R N Bearon
email: ar.bearon@liv.ac.uk

V Magar

Simple models of the chemical field around swimming plankton

Keywords: Swimming, nutrient uptake, exudation, low Reynolds Swimming, nutrient uptake, exudation, low Reynolds number

The chemical field around swimming plankton depends on the swimming style and speed of the organism and the processes affecting uptake or exudation of chemicals by the organism.

Here we present a simple model for the flow field around a neutrally buoyant self-propelled organism at low Reynolds number, and numerically calculate the chemical field around the organism. We show how the concentration field close to the organism and the mass transfer rates vary with swimming speed and style for Dirichlet (diffusion limited transport) boundary conditions. We calculate how the length of the chemical wake, defined as being the distance at which the chemical field drops to 10% of the surface concentration of the organism when stationary, varies with swimming speed and style for both Dirichlet and Neumann (production limited) boundary conditions. For Dirichlet boundary conditions, the length of the chemical wake increases with increasing swimming speed, and the self-propelled organism displays a significantly longer wake than the towed-body model. For the Neumann boundary conditions the converse is true; because swimming enhances the transport of the chemical away from the organism, the surface concentration of chemical is reduced and thus the wake length is reduced.

Introduction

Diverse swimming plankton interact with chemical fields in aquatic environments. The chemicals may be nutrients which are consumed or exudates which are produced by the organism. In both cases we want to understand the chemical field generated around the organism, and the rate at which chemicals are transferred between the organism and the environment. For example female copepods emit pheromones, generating trails which are tracked by their mates [START_REF] Doall | Locating a mate in 3D: the case of Temora longicornis[END_REF], and many phytoplankton are motile which can enhance their uptake rate of inorganic nutrients [START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF]. Other organisms, such as Phaeocystis, a motile phytoplankter which exudes DMSP, may also alter the spatial distribution of chemicals that they themselves emit (Laroche et al., 1999).

There have been extensive studies as to how fluid motion affects the chemical field around chemical sinks such as those reviewed by [START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF] in the context of nutrient uptake by aquatic single cells. By modelling swimmers as towed bodies, these studies have predicted how swimming affects the chemical field surrounding the organism and the masstransfer rate between the organism and the fluid environment. However, from simple force arguments, the towed body model is not a self-consistent model of a self-propelled swimmer.

To understand how propulsion affects the chemical field around a swimmer, it is useful to examine simple self-consistent models for swimming, such as those by Visser (2001) or [START_REF] Jiang | The flow field around a freely swimming copepod in steady motion. Part I: Theoretical analysis[END_REF]. Previous work has applied examples of these models to calculate the concentration of pheromone around a hovering copepod (Thygesen and Kiørboe, 2006), and the nutrient uptake by a model squirmer (Magar et al., 2003) which was based on a model for low Reynolds number swimming analyzed by [START_REF] Blake | Spherical envelope approach to ciliary propulsion[END_REF].

The aim of this paper is to examine how the chemical field around a self-propelled organism is affected by the organism's swimming speed and style, and how this field is affected by the choice of surface boundary conditions. The two types of boundary conditions discussed here depend on the biochemical activities taking place within the organism.

Method

Models of self-propelled swimmers

We model a swimmer by a sphere of radius a moving with speed U, and take a reference frame moving with the swimming organism. The key non-dimensional parameters of the problem are the Reynolds number (Re) which measures the relative importance of inertial to viscous forces, and the Péclet number (Pe) which represents the relative importance of advection (which here is due to swimming) to diffusion. Following [START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF], we define these parameters as follows:

(1) where ν is the kinematic viscosity of the water and D is the molecular diffusivity of the chemical.

In our calculations, we will assume that the swimmer is sufficiently small and slow so that we can assume the flow is governed by the Stokes flow equations, that is zero Reynolds number flows (Acheson, 1990). Previous studies have investigated how the flow field and resultant chemical field around a sinking sphere are modified when this assumption is relaxed [START_REF] Kiørboe | Fluid motion and solute distribution around sinking aggregates. I. Small-scale fluxes and heterogeneity of nutrients in the pelagic environment[END_REF], but here the novelty is to focus on how self-propulsion modifies the chemical field and we restrict attention to the simple case of Stokes flow, for which analytic solutions are available. The Stokes flow equations are a reasonable approximation for planktonic microorganisms [START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF], but are not valid in situations when plankton generate energetic, inertia-dominated events, for example the rapid escape jump of a copepod (Visser, 2001).

To simplify our analytic expressions for the flow field, we non-dimensionalise the system so that the swimmer is a sphere of unit radius moving at unit speed, and take a reference frame moving with the swimming organism. We also assume the swimming behaviour is axisymmetric so the velocity field can be described by the radial and azimuthal components U r and U θ . The simplest model for swimming is to consider a rigid sphere moving steadily through the fluid which has flow field given by Visser (2001): (3) which for the towed body model is given by (4) This flow field accurately represents a particle sinking under gravity, but is not a self-consistent model for a self-propelled body, as the hydrodynamic forces result in a net force acting on the body. Such a flow can only be realised if the drag force on the sphere is matched by an equal and opposite external force, for example gravity or some external `towing' force.

As discussed by Visser (2001), the net force on a neutrally buoyant, self-propelled body moving at uniform speed is zero, so that the thrust driving the body forward exactly matches the viscous drag. A simple model of this is the force dipole, or stresslet. The flow field generated by two equal and opposite forces F and -F separated by a vector b parallel to F is given at leading order in |b| by the stresslet, or dipole, flow field. Given that we choose to work in a frame of reference moving with the swimmer, the far field flow is a uniform flow of unit speed, giving the following total flow field:

(5) Note that for a sphere of unit radius moving with unit speed, the drag force is given by |F|= 6πµ, furthermore, if we assume the drag and propulsive forces act at points on opposite sides of the surface of the sphere of unit radius, we can assume |b|=2, and thus obtain the result given by Visser ( 2001 Although this model is a useful far-field representation of the flow around a self-propelled body, it does not represent the near-field behaviour. For example, the radial velocity on the surface of the body (r = 1) is typically non-zero.

In this paper we investigate the effect of self-propelled swimming on chemical transfer for a model swimmer which is the simplest we can think of which has two important properties: (1) the swimmer is a neutrally buoyant, self-propelled organism in that there is no net force acting on the organism; and (2) the organism has a well-defined shape that we can calculate the mass transfer across. The swimming model used by Magar et al. (2003) has these properties and generates the following flow field: (8) with streamfunction (9)

A key point for the purposes of this paper is that the radial component of the velocity field is zero on the surface of the sphere, and thus we can naturally consider the surface r = 1 to represent the surface of an organism whose surface is impenetrable to the surrounding fluid.

The model swimmer moves its surface steadily and tangentially to itself to generate propulsion. Specifically, at the surface r = 1, the tangential velocity is given by (10) In this model, the parameter q adjusts the imposed surface velocities as shown in Fig. 1; intuitively, it measures how much the swimming stirs the water and varying q allows an F o r P e e r R e v i e w investigation of different swimming styles. The model can be thought of as approximating a highly idealized planktonic organism such as a ciliate with infinitely small cilia or Volvox. In the case of a ciliate, varying the stirring parameter can also allow investigation into how localization of the cilia alters the swimming pattern, for example in Fig. 1 bii) the surface velocity is dominated by motions at the rear of the organism. However the primary reason for studying such an idealized model is to investigate how a self-propelled organism can create quite a different chemical field than a towed rigid body.

The flow field given by Eq. 8 was derived by considering imposed surface motions of the sphere and solving the flow field via an expansion in terms of Legendre polynomials and associated functions. Energy arguments were originally used to derive this flow field, based on the requirement that the total energy in the fluid is finite, which leads to formulae connecting the swimming speed to the surface velocities [START_REF] Blake | Spherical envelope approach to ciliary propulsion[END_REF]. Alternatively, the total force exerted by the swimmer due to the combination of pressure and viscous surface forces can be calculated as zero (e.g. Acheson, 1990), and thus the swimmer can be viewed as being selfpropelled in the same sense as the stresslet model.

Equations describing the chemical field

In the presence of a velocity field, u, generated by the swimming motions, around the organism the chemical is advected by the fluid and diffuses, and thus can be modelled by the following advection-diffusion equation with respect to non-dimensional variables: (11) In this paper we focus on computing the steady state chemical field, found by setting .

The considering the two boundary conditions as limiting cases, it is possible to assess the model predictions against experimental results to determine whether the underlying biology is transport or reaction limited [START_REF] Kiørboe | Fluid motion and solute distribution around sinking aggregates. I. Small-scale fluxes and heterogeneity of nutrients in the pelagic environment[END_REF].

For the Dirichlet boundary condition (diffusion limited transport), we can calculate how swimming enhances the mass transfer between the organism and its environment. In this case, we can non-dimensionalise the concentration field so that C = 1 on r = 1 and C = 0 as , and then define the Sherwood number, Sh, which quantifies the enhancement, as (Magar et al.,

In the absence of flow, C = 1/r, giving Sh = 2.

For the Neumann boundary condition, the total flux across the surface is assumed to have a fixed value, Q, hence

We choose to non-dimensionalise the concentration field so that the flux for both Neumann and Dirichlet boundary conditions are equal in the no-flow case, that is we take Q = 4π. This boundary condition is insufficient for obtaining a unique solution to the boundary value problem, and so we further assume that there is no angular variation in the surface concentration, that is we take where the value varies with Pe, so as to ensure Eq. 13 is satisfied. Thus the chemical field for the Neumann boundary condition is simply that calculated for the Dirichlet boundary condition rescaled by 2 / Sh.

The numerical solution for the chemical field is obtained using a finite volume method as described by Magar et al. (2003).

Results

Flow field around self-propelled swimmers

The flow fields generated by the model swimmers are plotted in Fig. 2. In order to compare the models, we note that the stresslet model agrees in the far field (up to 1/r 2 terms) with the squirmer model with stirring parameter q = -2 and can be compared in Fig. 2b and 2c(ii). In contrast to the towed body model (shown in Fig. 2a), we note that these models have a fore-aft asymmetry and there is a recirculation region ahead of the swimmer.

If we compare the expression for the stresslet given by Eq. ( 6) with the squirmer model, we see that these models agree up to 1/r 2 terms for a much wider range of values of the stirring parameter q. For example if we allow |b|, the distance between the application of the point forces in the stresslet model, to take values other than the diameter of the sphere we can obtain agreement for a much wider range of negative values of q. In Figs. 2c (i-vi), we further investigate how the choice of stirring parameter, q, affects the flow field. For q = 0 the flow field is fore-aft symmetric; as we increase the magnitude of the stirring parameter, a recirculation region appears behind or ahead of the squirmer, depending on the sign of q.

Chemical field

The chemical field around the organism depends on the Péclet number (Pe), the choice of swimming model, and the choice of boundary condition. Here we compare how the squirmer increasing values of the stirring parameter, q, the point at which the concentration boundary layer detaches from the organism moves forwards and, as shown in Fig. 3 b,iii,iv), there is a significant region of elevated concentration surrounding the rear half of the organism. Although the mathematical formalism is identical for both uptake and release of solutes from the swimming organism, we will separate our results to address two classical problems in plankton ecology, that is how swimming motions change nutrient uptake and how signal chemicals are spread from a swimming organism. In the first instance we focus on how nutrient uptake is enhanced due to swimming, that is investigate how Sherwood number is dependent on swimming speed (as expressed by Pe) and swimming style. This will be done by considering transport-limited (Dirichlet) boundary conditions at the surface of the organism. In the latter case of chemical exudation, our results will focus on how the chemical wake is dependent on swimming speed and swimming style. For the problem of chemical exudation we shall consider both the constant-leakage rate and transport-limited scenarios corresponding to Neumann and Dirichlet boundary conditions respectively.

Nutrient uptake

When we assume that nutrient uptake is limited by the rate of diffusive transport across the concentration boundary layer, that is we impose Dirichlet boundary condition, introducing fluid motion increases the rate of mass transfer as measured by the Sherwood number. In the basic case of a towed-body model, from Fig. 3a, i,ii) we see how increasing the translational speed of the organism through the fluid sharpens chemical gradients near the surface, thus increasing the diffusive flux of chemicals into the organism. This explains why the Sherwood number (Sh) is an increasing function of the Péclet number (Pe), as depicted in Fig. 4. Furthermore, the squirmer model introduces additional mixing near the surface of the organism which further enhances mass transfer relative to the towed body model moving at the same speed. As the stirring parameter, q, is increased, the enhancement of mass transfer is increased further. For moderate and large values of Pe, the organism trails a long chemical wake, an example of which is shown in Fig. 5a). This wake may provide important chemical signals, for example to mates, and so it is of interest to understand how the shape of the wake is affected by the swimming speed and style of the organism. The wake length as a function of Pe for the towed body model and the squirmer model for a range of stirring parameters is shown in Fig. 6. The length of the wake is defined as the distance behind the swimmer measured from the centre of the swimmer at which the non-dimensional concentration equals C = 0.1. For the Dirichlet boundary condition, where the surface concentration is fixed at C = 1 due to transport limitation, this corresponds to the concentration falling to 10% of the value specified at the surface of the organism. In this scenario, where fluid motions enhance the flux of chemicals away from the organism, the length of the chemical wake correspondingly increases for increasing values of Pe. Furthermore the squirmer model shows a greater enhancement in wake length compared to the towed body model because of the enhanced transport of chemicals away from the surface (Fig. 6). In contrast, we see that for the Neumann boundary condition, the length of the chemical wake decreases with increasing Pe (Figs. 5b,6). This can be explained because in this scenario there is a constant leakage rate of chemical, and so as fluid flow enhances the transport of chemicals away from the surface, the surface concentration is reduced for increasing values of Pe. Because of the reduction in the surface concentration with increasing Pe, we find that the wake length, defined as the distance behind the organism at which the concentration is given by C = 0.1 reduces with both increasing Pe and increasing values of q. Furthermore, for sufficiently high values of Pe and q, the surface concentration can drop below C = 0.1, and the wake disappears.

Chemical wake

Our results show that the chemical wake length is dependent on Pe, that is it varies depending on the swimming speed of the organism. This is true even when assuming constant leakage rate, that is taking the Neumann boundary condition. Previous authors, who modelled the organism as a point source emitting chemicals at a fixed rate, showed that the concentration distribution along the centre of the wake is identical to that of a non-moving point source; so that only the width and not the length of the chemical plume is affected by swimming speed (Bagoien, 2005). In contrast, by allowing the organism to have a finite size, our model produces a chemical plume with a length that is also dependent on swimming speed. We have shown how the chemical field around swimming plankton depends on swimming speed, style, and the type of boundary conditions imposed on the surface of the organism. We have also shown how the simple squirmer model for a self-propelled swimmer is related to a previously proposed self-consistent model of a swimmer, the stresslet model, and shown how using the squirmer model allows one to examine properties of chemical transport around a swimming organism.

Discussion and conclusions

For the Dirichlet boundary condition, the chemical transport is limited by the rate at which chemicals are transported across the diffusion boundary layer, and so fluid motion around the organism increases the flux of chemicals across the surface, as measured by the Sherwood number, and increases the length of the chemical wake. Furthermore, we have shown that, for a given swimming speed, the self-propelled swimmer can significantly enhance chemical transport, and increase the length of the chemical wake trailing the swimmer. Indeed our results agree with previous analytic and numerical results showing how at large Péclet numbers, Pe, the Sherwood number increases as the square root of Pe for a squirmer rather than as the cube root of Pe for a towed body (Magar et al., 2003).

In contrast, for the Neumann boundary condition, where transport is production limited and there is a fixed flux of chemicals across the surface, we see that although the fluid motion around the organism transports chemicals away from the surface, this has the effect of decreasing the surface concentration of chemicals and hence decreasing the length of the chemical wake.

The models presented in this paper are relevant to recent discussions regarding the distribution of DMSP surrounding algal cells [START_REF] Breckels | Modelling the concentration of exuded dimethylsulphoniopropionate (DMSP) in the boundary layer surrounding phytoplankton cells[END_REF]Strom and Fredrickson, 2010). In We note that the squirmer model presented here is highly idealized, and many plankton will have much more complicated morphologies and swimming patterns. However, simple models can often give useful insights, casting the problem in tractable terms and serving as a useful basis for comparison. Furthermore, recent work on the colonial alga Volvox suggests that the squirmer model can be used quite successfully to model real self-propelled organisms [START_REF] Drescher | Dancing Volvox: hydrodynamic bound states of swimming algae[END_REF]. Alternative simple models have recently been proposed to represent the motion of uni-flagellated organisms (Langlois et al., 2009). These appear more biologically realistic for plankton propelled by a single flagellum, but are more complex than the model considered in this paper as analytical expressions are not available for the flow field generated by the swimmer. As in this paper, Langlois et al. ( 2009) demonstrate that self-propulsion enhances nutrient uptake above the levels predicted for a towed-body, although the enhancement is not as significant as the results presented here for the squirmer model. More complex models of the flow field around cilia have also been proposed which consider in detail the near-surface forcing generated by the cilia [START_REF] Jiang | Hydrodynamic signal perception by the copepod Oithona plumifera[END_REF]; it would be interesting to examine whether our results on the chemical field around a very simple swimmer hold for these more complex models of swimmers.

A major simplification in our model has been to assume the organism is neutrally buoyant. As discussed by [START_REF] Jiang | The flow field around a freely swimming copepod in steady motion. Part I: Theoretical analysis[END_REF], typically plankton are denser than water and consequently will experience a net external force acting on them. This is important for hydromechanical signalling, as in the far field the flow only decays as 1/r rather than 1/ r 2 which is the case for neutrally buoyant, self-propelled models. In the context of mass transfer and chemical wakes, it would therefore be interesting to consider how an external gravitational force affects the local flow field. Another major simplification we have made is to assume the flow field is symmetric; however many organisms have an asymmetric physiology or swim with asymmetric beat patterns, for example which generate helical swimming paths [START_REF] Crenshaw | A new look at locomotion in microorganisms: Rotating and translating[END_REF] ) is purely tangential and given by Uθ , see Eq. 10, which varies with stirring parameter q: q = 0 solid line; q = 1 dot; q = 2 dash; q = 5 dash-dot. Velocity for negative values of q are obtained by reflecting about θ =π/2 (not shown). (b) Example surface velocity vectors for range of stirring parameters corresponding to flow fields shown in Fig. 2: (i) q = 0 (ii) q = -2 (iii) q = 1 (iv) q = -1 (v) q = 10 (vi) q = -10. Angle θ is defined from direction of swimming indicated by bold arrow. Only half the swimmer is shown as the model assumes symmetric behaviour. , and (c) squirmer, Eq. 9 , with stirring parameter (i) q = 0 (ii) q = -2 (iii) q = 1 (iv) q = -1 (v) q = 10 (vi) q = -10. Arrows indicate the swimming direction. Only half the swimmer is shown as the model assumes symmetric behaviour. 203x152mm (600 x 600 DPI) Wake length Plots of wake length, L against Pe for different swimmers: Thick solid line, towed body; thin solid line, squirmer, q = 0; dot line with circles, squirmer, q = -2; dot line with stars, squirmer, q = 2; dash line with circles, squirmer, q = -10; dash line with stars, squirmer, q = 10. The length of the wake is defined as being the distance from the swimmer's centre at which the concentration in the wake has decayed to 0.1. a) Dirichlet boundary conditions; b) Neumann boundary conditions; note the wake is undefined in situations where the surface concentration, C0, is less than 0.1. 148x111mm (300 x 300 DPI)

  flow field, streamlines can be represented as contour plots of the streamfunction, ψ, defined as (Acheson, 1990):

  boundary conditions describing the chemical concentration or chemical flux on the surface of the organism are key in determining the concentration field. Because the surface of the organism is not well defined for the stresslet model, we calculate the chemical field only for the towed body model and the squirmer. Here we consider two possible boundary conditions, either the concentration is constant over the surface of the sphere (Dirichlet boundary condition), or the total flux of chemical is constant over the sphere (Neumann boundary condition). The Dirichlet boundary condition is valid when the transport of chemicals is limited by the rate at can diffuse away from the concentration boundary layer. This boundary condition is useful to compute an upper bound on the importance of fluid dynamic effects, for example obtaining a maximum estimate of how swimming can affect the nutrient uptake rate[START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF], Magar et al., 2003). The Neumann boundary condition corresponds to the situation when the flux of chemicals is limited by the rate of biological processes. In their model of pheromone production, Thygesen andKiørboe (2006) assumed that the chemical was emitted from a point source at a constant rate Q independent of the flow field, which for a finite size organism would be most simply represented by a Neumann boundary condition. By

  the towed body model for a range of stirring parameters, q. Concentration contour plots close to the organism for a range of Pe and q for the Dirichlet boundary condition are plotted in Fig.3. As Pe increases, the diffusion boundary layer shrinks, and the chemicals are confined to a thin layer surrounding the organism except in the trailing wake. With

particular

  [START_REF] Breckels | Modelling the concentration of exuded dimethylsulphoniopropionate (DMSP) in the boundary layer surrounding phytoplankton cells[END_REF] calculated the steady concentration field around a stationary DMSP-producing algal cell, which using the results of this paper, could be extended to study motile or sinking cells. A key point in their calculations was the timescale of exudation: the timescale over which a fixed amount of chemical is released will be inversely proportional to the exudation rate and thus significantly affect the (steady) concentration distribution(Strom and Fredrickson, 2010). Furthermore, if chemicals are released rapidly, what may be of more interest is the unsteady solution to the advection-diffusion equation which predicts the transient distribution of the chemical as it is emitted.

Fig. 1

 1 Fig.1Surface velocity of model squirmer (a) Velocity on surface of organism (r = 1) is purely tangential and given by Uθ , see Eq. 10, which varies with stirring parameter q: q = 0 solid line; q = 1 dot; q = 2 dash; q = 5 dash-dot. Velocity for negative values of q are obtained by reflecting about θ =π/2 (not shown). (b) Example surface velocity vectors for range of stirring parameters corresponding to flow fields shown in Fig.2:(i) q = 0 (ii) q = -2 (iii) q = 1 (iv) q = -1 (v) q = 10 (vi) q = -10. Angle θ is defined from direction of swimming indicated by bold arrow. Only half the swimmer is shown as the model assumes symmetric behaviour. 148x111mm (300 x 300 DPI)

Fig. 2

 2 Fig.1Surface velocity of model squirmer (a) Velocity on surface of organism (r = 1) is purely tangential and given by Uθ , see Eq. 10, which varies with stirring parameter q: q = 0 solid line; q = 1 dot; q = 2 dash; q = 5 dash-dot. Velocity for negative values of q are obtained by reflecting about θ =π/2 (not shown). (b) Example surface velocity vectors for range of stirring parameters corresponding to flow fields shown in Fig.2:(i) q = 0 (ii) q = -2 (iii) q = 1 (iv) q = -1 (v) q = 10 (vi) q = -10. Angle θ is defined from direction of swimming indicated by bold arrow. Only half the swimmer is shown as the model assumes symmetric behaviour. 148x111mm (300 x 300 DPI)

Fig. 3 Fig. 4 Fig 5 :

 345 Fig. 3 Near field chemical concentration with Dirichlet boundary condition Contour plot of chemical field surrounding swimmer for range of Pe and swimming styles: (a) towed body for (i) Pe = 10, (ii) Pe = 100; (b) squirmer for (i) Pe = 10, q = -2 (ii) Pe =100, q = -2, (iii) Pe = 10, q = 10 (iv) Pe = 100, q = 10. Only half the swimmer is shown as the model assumes symmetric behaviour. 203x152mm (600 x 600 DPI)

  Langlois, V.J., A.Andersen, T.Bohr, A.W. Visser and T.Kiørboe (2009). Significance of swimming and feeding currents for nutrient uptake in osmotrophic and interception-feeding flagellates. Aquat.Microb. Ecol. 54,[35][36][37][38][39][40][41][42][43][44]

	Laroche, D., A.Vezina, M.Levasseur, M.Gosselin, J.Stefels, M.Keller, P.Matrai and R.Kwint
	(1999). DMSP synthesis and exudation in phytoplankton: a modeling approach. Mar. Ecol.
	Prog. Ser. 180, 37-49.
	Magar, V., T.Goto and T.J. Pedley (2003). Nutrient uptake by a self-propelled steady squirmer.
	Quart. J. Mech. App. Math. 56, 65 -91.
	F o r F o r
	P P
	e e
	e r e r
	R R
	e e
	v i e v i e
	w w
	. Although our
	model cannot predict how asymmetries alter the chemical field, it is plausible that such motions
	induce a circulating component to the flow which may reduce chemical transport at large values
	of the Péclet number (Karp-Boss et al., 1996).
	References

Strom, S.L. and K.

A. Fredrickson (2010)

. Modelling the concentration of exuded dimethylsulphoniopropionate (DMSP) in the boundary layer: reply. J. Plank. Res. 32, 259-260.

Thygesen, U.H. and T.

Kiørboe (2006)

. Diffusive transport in Stokeslet flow and its application to plankton ecology. J.

Math. Biol. 53, 1 -14. Visser, A.W. (2001)

. Hydromechanical signals in the plankton. Mar. Ecol. Prog. Ser. 222, 1 -24.

http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research


Thorfinnur Gunnlaugsson

Daniel Omar Frimannsson

Thomas ; Mccabe

Mark ; Lawler

Wolfgang Schmitt

Synthesis and crystallographic analysis of short pyridine based oligoamides as DNA targeting supramolecular binders

Keywords: Oligoamides, polyamides, pyridine, DNA binders, X-ray crystallography _cell_measurement_reflns_used datum .... ? PLAT195_ALERT_1_C Missing _cell_measurement_theta_max datum .... ? PLAT196_ALERT_1_C Missing _cell_measurement_theta_min datum .... ? Oligoamides, pyridine, DNA binders, X-ray crystallography

come

The following ALERTS were generated. Each ALERT has the format test-name_ALERT_alert-type_alert-level. Click on the hyperlinks for more details of the test. based structures, based on the use of pyrrole-imidazole polyamides and bicyclic ring structures, have also been synthesized with the view of mimicking hydrogen bonding driven protein-DNA interactions and for studying the binding and cellular uptake of such polyamdies. 4,5,6 We have developed several examples of luminescent imaging and potential therapeutic agents as DNA targeting molecules. These have been based on the use of simple heterocyclic structures, 7 naphthalimide derived Ru(II) complexes 8 and Tröger's bases, 9 or by using mixed lanthanide (Yb(III) and Nd(III)) transition-metal (Ru(II)) complexes, 10 In this communication, we present the synthesis of five bis-pyridines di-peptides 1-5, based on the use of acetyl-protected amino-picolinic acid, and the coupling of Boc-protected pyridine diamine to pyridine di-carboxylic acid. The solid state structure of both 1 and 5 is also presented, which display complex crystallographic packing consisting of both intra-and intermolecular hydrogen bonding networks. These building blocks were synthesized with the aim of investigating their ability to form a helical construct, a structural feature important for DNA interactions (See Graphical Abstract), by incorporating them into pyridine based oligoamides. 11 To the best of our knowledge, only a few such pyridine di-amines have been developed to date. [12][13][14][15][16][17][18][19] acid. However, structures 1-5 have not been previously formed as building blocks for the incorporation into such pyridine based oligoamides.

Alert level B

3 N N H 2 N N H 2 N N H O OH O N N H O OH O N N H 2 R1 R2 N N H 2 NH 2 N BocHN NH 2 N O O O H OEt N BocHN R1 R2 N N H 2 R1 R2 N N H 2 R1 R2 i-ii i-ii

RESULTS AND DISCUSSION

Synthesis: The synthesis of 1-4 required the initial preparation of the acetal protected pyridine amino acid units, possessing the carboxylic acid functionality in either the fifth or the sixth positions of the pyridine ring. For the synthesis of 1 and 2, a two-step synthesis for the N-acetyl-2-amino-6-picolinic acid 8 and 9 was undertaken, 17b Scheme 1, which initially involved the protection of 2-amino-6-picoline (6) and 2-amino-5-picoline (7) with acetic anhydride in toluene overnight at 95 ºC, which yielded the two structural isomers in 87 and 90% yield, respectively. 18 The protection of the amine was employed to prevent oxidation of this moiety and increase the stability of the resulting pyridine amino acid. and NaOH overnight, followed by acidification to pH 4 using conc. HCl. These syntheses were also repeated using Boc anhydride, as the amino protecting group, with the aim of facilitating their incorporation into oligoamides using peptide coupling protocols. While the two Boc-protected picolines were obtained in good yield, the oxidation of these structures to their corresponding acids 12 and 13 was unsuccessful using either KMnO 4 , or K 2 CrO 7 , only resulting in the recovery of the two starting materials. With the aim of forming the free amine, which could then be coupled to 10 or 11, the hydrolysis of 8 and 9 was also undertaken by refluxing these derivatives in 10% aqueous NaOH solution for 6 hours, followed by acidic workup. This gave the corresponding pyridine amino acids 14 and 15 in 58% and 46% yields, respectively. By reacting either 11 or 12 in a 3:20 mixture of H 2 SO 4 and ethanol under reflux for five hours under argon, 19 followed by aqueous NH 4 + workup, resulted in deprotection of the N-acetyl group and in the esterification of the carboxylic acids, giving 16 and 17 in ca.

50% yields.

The two amino acids 14 and 15 were also reacted in an excess of oxalyl chloride in an alternative hydrogen donating pattern, the synthesis of the orthogonal protected 5 was also undertaken, commencing with the mono-protection of 18, to give 19. 25 As for the synthesis of bis-pyridine di-peptides 1-4, the 19 and the mono protected di-acid 20 26 were coupled together using HOBt, DCC and triethylamine in dry THF at 0 ºC overnight at room temperature. The resulting residues were purified by column chromatography on flash silica (using 40/60 EtOAc/CH 2 Cl 2 ), giving the protected di-peptide 5 as a white solid in 27% yield.

X-Ray Crystallographic Analysis: Crystals suitable for X-ray crystallographic analysis of 2

were grown from DMSO solution, [START_REF]a 0.30mm quartz fibre tip and immediately placed on the goniometer head in a 121K N 2 gas stream. The data set was collected using Crystalclear-SM 1.4.0 software and[END_REF] and crystallises in the monoclinic crystal system in the space group P2 1 /c. Crystallographic data and details of the structure refinement for 2 are provided in Table 1. The structure shows that the pyridine units are anti to each other in the solid state, and that the angle between the amide hydrogen atoms is 105º. Viewing the crystal structure through the 2,6-substituted pyridine plane, the second pyridine unit appears slightly twisted out of plane. The angle between the planar pyridine units is equal to the sum of three torsion angle deviations in the bonds that connect the pyridine units (-7.3º + 4.7º + 0.2º) and was calculated as -2.4º. In a similar manner, the pyrrole ring planes in the well known DNA grove binding drug netropsin have been reported to have a twisted angle to one another. This angle is slightly larger than that seen herein. 25 The difference between the pyrrole and the pyridine units in Figure 2, we believe, arises from the non-planarity of the pyrrole units, which is ascribed to the steric clash between the pyrrole CH and the adjacent amide group. [START_REF] Kopka | [END_REF] In addition, the angle formed between the amide hydrogens was measured as 105º, suggesting that repeated units of the 2,6-substituted pyridines would span 360º after nearly 5 repeating units (360º/75º = 4.8), while DNA has about 10 base pairs per turn. The results for 2, suggest angles could change upon binding in the minor groove as has been shown to be the case for netropsin. 25 The supramolecular arrangement of 2 and the involved intermolecular forces might give further indications of how this compound could interact with DNA, and these were Crystals suitable for X-ray crystal structure analysis of 5 were obtained by slow evaporation form a methanol solution. Compound 5 crystallises in the triclinic crystal system in the space group P-1. The resulting structure is shown in Figure 3a. Crystallographic data for 2 are provided in Table 1. The crystal structure consists of isolated molecules, held together through weak dispersion forces. In contrast to the crystal structure of 2, shown in Figure 2a, the crystal structure of 5 shows the pyridine rings syn to each other. This orientation and binding pattern (H-donor/H-accepter) could preferentially target i) G sequences, by hydrogen bonds between the pyridine nitrogen and the NH 2 of G and by ii) the amide hydrogen and the lone pair of the G base (See Graphical Abstract). As for the crystal structure of 2, the angle between the relatively planar pyridine aromatic rings was calculated from the three torsion angles that connect the pyridine units (-1.28º -0.56º + 1.48º) as -0.36º, which is of similar magnitude to that seen for 2, demonstrating a small rise between the two pyridine units. Furthermore, the angle formed between the two 'amide' protons was determined as 130º, which is somewhat larger than seen for 2. These results indicate that the configuration of the 2,6-diaminopyridine units would span 360º after 7 units (360º / 50º = 7.2), an improvement on that seen for 2. The supramolecular arrangement of 5 was also To evaluate if these structures could bind to DNA in situ or/and induce apoptosis in cancer cell-lines, the cytotoxicity of the di-peptides 1-4 were investigated against drug resistant chronic myeloid leukemia K562 cell-line. These acetate protected di-peptides were only found to be moderately soluble in water. However, the incubation of 2 with K562 for 48 We have further shown that in the case of 2, the binding of the oligopyridine units stabilizes the DNA structure at the same time as we demonstrated that 2 entered cells and gave rise to apoptosis with an EC 50 value of 82 µM in a drug resistant chronic myeloid leukemia K562 cell-line. We are currently undertaking the synthesis of other oligoamid derivatives based on these building blocks presented herein and investigating their binding to DNA as well as their cytotoxicity in cancer cell lines. with a Gateway 2000 4DX2-66 workstation. 1 H NMR spectra were recorded at 400 MHz using a Bruker Spectrospin DPX-400 instrument. 13 C NMR spectra were recorded at 100 MHz using a Bruker Spectrospin DPX-400 instrument.

Synthesis:

6-Acetylamino-pyridine-2-carboxylic acid (6-methyl-pyridin-2-yl)-amide (1) Compound 8 (0.28 g, 1.57 mmol) along with 2-amino-6-picoline (0.21 g, 1.88 mmol) and HOBt (0.25 g, 1.84 mmol) were dissolved in DMF. To this solution DCC (0.35 g, 1.67 mmol) was added at 0 °C. The reaction was stirred overnight under argon at room temperature. The reaction mixture was then kept at 4 °C for 3 hours before filtering it through celite. The solvent was evaporated under reduced pressure and the product was obtained as a white solid dried (0.06 g, 0.22 mmol, 14%) that was dried under high vacuum. m.p. 205-207 ºC; 1

6-Acetamido-N-(5-methylpyridin-2-yl)nicolinamide (3):

To a stirred solution of 9 (0.18 g, 1.0 mmol), 2-amino nicoline (0.12 g, 1.1 mmol) and HOBt (0.17 g, 1.3 mmol) was suspended in DMF and, DCC (0.26 g, 1.3 mmol) and triethylamine were added and stirred for 30 minutes at 0-4 ºC under argon and then at room temperature overnight. The reaction was filtered through celite and the solvent was evaporated under reduced pressure. After purification by silica flash column chromatography eluting with 20% EtOAc/CH 2 Cl 2 → 50% EtOAc/CH 2 Cl 2 the product was obtained as a white solid. (0.06 g, 0.24 mmol, 24% yield). m.p 217-218 °C; 1 2-amino-5-picoline (0.15 g, 1.0 mmol) and HOBt (0.15 g, 1.1 mmol) was weighed out and dried under reduced pressure before dissolved in dry THF (10 mL) and cooled in an ice bath for 30 minutes. EDCI. HCl (0.21 g, 1.1 mmol) was added and stirred for further 30 minutes at 0 ºC and then stirred overnight at R.T. The solvent was removed and the solid dried under reduce pressure before CH 2 Cl 2 was added. The mixture was washed three times with saturated NaHCO 3 and the organic layer was dried over MgSO 4 , filtered and evaporated to give white crystals (0.04 g, 0.16 mmol, 16% yield); m.p 216-217 °C; 1

Figure 1 .

 1 Figure 1. Structure of di-pyridine peptides 1-5

Scheme 1 .

 1 Scheme 1. (i) Acetic anhydride, toluene, 95 ºC. (ii) KMnO 4 , NaOH, H 2 O, reflux. (iii) DCC (or EDCI) HOBt, DMF, RT. (iv) Boc anhydride, THF, room temperature, (v) DCC, HOBt, Et 3 N, THF, RT.

CHCl 3 5 from

 35 for four days at 45 ºC under anhydrous conditions, this gave their corresponding acid chlorides. Further treatment of this solution with methanol, gave the corresponding methyl esters protected pyridine amino acids analogues of 16 and 17, and in ca. 50% overall yield.Having obtained several pyridine amino acid building blocks, we next formed the bis-pyridine di-peptides 1-4. Initially, the acid derivatives 8 and 9 were treated with SOCl 2 , resulting only in the recovery of the starting materials. Consequently, the peptide coupling reagents dicyclohexylcarbodiimide (DCC) or 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide hydrochloride (EDCI) were employed. Here, 10 and 11 were coupled to 8 or 9 in DMF at room temperature using these reagents, which after aqueous base workup and crystallization EtOAc, gave the desired di-peptides 1-4 in moderate yields. These were all characterised using conventional methods. With the view of coupling 1-4 to another di-peptide possessing

Figure 2 .

 2 Figure 2. a) Structure of 2. b) and c) H-bonded 1D chain structure of 2. d) and e) Packing diagrams with view in the direction of the crystallographic c-and b-axis

  at the packing arrangement of the molecule. The supramolecular structure of 2 is characterised by intermolecular H bonds involving the N and O atoms of the amide functionality that bridges the aromatic moieties. This hydrogen bond results in a 1D chain structure that runs in the direction of the crystallographic c-axis. The distance between the hydrogen donor N atom and the O acceptor atom is 2.935(5) Å. The alkyl moiety of the terminating ester functionality and the methyl subsistent of the aromatic ring prohibit strong π-π interactions in the crystal structure. The Figures 2b and 2c show the 1D H-bonded chain structure; Figure 2d and Figure 2e highlight the arrangement of the chain within the crystal structure.

Figure 3 .

 3 Figure 3. a) Structure of 5. b) Packing of 5 in the crystal structure with view in the direction of the crystallographic b-axis (C grey, O red, N blue, H white).analysed. Figure3bshows the packing of 5 with view in the direction of the crystallographic

Figure 4 .

 4 Figure 4. Thermal denaturation curves of ct-DNA (150 µM) in 10 mM phosphate buffer, pH 7.4, in the absence (green) and in the presence of 2 (purple) at a P/D of 10.

  we have undertaken the synthesis of several pyridine amino acid derivatives and bis-pyridine di-peptides for the formation or the incorporation into pyridine based oligoamides. We have shown, using X-ray crystal structure analysis, that the bis-pyridines dipeptides 2 and 5 have curved molecular structures, which could facilitate the formation of helically shaped pyridyl oligoamides. The supramolecular structure of 2 is characterised by 1D chains which run parallel to the crystallographic c-axis and in which single molecules are connected through H-bonds. In contrast to 2,.the arrangement of N-atoms in 5, does not allow the molecule to be engaged in strong intermolecular H-bonds. The packing of 5 in the crystal structure is characterised by isolated molecules held together through weak dispersion forces.

  H NMR δ H (400 MHz, DMSO-d 6 ): 10.9 (1H, s, Ar-NH-CO-Ar), 10.2 (1H, s, Ar-NH-CO-CH 3 ), 8.34 (1H, d, J = 8.4 Hz, Ar-H), 8.11 (1H, d, J = 8.0 Hz, Ar-H), 8.03 (1H, t, J = 8.0 Hz, Ar-H), 7.86 (1H, d, J = 7.6 Hz, Ar-H), 7.77 (1H, t, J = 7.8 Hz, Ar-H), 7.05 (1H, d, J = 7.2 Hz; Ar-H), 2.43 (3H, s, CO-CH 3 ), 2.16 (3H, s, Ar-CH 3 ); 13 C NMR δ c (100 MHz, DMSO-d 6 ), 169.6, 161.6, 157.0, 151.0, 150, 146.8, 140.4, 139.0, 119.5, 117.0, 116.9, 110.0, 24.0, 23.6; Accurate MS (m/z) Calculated: 271.1190 (M+H) + , Found: 271.1195; IR υ max (cm -1 ): 3354 (NH), 3251 (NH), 1689 (C=O), 1668 (C=O), 1449 (Pyridine). 6-Acetamido-N-(5-methylpyridin-2-yl)picolinamide (2): Compound 8 (0.61 g, 3.4 mmol), 2-amino-5-picoline (0.44 g, 4.1 mmol) and HOBt (0.15 g, 4.0 mmol) were dissolved in DMF.DCC (0.76 g, 3.7 mmol) was dissolved in CH 2 Cl 2 and added at 0 ºC under argon. The reaction mixture was stirred for 2 hours and then additional DCC (0.38 g, 1.8 mmol) was added and the reaction allowed to stir overnight at room temperature. The reaction mixture was then and the filtrate was evaporated under reduced pressure. After purification by silica flash column chromatography eluting with 20% EtOAc/ CH 2 Cl 2 , the product was obtained as a white solid (0.15 g, 0.56 mmol, 16%). m.p. 204-206 ºC;1 H NMR δ H (400 MHz, DMSO-d 6 ): 10.9 (1H, s, NH-CO-Pyr), 10.4 (1H, s, CO-NH-Me), 8.34 (1H, d, J = 8.0 Hz, Ar-H), 8.24 (1H, s, Ar-H), 8.20 (1H, d, J = 8.4 Hz, Ar-H), 8.06 (1H, t, J = 8.0 Hz, Ar-H), 7.87 (1H, d, J = 7.6 Hz, Ar-H), 7.73 (1H, dd, J = 8.4; 2.0 Hz, Ar-H), 2.30 (3H, s, Ar-CH 3 ), 2.16 (3H, s, CH 3 -CO); 13 C NMR δ C (100 MHz, DMSO-d 6 ): 169.6, 161.4, 156.6, 151.0, 148.2, 146.8, 140.5, 139.0, 119.5, 117.0, 116.9, 110.0, 24.0, 23.6; IR υ max (cm -1 ): 3354 (NH), 3252 (NH), 1689 (C=O), 1668 (C=O).

3 ( 1 H

 31 H NMR δ H (400 MHz, DMSO-d 6 ): 10.9 (1 H, s, Ar-NH-CO-Pyr), 10.2 (1 H, s, Ar-NH-CO-Me), 8.34 (1 H, d, J = 8.4 Hz, Ar-H), 8.11 (1 H, d, J = 8.0 Hz, Ar-H), 8.0., t, J = 8.0 Ar-H), 7.86 (1 H, d, J = 7.6 Hz, Ar-H), 7.77 (1 H, t, J = 8.0 Hz, Ar-H), 7.05 (1 H, d, J = 7.2 Hz, Ar-H), 2.43 (3 H, s, Ar-CH 3 ), 2.16 (3 H, s, CH 3 -CO); 13 C NMR δ C (100 MHz, DMSO-d 6 ): 170.0, 161.6, 157.0, 151.0, 150.0, 146.8, 140.3, 139.0, 119.5, 117.2, 116.9, 110.0, 24.0, 23.6; Accurate MS (m/z) Calculated: 293.1008 (M+Na) + , Found: 293.1014; IR υ max (cm -1 ): 3355 (NH), 3251 (NH), 1689 (C=O), 1668 (C=O), 1447 (pyridine).

6 -

 6 Acetamido-N-(6-methylpyridin-2-yl)nicotinamide (4): Compound 9 (0.18 g, 1.0 mmol),

4 ;

 4 Schemes and Figures

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Scheme 1. (i) Acetic anhydride, toluene, 95 ºC. (ii) KMnO 4 , NaOH, H 2 O, reflux. (iii) DCC (or EDCI) HOBt, DMF, RT. (iv) Boc anhydride, THF, room temperature, (v) DCC, HOBt, Et 3 N, THF, RT.

checkCIF/PLATON report No syntax errors found. CIF dictionary Interpreting this report Datablock: df1m

	Bond precision: C-C = 0.0069 A	Wavelength=0.71073
	Cell:	a=11.771(3)		b=11.903(3)	c=9.525(2)
		alpha=90		beta=92.399(5)	gamma=90
	Temperature:	121 K
	F o r Calculated F 1333.4(6) o Space group Volume P 21/c r Hall group -P 2ybc		Reported 1333.4(6) P21/c ?
	P e e r Moiety formula C14 H14 N4 O2 Sum formula C14 H14 N4 O2 P Mr 270.29 Dx,g cm-3 1.346 e Z 4 e Mu (mm-1) 0.094 r F000 568.0	? C14 H14 N4 O2 270.29 1.346 4 0.094 568.0
	F000' h,k,lmax Nref Tmin,Tmax Tmin'	568.23 13,14,11 2347 0.978,0.981 0.968	R e v i e w 13,14,11 R 2346 e w e i v
				O n O n
					l y l y

Correction method= Not given Data completeness= 1.000 Theta(max)= 25.000 R(reflections)= 0.0736( 1194) wR2(reflections)= 0.2164( 2346) S = 1.018 Npar= 183

Table 1 .

 1 Summary of crystal data, data collection and structure refinement for X-ray diffraction study of 2 and 5. __________________________________________________________________________

		Compound 2	Compound 5
	Chemical formula	C 14 H 14 N 4 O 2	C 18 H 20 N 4 O 5
	Formula weight (gmol -1 )	270.29	372.38
	Crystal system	monoclinic	triclinic
	Space group	P21/c (no. 14)	P-1 (no. 2)
	a (Å)	11.771(3)	7.2480(16)
	b (Å)	11.903(3)	7.7467(19)
	c (Å)	9.525(2)	16.949(4)
	α (°)	90	82.609(6)
	β (°)	92.399(5)	80.612(8)
	γ (°)	90	81.488(8)
	V (Å 3 )	1333.4(6)	923.2(4)
	Z	4	2
	D calc (g cm -3 )	1.346	1.340
	µ (mm -1 )	0.094	0.100
	F (000)	568.0	392.0
	Radiation (Å)	0.71073	0.71073
	Temperature (K)	121(2)	121(2)
	2θ Range (°)	50	50
	h, k, l Ranges	-13→ 13, -13→14, -11→11	-8→ 8, -9→9, -20→20
	Measured reflections	10397	14143
	Unique reflections	2346 (R int = 0.0910)	3247 (R int = 0.0257)
	Reflections used [I> 2σ (I)]	1194	2927
	Parameters refined	183	245
	GoF (on F 2 )	1.018	1.120
	R1 [I> 2σ (I)]	0.0736	0.0408
	wR2 [I> 2σ (I)]	0.1727	0.1201
	(∆ρ) max / (∆ρ) min (e Å 3 )	0.315/-0.480	0.062/-0.357

  H NMR δ H (400 MHz, DMSO-d 6 ): 10.84 (1H, s, CO-NH-Pyr), 10.83 (1H, s, CO-NH-Me), 8.93 (1H, d, J = 2.4 Hz, Ar-H), 8.37 (1H, dd, J = 8.8; 2.8 Hz, Ar-H), 8.22 (1H, s, Ar-H), 8.16 (1H, d, J = 8.8 Hz, Ar-H), 8.08 (1H, d, J = 8.4 Hz, Ar-H), 7.67 (1H, dd, J = 8.4; 2.0 Hz, Ar-H), 2.29 (3H, s, Ar-CH 3 ), 2.14 (3H, s, CH 3 -CO); 13 C NMR δ C (100 MHz, DMSO-d 6 ): δ 169.8, 163.9, 154.2, 149.9, 148.3, 147.7, 138.5, 138.1, 128.9, 125.0, 114.3, 112.0, 24.1, 17.

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

ACKNOWLEDGMENTS

We thank Trinity College Dublin, CSCB, SFI and IRCSET for financial support. We also like to thank Dr. John E. O'Brien for his help and Dr. Tony McElligot for his initial help and support, and Dr. Emma B. Veale for helping out with analysis of DNA denaturation studies.

EXPERIMENTAL

General: Reagents (obtained from Aldrich) and solvents were purified using standard techniques. Melting points were determined using a GallenKamp melting point apparatus.

Infrared spectra were recorded on a Mattson Genesis II FTIR spectrophotometer equipped