Drug Transport Microdevice Mimicking an Idealized Nanoscale Bio-molecular Motor

Journal of Bionic Engineering 8 (2011) 455–463

Drug Transport Microdevice Mimicking an Idealized Nanoscale Bio-molecular Motor Jae Hwan Lee, Ramana M. Pidaparti Department of Mechanical Engineering, Virginia Commonwealth University, Richmond, VA 23284, USA

Abstract Molecular motors are nature’s nano-devices and the essential agents of movement that are an integral part of many living organisms. The supramolecular motor, called Nuclear Pore Complex (NPC), controls the transport of all cellular material between the cytoplasm and the nucleus that occurs naturally in biological cells of many organisms. In order to understand the design characteristics of the NPC, we developed a microdevice for drug/fluidic transport mimicking the coarse-grained representation of the NPC geometry through computational fluid dynamic analysis and optimization. Specifically, the role of the central plug in active fluidic/particle transport and passive transport (without central plug) was investigated. Results of flow rate, pressure and velocity profiles obtained from the models indicate that the central plug plays a major role in transport through this biomolecular machine. The results of this investigation show that fluidic transport and flow passages are important factors in designing NPC based nano- and micro-devices for drug delivery. Keywords: molecular motors, nuclear pore complex, geometry, design, simulation Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60055-3

1 Introduction Molecular motors, pumps and sorters are nature’s nanomachines. They are the essential agents of movement and integral parts of many living organisms. The performance of these molecular motors, in terms of mechanical efficiency, is unparalleled by any man-made motors. The Nuclear Pore Complex (NPC) is one such nanoscale supramolecular machine. The NPC controls the transport of all cellular material between the cytoplasm and the nucleus that occurs naturally in biological cells of many organisms including yeast, vertebrate and others. It consists of a large number of spatially organized proteins that, together with soluble transport factors, manage to export, import and exclude proteins with remarkable speed (few hundred translocations per NPC in one second) and fidelity[1]. The overall structure of the NPC spans the nuclear envelope with a tripartite structure consisting of a complex central cylindrical body embedded between cytoplasmic and nuclear octagonal rings[2–6]. Each ring consists of eight spokes positioned in a symmetrical array Corresponding author: Ramana M. Pidaparti E-mail: [email protected]

around a central tube spanning the cytoplasmic and nucleoplasmic surfaces of the nuclear envelope. The central body contains a channel that consists of a central plug, and eight radiating spokes connecting the spokes of the rings to the central plug, in a characteristic eight-fold symmetry[2,4–10]. The actual dynamics of molecular transport across the NPC is not known completely. Small metabolites (ions, small molecules and proteins < 30 kD – 40 kD) diffuse freely through the NPC whereas large proteins and RNA flow selectively through the NPC. The transport signals are recognized by mobile receptors called importins, exportins and transportins, which interact with proteins of the NPC to translocate and shuttle cargo between the cytoplasm and the nucleoplasm. Due to NPC’s complex architecture, the structural design changes during transport and their properties remain incompletely understood. In recent years there has been an increasing interest in developing microdevices for fluidic transport for many applications, including drug delivery systems[11–13], insulin injectors[14], liquid cooling systems[15–16], fuel cells[17], space missions[18,19], as well as macromolecule

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In yeast and vertebrates, the NPCs are distinctive structures of the nuclear envelope, with molecular weights of ~ 50 MDa – 125 MDa, and typical dimensions of 100 nm – 200 nm. The NPC spans the nuclear envelope with a tripartite structure consisting of a central cylindrical body embedded between cytoplasmic and nuclear octagonal rings as shown in Fig. 1. The various components of the NPC are shown schematically in Fig. 1, including the cytoplasmic ring, central plug, spoke complex, nucleoplasmic ring, basket, and distal ring.

x y z

10.5

2 Idealized nanoscale biomolecular motor

mechanisms. As a first step in understanding the role of geometrical components of NPC for transport of material/fluid, design analysis is conducted using a coarsegrained model through the finite element method.

8.1

and cell analysis[20]. Due to the NPC’s interesting and unique geometric architecture, various components play an important role in controlling the transport of material. Even though there are several unknowns in the transport process through this bio-molecular motor, we believe that a better understanding of mechanical and transport characteristics through computer simulation may provide us with unique solutions for designing nano/microscale machines for engineering applications. In order to understand the design characteristics of the NPC, we developed a microdevice for drug/fluidic transport mimicking the coarse-grained representation of the NPC geometry through computational fluid dynamic analysis and optimization. Specifically, the role of the central plug in active fluidic transport and passive transport (without central plug) was investigated.

10

456

10.8

7.7

(a) Geometry without Central Plug (CP)

x y z

Cytoplasmic filament

Inner diameter of Central Plug (CP)

Cytoplasmic ring

Total height: 20 mm (=200 nm)

1

Spokes

4 1

Central plug 12

10 mm (=100 nm)

Nuclear ring x1= 1 Nuclear basket

x2 = 9 20

(b) Original geometry

Fig. 1 A schematic of an idealized NPC structure.

The geometric models of NPC along with overall dimensions considered for developing the micro-device for drug transport are presented in Fig. 2. One geometry is without the central plug as shown in Fig. 2a and the other geometry is with central plug in which the transport facilitated through both active and passive diffusion

Fig. 2 Geometric details of an idealized simulation models of NPC (all dimensions are in mm), illustrated the central plug position for x1 and x2 in Fig. 2b).

3 Materials and methods The design analysis model and the optimization method used in the present study are briefly described below.

Lee et al.: Drug Transport Microdevice Mimicking an Idealized Nanoscale Bio-molecular Motor

3.1 Finite element model A two-dimensional finite element model of the geometric models shown in Fig. 2 is considered for design simulations. The design simulations for the micro-device for drug/fluidic transport require solving the conservation equations for mass, and momentum. The governing equations for the laminar flow are described by the conservation and momentum, which can be expressed as

wui wxi wui wu  uj i wt wx j



0,

1 wp w  U wxi wx j

ª § wu wu j · º «v ¨¨ i  ¸» , «¬ © wx j wxi ¹¸ »¼

(1) (2)

where ui is the velocity vector in two coordinate directions, i.e., i = 1 and 2 and for this two-dimensional problem, wui / wt is the derivative of velocity with respect to time, p is the pressure, ȡ is the fluid density, and Ȟ is its kinematics viscosity. Using the incompressible steady flow, and assuming that the fluid density is homogenous, the mass conservation equation (Eq. (1)) is the divergence of flow velocity vector. The Solidworks software (Solidworks Corp., Massachusetts, USA) was used to develop the models and extract the volumetric fluid area from the models. These models were converted into parametric-solid format in order to be imported by the ANSYS software (ANSYS, Inc., Houston, PA, USA). Finite Element (FE) analysis using ANSYS-FLOTRAN module was performed for the design simulations. Half of the model was axisymetrically simulated in order to obtain an obvious contour at the central area that includes the central plug and the inner walls. The working fluid of the micro-device was assumed to be water with density (ȡ) of 1000 kg·mí3, and viscosity of 1eí3 kg·mí1·sí1. The design geometry was modeled using 3D Fluid 12 – tetrahedral finite elements to study the steady and transient state fluid analysis. The boundary conditions used for the model include inlet velocity of 0.01 m·sí1 (uniform) and an outlet pressure of 0 Pa (ambient). Further, a no-slip boundary condition was used on the walls of the fluid region and the velocity in x-direction was imposed as zero along the line of symmetry (y-axis) of the design model. Even though the NPC is at a nano-scale, a scaled up micro-device from nano-scale to millimeter scale model

457

(from 100 nm to 10 mm) was developed in this investigation. This is due to our ultimate objective of developing a micro-fluidic drug delivery device (1 ȝL·sí1 – 100 ȝL·sí1) mimicking the overall NPC geometry configuration. Table 1 lists the various parameters (length, diameter, water density, and viscosity) of the nano-scale biological motor and the present micro-device model. The similarities between the nano-scale biological motor and the present micro-device are shown in Table 2. As can be seen from Table 2, we kept the geometric similarity to be 1 so that the continuum mechanics hold and the results of velocities and flow rate will be similar between the nano-scale and mille-meter scale considered in this study. In order to further illustrate this aspect, a nano-scale model was built and the results were compared to those obtained by the micro-scale model. In general, the results from the millimeter micro-device model were applicable to the nano-scale model with the scaled up ratio (Ȝ) and the trends were also similar. Table 1 Various parameters used in the present micro-scale device in comparison to nano-scale biological motor Variable

Nano-scale biological motor

Present micro-scale device

Ratio (Ȝ)

Units

Length(L)

0.2 ×10í3

20

105

mm

í3

12

105

mm

Diameter(D)

0.12×10

Density of water(ȡ)

1000

1000

1

kg·mí3

Dynamic viscosity(μ)

1.0 ×10í3

1.0 ×10í3

1

Pa·s (N·s·mí2)

Table 2 Dimensionless similarities between nano-scale biological motor and the present micro-scale device Similarity Geometric Kinematic Dynamic

Variable Rate of diameter over length (D/L)

Scaled ratio (Ȝ)

Velocity (V)

105

Flow rate (Q)

1015

Pressure (P)

1010

1

A convergence study was conducted to make sure the accuracy of the results is within 5% by successively refining the computational mesh. Three different meshes (with elements ranging from 120,000 to 390,000) were used to model the geometries shown in Fig. 2. Results of velocities and pressure distributions are obtained as part of the analysis. 3.2 Optimization method Due to various design parameters, specifically the central plug location and the diameter of the central plug,

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affecting the fluidic transport through the micro-device, an optimization analysis is performed to obtain the optimum location of the central plug for achieving maximum velocity at the outlet of the device. Fig. 3 shows the optimization flow chart used in this study. Initially, six different designs for central plug location (x1 and x2) were selected (see Table 3), and the finite element analysis (ANSYS-FLOTRAN) was carried out to obtain the maximum velocities at the outlet of the device. Then using the finite element results, the objective function, and derivatives of objective function with respect to velocity, boundary conditions, and constraints were defined. We used a “Polyfitn” function in MATLAB to obtain a solution for the optimized design.

The objective function considered in this study, is to maximize the velocity (V) at the outlet of the device with inlet velocity of 10 mm·sí1 under incompressible and laminar steady-state flow conditions. The set of design variables (x1, x2) for the development of the micro-device with optimum location for the central plug is shown in Fig. 2. Mathematically, the optimization problem can be written as, Maximize Velocity V

f  x1 , x2 A  Bx1  Cx2  Dx1 x2  2

3 1

3

Subject to constraints

x1L d x1 d x1H , x2L d x2 d x2H . Start

0.5 mm ” x1 ” 3 mm, 8 mm ” x2 ” 11mm.

Set a Objective Function

Define Derivatives of Objective Function with respect to Velocity New velocities and meshed elements

Polyfit functionMTLAB New Geometry

Vary central plug position

(4)

Due to design limitations of the device, the constraints are

Develop Microdevice Geometry

ConstraintsBoundary Conditions

(3)

Ex  Fx2  Gx  Hx2 , 2 1

Simulation; ANAYSFlotran

Optimized Design

Fig. 3 Flow chart of the procedure used in geometry optimization of the NPC. Table 3 Comparison of maximum velocities (mm·sí1) obtained for various central plug positions (x1, x2 in mm) of Fig. 2b Maximum velocities x1=0.5

x1=1.0

x1=1.5

x1=2.0

x2=8.5

9.894

12.450

11.766

x2=9.0

10.463

12.458

11.216

x2=9.5

8.804

10.992

x2=10.0

10.410

x2=10.5 x2=11.0

x1=2.5

x1=3.0

10.648

11.749

11.656

11.924

12.225

12.011

10.751

12.451

11.934

11.134

11.145

10.798

12.140

12.074

11.878

9.527

11.678

11.122

11.714

11.986

11.587

10.291

10.226

11.153

12.094

12.387

11.388

(5)

Constants A, B, C, D, E, F, G, and H were determined through the polynomial fit function in MATLAB and then the optimization solution was obtained. The position of the central plug and fluid velocities can potentially represent any input-output relationship with a finite number of discontinuities, assuming that there is a solution to satisfy the constraints.

4 Results and discussion The results of two finite element models (mesh #1 – 83,231 elements and mesh #2 – 371,371 elements) and the respective pressure contours for the NPC design without the central plug (Fig. 2a) are shown in Fig. 4. It can be seen from Fig. 4 that mesh #2 has uniform flow through the device as well as higher pressure/velocity at the outlet in comparison to mesh #1. Fig. 5 shows the results of two finite element models (mesh #1 –149,225 elements and mesh #2 –306,548 elements) and the respective pressure contours for the NPC design with the central plug (Fig. 2b). It can be seen from Fig. 5 that mesh #2 has a higher number of finite elements, and higher pressure and higher velocity at the outlet in comparison to mesh #1. Most of the flow is concentrated near the central plug and a uniform flow exits towards the outlet of the device. It can be seen from Figs. 4 and 5 that the flow velocities and magnitudes are different between the two designs (one with and the other without the central plug).

Lee et al.: Drug Transport Microdevice Mimicking an Idealized Nanoscale Bio-molecular Motor

V = í54.7297 + 2.2474x1 + 20.6109x2

Vector plot

Mesh

459

Vwall=0

+0.144x1x2 í1.2634x12í2.1886 x22

(6)

+0.1336x13+ 0.0759x23

Vin Pout = 0 Mesh#1 - 38, 231 elements

Mesh#2 - 371, 371 elements 1.5

3.0

4.5

6.0

7.5

9.0

10.5 12.0

Fig. 4 Finite element meshes, boundary conditions and the corresponding contour plots of flow rates for Fig. 2a geometry – without the central plug.

The objective function (V) given by Eq. (6) is plotted in Fig. 6. After the optimization, the best position for the central plug was found to be at x1 = 2.038 mm and x2 = 8.885 mm with a maximum velocity of 11.91 mm·sí1. The optimized geometry, the finite element model (mesh size = 367,431 elements) and the velocity contour plot are shown in Fig. 7. It can be seen from Fig. 7 that the velocity contour produces less pressure in front of the central plug and smooth flow (fewer vortexes) at the inner channel of the central plug.

Vector plot

Mesh Vwall=0 Vin Pout = 0 Mesh#1 ̢ 149,225 elements

Mesh#2 - 306, 548 elements

1.5

3.0

4.5

6.0

7.5

9.0

10.5 12.0

Fig. 6 Result of velocity variation with design variables through a polyfitn function analysis.

Fig. 5 Finite element meshes, boundary conditions and the corresponding contour plots of flow rates for Fig. 2b geometry – with the central plug.

8.87 2.08

4.1 Optimized design

In order to see how the drug/fluid is transported with variations in the central plug locations (x1 and x2) of the micro-device, several finite element runs were carried out and the results of flow rate and the maximum velocity were obtained. The results are summarized in Table 3. It can be seen from Table 3 that the maximum velocity changes by about 13%. Since the central plug affects the flow rate, an optimization study as discussed in section 3 was carried out to estimate the best central plug location for the micro-device design. By using the “polyfitn” function in MATLAB (“Polyfitn” is provided by Matlab central), the constants in the objective function (V) given by Eq. (3) were found to be, A = í54.7297, B = 2.2574, C = 20.6109, D = 0.144, E = í1.2634, F = í2.1886, G = 0.1336, and H = 0.0759. Using the constants, the objective function (V) can be written as

0

1.5 3.0

4.5 6.0 7.5

9.0 10.5 12.0

Fig. 7 Geometry (top), finite element model (middle) and flow rate contour plot (bottom) for the optimized design of micro-device of Fig. 2b.

Journal of Bionic Engineering (2011) Vol.8 No.4

4.2 Effect of central plug diameter In order to determine how the diameter of the plug may affect the pressure and velocity profiles, three cases were simulated with different diameters of the central plug (d = 1 mm, 2 mm, and 3 mm) and various inlet pressure/velocity combinations for the optimized NPC design geometry. The results of flow rate and maximum velocity at the outlet are summarized in Table 4. It can be seen from Table 4 that with increasing central plug diameter, both the flow rate and the maximum velocity decrease and the decrease is about 13% when the diameter is increased about 3 times from the optimized design. The results of velocity contours for optimized design with 1 mm and 3 mm diameters of central plug are shown in Fig. 8. It can be seen from Fig. 8 that more fluid freely diffuses through the plug with higher diameter. The velocity in the central plug is approximately 16% higher with higher diameter.

of the micro-device without the central plug in comparison to the optimized design. With increasing inlet velocity, the pressure difference between the optimized design and design without the central plug (Fig. 2), decreases (about 66% when the velocity is increased 4 times) and then increases afterwards. The results of pressure difference and the flow rate are shown in Fig. 10b. For both the optimized design as well as the design without the central plug, the flow rate increases with increasing pressure difference. There is no significant difference between the flow rate and pressure drop for both the cases. The results presented in Figs. 9 and 10 indicate that the NPC geometry design greatly affects the fluid transport from inlet to outlet. 0.12 Flow rate(cm3·sí1)

460

Table 4 Comparison of flow rate and maximum velocities obtained for various central plug diameters for an optimized design Inner diameter (mm)

Flow rate (mm3·sí1)

Maximum velocity (mm·sí1)

1

0.10377

13.212

2

0.09906

12.613

3

0.08635

10.994

0.06 0.04 0.02 W/o CP

Orignial

Optimized CP

Fig. 9 Comparison of flow rates among various design configurations of the NPC geometries. Velocity vs. Pressure differences Optimized Without plug

2

4

6 8 Velocity (mm·sí1)

10

12

(a) Variation of pressure difference between inlet and outlet versus with inlet velocity for optimized design and original design without central plug

Flow rate (mm3·sí1)

Fig. 9 shows the comparison of flow rates for various NPC geometric configurations. It can be seen from Fig. 9 that the optimized design with central plug increases the flow rate by 12.3% whereas the non-optimized design without the central plug geometry reduces the flow rate by 9%. In order to see how the NPC designs effects the fluidic transport, the pressure difference between inlet and outlet is used as a measure for the performance. The results of pressure difference with increasing input velocity are shown in Fig. 10a. For both cases, the pressure drop increases with increasing velocity, however, the pressure drop is higher in the case

+12.3%

í9%

0.08

0

180 160 140 120 100 80 60 40 20 0

Fig. 8 Effect of increasing the central plug diameter on the flow rate for the optimized design.

0.10

Flow rate

(b) Variation of flow rate versus pressure difference between inlet and outlet for optimized design and original design without central plug

Fig. 10 Comparison of flow rates and pressure differences between optimized design and original design without central plug of the NPC geometries.

Lee et al.: Drug Transport Microdevice Mimicking an Idealized Nanoscale Bio-molecular Motor

4.3 Comparison between micro-scale and nano-scale models In order to illustrate the differences between micro-scaled and nano-scaled models for drug particle transport through the device, a nano-scaled model of the device was built and analyzed using ANSYS-Fluent software. Particles of different sizes (10 μm and 100 μm for the micro-device, and, 10 nm and 100 nm for the nano-device) were injected at a velocity of 0.01 μm·sí1 at the inlet in order to see how the drug particles transport through the device either micro-scale or nano-scale. The velocity of 0.01 μm·sí1 used in the analysis is based on the observation and findings on single-molecule trajectories of Imp ȕ1 through the NPC[21]. The results of particle trajectories between the micro-scale model and the nano-scale model are presented in Figs. 11 and 12 for both the designs (with and without the central plug). It can be seen from Fig. 11 that in the case of design device without the central plug, the trends of particle trajectories between the micro-scale model and the nano-scale model are similar. However, the nano-scale model resulted in close particle trajectories. Interestingly, in comparing the results of particle trajectories between the 2.83 eí01 2.68 eí01 2.54 eí01 2.40 eí01 2.26 eí01 2.12 eí01 1.98 eí01 1.84 eí01 1.70 eí01 1.55 eí01 1.41 eí01 1.27 eí01 1.13 eí01 9.89 eí02 8.48 eí02 7.06 eí02 5.65 eí02 4.24 eí02 2.83 eí02 1.41 eí02 0.00 e+00

461

micro-scale and nano-scale models for the design with the central-plug, the micro-scale model resulted in more uniform particle trajectories. The results of particles velocity through the nano-scale device for both the designs (without and with central plug) presented in Fig. 13 shows that the particle velocity distribution in and around the central plug is different. The majority of particles flow occurred around the central plug while a small amount of passive flow was generated through the central plug. The results of velocity ratio between outlet and inlet for both the micro-scale and nano-scale model designs are presented in Table 5. In general, the trends of particle trajectories, diffusivity (shown in Table 5 and calculated based on a typical Rhodamine B molecule) and velocity ratio are similar between the micro-scale and nano-scale models. This suggests that the results obtained for micro-scale model are equally valid for the nano-scale model as well. Overall, the results presented in Table 5 along with Figs. 11–13 which discuss the particle trajectories between micro-scale and nano-scale models demonstrate the trends of particle/molecule transport for the cases of with and without central plug in the device design.

Microscale - 10 ȝm particles

Fig. 11 Comparison of particle trajectories between micro-scale (left) and nano-scale (right) models for the optimized design without the central plug. 2.60 eí06 2.47 eí06 2.34 eí06 2.21 eí06 2.08 eí06 1.95 eí06 1.82 eí01 1.69 eí01 1.56 eí01 1.43 eí01 1.30 eí01 1.17 eí01 1.04 eí01 9.11 eí01 7.81 eí02 6.51 eí02 5.21 eí02 3.91 eí02 2.60 eí02 1.30 eí02 0.00 e+00

Nanoscale - 10 nm particles

Fig. 12 Comparison of particle trajectories between micro-scale (left) and nano-scale (right) models for the optimized design with central plug.

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develop micro-devices for drug transport that can mimic the transport characteristics of NPC for engineering applications.

Velocity magnitude (10í6m·sí1)

3.00 2.50

5 Conclusion

2.00 1.50 Without central plug

1.00 5.00

0.00 í2.5 í2.0 í1.5 í1.0 í0.5 0 0.5 1.0 2.0 1.5 2.5 Position (10í4mm)

Velocity magnitude (10í6m·sí1)

3.00 2.50 2.00 1.50 With central plug

1.00 5.00

0.00 í2.5 í2.0 í1.5 í1.0 í0.5 0 0.5 1.0 2.0 1.5 2.5 Position (10í4mm)

Fig. 13 Comparison of particle velocity distributions along the length of the nano-scale device for both designs (without and with central plug). Table 5 Comparison of raynolds number and diffusivity between micro and nano-scale models Model

Re = ȡVD / μ

Diffusivity (kmol·sí1)

Micro-scale

200

5.2452×10í12

§ 2.9

5.2452 ×10í2

§ 2.6

Nano-scale

í8

2.0 × 10

Velocity ratio (Vout/Vin)

Further, as our objective in this study is to evaluate the role played by the central plug in drug transport in the micro-device design, the results presented in Figs. 8–13 illustrate the importance of the central plug in drug delivery transport mechanisms. This finding is consistent with the observations and findings outlined by Terry et al.[22], and recognize the importance of nucleocytoplasmic transport occurring at multiple levels involving NPC transport channel, the transport receptors, and the transport cargo itself. The hierarchical role of the NPC in nucleocytoplasmic transport mechanisms, and the need for multidisciplinary approaches, specifically models that can accommodate documented active and diffusive transport capacities of the NPCs were also emphasized by Terry et al.[22]. The present study is an attempt to

A micro-device was developed in this study mimicking the coarse-grained representation of the NPC that controls the transport of all cellular material between the cytoplasm and the nucleus. The micro-device design geometry was developed and optimized through computational fluid dynamics methods. The role of the central plug in the micro-device in active fluidic transport and passive transport (without central plug) was investigated. Results of flow rate and velocity profiles obtained from the models indicate that the central plug plays a major role in drug/fluidic transport in the micro-device. The results of this investigation show that fluidic transport and flow passage are important factors in the design of NPC based nano- and micro-devices for drug delivery. The optimized design developed in this study will be used to develop a prototype to further investigate fluidic transport in the future.

Acknowledgments The authors thank the US National Science Foundation for sponsoring the research reported in this study through a grant ECCS-1058067.

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