 Research
 Open Access
A framework for an organellebased mathematical modeling of hyphae
 Rudibert King^{1}Email author
 Received: 15 April 2015
 Accepted: 27 June 2015
 Published: 21 July 2015
Abstract
Background
Although highly desirable, a mechanistic explanation for the outstanding protein secretion capabilities of fungi such as Aspergilli is missing. As a result, a rational and predictive design of strains as cell factories for protein production is still out of reach. The analysis of the secretion apparatus is not only hampered by open issues concerning molecular cell biological processes, but as well by their spatial fragmentation and highly dynamic features. Whereas the former issues are addressed by many groups, an account of the space and timedependent processes, which is best done by means of mathematical models, is lacking. Up to now, mathematical models for hyphal organisms mainly focus on one of two extremes. Either macroscopic morphology, such as pellet or mycelium growth, is addressed, or a microscopic picture is drawn predicting, for instance, the form of a hyphal tip. How intrahyphal transport and organelle distribution works, however, has not been tackled so far mathematically.
Results
The main result of this contribution is a generic modeling framework to describe the space and timedependent evolution of intracellular substances and organelles. It takes intrahyphal, passive and active transport of substances into account and explains exponential and then linear length growth by tugordriven uptake of water. Experimentally observed increasing concentration levels of organelles towards the tip can be well explained within the framework without resorting to complex biological regulations. It is shown that the accumulation can be partly explained by geometrical constraints, besides a necessary deceleration of the active transport velocity. The model is formulated such that more intricate intracellular processes can be included.
Conclusions
Results from steadystate experiments are easy to be interpreted. In a hyphal network, however, new branches are produced at an exponential rate. Moreover, passive and active transport processes give rise to a spatial distribution of organelles and other cytoplasmatic constituents inside hyphae. As a result, most of the data obtained in experiments will be from a nonsteady and space dependent state. A quantitative and mechanistic explanation of the processes occurring will only be possible if these dependencies are taking into account while evaluating experimental findings.
Keywords
 Morphological model
 Hypha
 Vesicles
Background
The ecological and technical relevance of fungi is outstanding. They are integrated in most ecosystems, act as detrimental agents for plants and humans, decompose waste materials, and are exploited in the synthesis of valuable products [1, 2], to name just a few areas in which they play a major role. Their most striking feature is polarized growth and branching which leads to more or less dense mycelia or pellets [3, 4]. Concomitantly, Aspergilli such as A. niger, A. oryzae and A. terreus have astounding capabilities to secrete interesting enzymes, mainly through the apical region [5]. A rational design to obtain modified strains as optimized cell factories, however, is still limited by the incomplete picture of their growth, production and secretion machinery. By the very nature of living cells all occurring processes are highly dynamic and the behavior of a cell does not only depend on the actual stimuli but what had happened to the cell in the past. For fungal organisms, interpretation of physiological data is even more challenging. Besides the compartmentalization of biological functions in distinct organelles, space and timedependent distributions occur. This relates to organelles, other cytoplasmic compounds, and stimuli in and around a mycelium, and, therefore, impede the deduction of meaningful knowledge and hypotheses. All of which could be addressed in the context of mathematical models.
A more detailed account of microscopic features of individual hyphae is given by a last group of models to describe, for instance, the shape of a tip, or the growth in length. A wellknown example for the first class of problems, which will be used in what follows, is given by BartnickiGarcia et al. in [27]. Here, the geometrical form of an apex is predicted with a simple model. It is based on a set of hypotheses how vesicles are transported ballistically from the Spitzenkörper to the wall. The model has been refined in future works to better account for the threedimensional shape of a tip, the way vesicles are transported to the wall by diffusion, by representing the cell wall as a flexible membrane, or by a better account of vesicle fusion with the cell wall [28–32].
Mathematical descriptions of the growth in length, as the second class of microscopic features, were given recently with two different approaches. In [33], the longrange transport of material in hyphae is depicted by a particle transport along a single, hypothetical microtubule extending over the whole length of a hypha. The amount of material reaching the tip of the hypha determines length growth. A changing velocity, however, is neither considered nor a movement of the microtubules with the cytoplasmatic flow. In contrast, [34] explain length growth of Phanerochaete velutina mathematically by a turgor driven intrahyphal flow towards the tip. In all these approaches a constant length growth rate is considered which is not true for the germ tube. Moreover, new branches of a mycelium very often show a lower initial velocity as well. As a result, and as a mycelium grows exponentially by an exponential production of new branches, a significant part of a mycelium will not be in a kind of quasisteady state which is assumed above. In major parts of a mycelium, organelles and intracellular substances have not yet reached their quasisteady state distribution, which might be important for a quantitative prediction of the growth of the mycelium. Likewise, if septa are closed and opened by Woronin bodies, intrahyphal flow has to stop or will resume resulting in even more complex situations.
In our former works [11, 19], we explained the initially observed exponential and then linear growth with the limitation by a hypothetical intracellular compound. We had to resort to a hypothetical compound at that time as neither for fungi nor for actinomycetes details about the mechanism were known. Especially for fungi, this situation has changed drastically in recent years. Molecular methods, bioinformatics and image analysis have provided us with a whelm of information if not give rise to a ‘Big Data Tsunami’ [35]. More specifically, for the processes addressed here, which are responsible for length growth and (product) secretion, much more is known today. Excellent recent reviews about growth in length of fungal hyphae are given, e.g., by [36, 37], and about secretion in [32]. The importance of turgor driven length extension is stretched by [38, 39] in a series of papers.

First of all, it represents a basic model structure, with which the initial exponential and then linear growth of a hypha can be described with a minimal amount of assumptions. This turgor driven evolution of the intrahyphal flow forms the ‘backbone’ for all other processes occurring in a hyphae, and, therefore, has to be considered first. To be more specific, besides the postulation of some kinetics, no further biological regulations will be introduced to describe the experimentally observed growth evolution. If this is possible, already simple physical transport processes combined with implicitly formulated regulations through kinetic expressions can be used to explain the observed behavior without resorting to complex biological mechanisms. This, of course, does not rule out such regulations which additionally may occur. If experimental evidence is given, these processes can be included readily.

Secondly, the model structure derived serves as a basis for future work when experimental data is interpreted and condensed in a mathematical framework. As an example, the distribution of vesicles in a hypha will be considered here which shows a distinct profile along the length of a hypha. Again simple physical arguments, mainly with respect to the active transport velocity and the geometry of the tip, will be enough to explain experimental data where a significant increase in concentration is observed toward the tip.
In the long run, such kind of models might help in answering questions raised in the endeavor toward a rational strain design. Examples are [5]: How many vesicles carrying proteins of interest can be used without interfering with vesicles for growth? Where are the bottlenecks in vesiclemediated protein secretion? How many proteins can be channeled through the secretory pathway in order to provide each protein sufficient time to become correctly folded? Extending this list of questions will naturally occur when a model is at hand.
The rest of the paper is organized as follows. After a problem statement in the next section, the model of BartnickiGarcia et al. is revisited to determine the volume and surface area in the tip region. This will then be used to correct experimental data. The general model is formulated next. As a first application, length growth by turgor driven water uptake is described. Extending the model with vesicles allows for a comparison with experimental data in the last section before the paper finishes with some conclusions.
Problem statement
The general model will be obtained by a formulation of balance equations applied to an infinitesimal intrahyphal balancing volume of length dx, see the space between shaded areas in Figure 2. Terms of production, consumption and transport via the cell membrane, the consequences of an intrahyphal flow, and, finally, due to active translocation will be considered. Spacedependent uptake of nutrients could be included readily, but is not done here. Most importantly, a constant physiological and functional state is assumed along a hypha. If this is not the case, the model developed in this contribution has to be combined with approaches proposed, e.g., by Nielsen and Villadsen [18]. As the transport from and to the environment will be proportional to the local surface area and production and consumption rates will be given based on the local volume, these quantities have be determined first.
Geometrical model of the tip
For simplicity, the 2Dmodel proposed in [27] describing the form of a hyphal tip is used to derive expressions for the local surface area \(A(x_t)\) and volume \(V(x_t)\). Although the initial conjecture that the actual 3Dform of a hypha can be produced by a rotation of the solution of the 2Dmodel was corrected in [28], see as well [29], the simpler approach is used here. This is motivated by the fact that the actual differences in the forms obtained are small while the calculation of the 3Dform is rather involved.
In [27] it is proposed that: (1) the cell surface expands from materials discharged by walldestined vesicles, (2) vesicles are released from a postulated vesicle supply center (VSC), (3) vesicles move from the VSC to the surface in any random direction. Based on these propositions, they derive the following model.
For the calculation of the volume \(V(x_t)\) and surface area \(A(x_t)\), \(r_t(x_t)\) is locally approximated by a straight line of length \(l_t\) connecting \(r_t(x_t)\) and \(r_t(x_t+dx)\). Rotating this line defines the area and the enclosed volume.
Experimental data
Although the main goal of this contribution lays in the derivation of a generic model structure, some comparisons with experimental data will be done.
In the experiments described above, hyphal length was not measured for A. niger as a function of time for newly developing branches. Typically, an initial exponential growth will be observed followed by a linear one. As the model will be able to describe this, another set of data is used here for comparison. Experimental results from [41] are exploited. This rather old set of data was already used by us in [11]. In this former work, a much simpler model was proposed to describe length evolution. Using the data again, both approaches can be compared. Fiddy and Trince [41] measured the evolution of a primary branch of Geotrichum candidum extending out of an intercalary compartment just behind a septum. They observed a correlation of the decreasing length extension rate of this branch with septation occurring after some time in this branch. However, the extension rate of the primary branch continued to increase, despite septation, until a length of about 700 μm was reached. From Figure 3b in [41], a maximal extension rate of 2.5 μm/min can be estimated. The data will be given later together with the results of a simulation.
Generic model
To derive a generic model, a substance \(\mathcal{S}_i\), \(i=d,t\) is balanced in a segment extending from \(x_i\) to \(x_i+dx\), see Figure 2. In what follows, \(\mathcal{S}_t\) represents vesicles (\(\mathcal{V}_t\)), Spitzenkörper (\(\mathcal{K}_t)\), osmolytes (\(\mathcal{O}_t\)), etc. in the tip, i.e., \(\mathcal{S}_t = \{\mathcal{V}_t, \mathcal{K}_t, \mathcal{O}_t, \ldots \}\). Accordingly, \(\mathcal{S}_d = \{\mathcal{V}_d, \mathcal{K}_d, \mathcal{O}_d, \ldots \}\) will denote variables in the distal part. A radial distribution inside the hypha and diffusion in all directions are neglected. Diffusion in the xdirection could be included readily without complicating the numerical solution much. It would, however, make less sense for organelles. See [39] for a discussion of diffusion coefficients of different cellular components compared against the intrahyphal flow velocity.
The last two terms in Eq. 9 represent intrahyphal flow in and out of the balance volume, i.e., flow through the shaded areas in Figure 2. As due to turgor pressure hyphae take up water from the surroundings, and as only the apical region can extend in the real hypha, an intrahyphal flow is set up. Hence, the volumetric flow rate \(Q_i(x_i,t)\) is both a function of space \(x_i\) and time \(t\).
 1.
The last term of Eq. 9 and the second term of the righthand side of Eq. 11 are expanded in a Taylor series, neglecting all terms in \((dx)^n\), \(n \ge 2\).
 2.
All equations are combined.
Generic model of the constant, distal part
Before specifying the individual production and consumption rates this generic model equation will be adapted to the nonconstantarea and nonconstantvolume case seen in the tip.
Generic model of the tip
Model of length growth
Complete generic model
 1.
Integration of Eq. 16 determines the intrahyphal flow rate at \(x_d=L_d\) which sets the boundary condition for Eq. 23.
 2.
Integration of latter equation leads to the hypothetical flow rate at the tip, \(Q_{tp}(L_t,t)\),
 3.
and, with Eq. 22, to the actual extension rate \(\dot{L}\).
 4.
In a moving boundary framework, as L(t) grows, Eqs. 14 and 18 are solved to determine \(\mathcal{S}_i\), \(i=d,t\).
Initially, only the tip region exists. Hence, Eqs. 16 and 18 are not needed.
Modeling pressure regulation via osmolytes
We consider osmolytes which are responsible for maintaining a certain pressure and pressure gradient inside hyphae, see [38]. It is assumed that osmolytes are produced until a certain pressure is obtained for which intracellular sensors must exist. For the MAPK pathway, OS1 is discussed as a sensor in [39]. For simplicity, an intracellular substance called osmolyte \(\mathcal{O}_i\), \(i=d, t\), is introduced, which represents both the osmolyte, and, indirectly, the pressure. To obtain a mass flow toward the apex, its concentration must be higher in subapical parts. Using the equations derived above, \(\mathcal{S}_i\) is now replaced by \(\mathcal{O}_i\).
With these kinetics and the generic equations derived in the last section, the models of the distal part and the tip can be formulated. They are omitted here for brevity. Furthermore, to reduce the number of kinetic parameters and to ease parameter identification, a normalization is done with \(o_i = \mathcal{O}_i / \mathcal{O}_{max}\) and \(q_i = Q_{i,cyt} / Q_{max}\), and \(o_i, q_i \in [0,1]\) for \(i=d,t\), \(\dot{L}_{max}=Q_{max}/(\pi R^2) = Q_{max} \rho _{d2}\), \(\theta _1 = k_1\), \(\theta _2 = k_2 \mathcal{O}_{max}\), \(\theta _3 = k_3 \mathcal{O}_{max}\).
For the numerical solution of the partial differential equations, the spatial coordinate is discretized equidistantly with a step size of \(\Delta x=0.2\) μm. The method of lines is applied for the equation describing the evolution of the osmolytes approximating the spatial derivatives by a firstorder backward difference operator. In the beginning, when only the tip exists, the left most discretization segment of the tip, see Figure 2, is allowed to grow in length according to \(\dot{L}\) until it exceeds a length of 0.3 μm. After the distal part is formed, its right most discretization segment takes over this task and grows accordingly until it exceeds a length of 0.3 μm. Then, this segment is split up into a segment of constant length (0.2 μm) and a growing one with an initial length <0.2 μm and the calculations are continued as before. The normalized flow rate \(q_i\) is obtained accordingly from Eqs. 16 and 23 exploiting the trapezoidal rule. Parameters \(\theta _i\) have to be chosen such that \(q_i \le 1\) is guaranteed.
As this work concentrates on the formulation of the generic model and not on a parameter fit or selection of appropriate kinetic expressions to describe a very specific problem, a simple approach was chosen to find kinetic parameters for simulation studies. Measurements performed by Spanhoff et al. were done with hyphae growing with approximately \({\dot{L}}_{max} = 3 \mu\)m/min. The parameters \(\theta _{1,2,3}\) are determined such that L(t) shows an initial exponential increase followed, after a transition, by a phase of constant growth velocity of approximately \({\dot{L}}_{max} = 3 \mu\)m/min. To this end, an optimization problem was formulated. In lack of real data for this first study, a ’desired’ evolution \(L_{des}(t)\) was determined to allow for an adaption of the \(\theta _i\)’s.
In the simulations given in Figure 8, growth is followed up to a length of about 170 μm, i.e., in the end more than 850 discretization segments are used. Due to production and consumption of the osmolyte, and due to length growth, the osmolyte profiles change dynamically over time. These osmolyte profiles, on the other hand, determine the overall volume production, see Figure 9, and, hence, length increase. As a result, all processes are highly interwoven.
Using the same volume production parameters \(\theta _2=\theta _3=0.2616\) in the distal and tip region results in a much too slow growth corroborating different uptake rates. These simple studies show that an intricate relation exists between parameters and the processes described by the model. Predicting the outcome of parameter or kinetic variations is difficult. Therefore, numerical studies are indispensable.
Modeling of vesicle distribution
As a first example of an organelle, vesicles are considered which have been already used above to get an idea of the form of the tip. They are, beside other functions, responsible for material transport to the growing tip. According to the model of [27], vesicles are used up in the apical part of a hypha. Therefore, the consumption rate in the tip will be \(\mu _{\mathcal{V}_tc} \ne 0\), while \(\mu _{\mathcal{V}_dc} = 0\) is assumed in the distal region.
The equations are normalized again with \(v_i = \mathcal{V}_i /\mathcal{V}_{max}\) and \(\theta _4 = k_4\).
With these parameters the experimentally observed accumulation profile of vesicles can be reproduced. A keyenabler is the model assumption that the transport velocity must decrease towards the tip. This could be tested in future experiments.
Conclusions
As constantly new branches and septa are produced in a mycelium and a flow of cytoplasma towards the different apices occurs, quasisteady state concentration profiles of intracellular substances and organelles hardly establish. Moreover, due to these processes and due to active and cytoplasmatic flows intracellular components are not only a function of time but as well a function of space inside a hypha. A comprehensive and quantitative interpretation of experimental data of individual hyphae will therefore only be possible when these space and timedependent processes are taken into account. To this end, a generic mathematical model is proposed here which first of all describes turgor driven length extension. By this, an initially accelerated and then linear growth can be predicted as seen in microscopic experiments. A much simpler model, see [11], could be used to predict the length evolution though. That model, however, cannot to be extended so easily to describe other constituents of a hypha as it is done here. The turgor driven length extensions forms the ‘backbone’ of a generic model to study, for instance, the timedependent distribution of organelles and other compounds. These may be transported actively or passively towards the tip. Diffusion is not considered yet, but can be included readily. For a complete specification of the model, kinetic expressions have to be stated. In this contribution, very little effort is invested to study the effect of different kinetics and parameters, e.g., with respect to osmolyte or vesicle production and consumption. The emphasis is rather on the formulation of a generic model. Effects of different kinetics will be tackled in future works when more experimental data is available. However, even with simple kinetics chosen here it can be shown, for example, that the experimentally observed accumulation of vesicles near the tip can be explained. A crucial assumption to be able to do this is the postulation of a decreasing active transport velocity in the tip region. Without this, the experimentally observed apical accumulation of vesicles cannot be described in the chosen setting. The model structure can be readily extended to study the effect of different organelles and cytoplasmatic compounds. Before doing so, however, the numerical solution of the partial differential equations with a moving boundary should be revisited to hopefully decrease the computational burden. This was not done yet, as septation and branching have to be included in future works.
Declarations
Acknowledgements
The author would like to thank Frederik Spanhoff, Arthur Ram and Vera Meyer for supplying unpublished vesicle measurements and VM who inspired this work.
Compliance with ethical guidelines
Competing interests The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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