk

Seguin

2016

ESAIM: M2AN

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Abstract. In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall's constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reactiondiffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.

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Derivation of a macroscopic model for nutrient uptake by hairy-roots

Cited by 12 publications

References 17 publications

A dynamic model of nutrient uptake by root hairs

A dynamic model of nutrient uptake by root hairs

Homogenization of a viscoelastic model for plant cell wall biomechanics

Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

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