An In Silico Transwell Device for the Study of Drug Transport and Drug–Drug Interactions

Abstract

Purpose

Validate and exemplify a discrete, componentized, in silico, transwell device (ISTD) capable of mimicking the in vitro passive transport properties of compounds through cell monolayers. Verify its use for studying drug–drug interactions.

Methods

We used the synthetic modeling method. Specialized software components represented spatial and functional features including cell components, semi-porous tight junctions, and metabolizing enzymes. Mobile components represented drugs. Experiments were conducted and analyzed as done in vitro.

Results

Verification experiments provided data analogous to those in the literature. ISTD parameters were tuned to simulate and match in vitro urea transport data; the objects representing tight junction (effective radius of 6.66 Å) occupied 0.066% of the surface area. That ISTD was then tuned to simulate pH-dependent, in vitro alfentanil transport properties. The resulting ISTD predicted the passive transport properties of 14 additional compounds, individually and all together in one in silico experiment. The function of a two-site enzymatic component was cross-validated with a kinetic model and then experimentally validated against in vitro benzyloxyresorufin metabolism data. Those components were used to exemplify drug–drug interaction studies.

Conclusions

The ISTD is an example of a new class of simulation models capable of realistically representing complex drug transport and drug–drug interaction phenomena.

Appendix

Given the five expectations in the Introduction, we worked backwards and specified capabilities needed for their achievement, as done by Grant et al. (29), Tang et al. (30), and Hunt et al. (5) in other modeling contexts. It will require the design and validation of new types of in silico devices that can exhibit at least nine capabilities.

• 1. Turing test: In silico “transport” data are, to a domain expert (in a type of Turing test), experimentally indistinguishable from the referent wet-lab transport data; achieving this capability requires the in silico device to be suitable for experimentation.

• 2. Phenotype overlap: There is meaningful overlap between in silico and in vitro system phenotypes. Consequently, the events occurring during simulated transport, for example, accurately represent corresponding in vitro events at a level of detail (granularity) sufficient to mimic or predict specific phenotypic attributes. The concepts underlying designing analogues to achieve phenotype overlap (29,30) are similar to those on which Pattern-Oriented Modeling is based (31).

• 3. Mappings: Observables in silico are designed to be consistent with in vitro observables. Doing so enables clear mappings between in vitro and in silico components and mechanisms.

• 4. Transparency: Simulation details as they unfold need to be visualizable, measurable, and comparable where feasible to those of the in vitro system.

• 5. Flexibility: It must be relatively simple to increase or decrease detail in order to simulate an additional phenotypic attribute or change usage and assumptions, without requiring significant reengineering. Achieving this capability will enable cycles of scientific testing and device refinement, as new wet-lab data becomes available. Having flexible methods will lead to improved, more realistic and heuristic analogue systems.

• 6. Reusability: Device components can be designed to be autonomous and thus can be easily reconfigured to represent different cell types, experimental conditions, and compounds, alone or in combination. It must be easy to simulate and analyze outcomes of several different in silico experiments in a fraction of the time (and at a fraction of the cost) required to complete the wet-lab experiments.

• 7. Adaptability: In addition to flexibility and reusability, the components must be constructed so that they can be adapted to function as components in physiologically based, intestine and whole organism models.

• 8. Components articulate: Because the system attributes of interest may result from interactions of autonomous components, to study alternative mechanisms it must be easy to join, disconnect, and replace them without having to reengineer the in silico device or its components.

• 9. Local mechanisms: as in vitro, the behaviors that emerge during a simulation are the consequences of local mechanisms–local component interactions.

To enable these capabilities, the in silico device must be synthetic, as is the in vitro model (4,5), and its framework must use discrete interactions. Temporally and spatially discrete models offer theoretical and practical advantages because they allow, even facilitate achieving the preceding capabilities. Moreover, in the case where a continuous model is desirable, a discrete model can simulate or approximate it to arbitrary precision (25).

The equations to determine D (including D _a and D _m) are:

$$ D = \varpi \frac{{K_{B} T}} {{6\pi \eta r}},\quad {\text{where}}\quad r = \sqrt[3]{{\frac{{3V}} {{4\pi }}}}\quad {\text{and}}\quad V = \lambda . \cdot \operatorname{MW} . $$

The first equation is the Stokes–Einstein equation to calculate the diffusion coefficient for Brownian spheres. K _B is the Boltzmans constant, ϖ is a remedy coefficient (ϖ = 1 for the cell membranes and 1.67 for aqueous spaces). η is the viscosity at 37°C, η = 5e − 3 Pa s for membranes and 0.69e − 3 Pa s for aqueous spaces). r is the radius of the referent molecule assuming it is an ideal sphere and V is the molecular van der Waals volume. V is proportional to MW with λ = 1.6755 according to the data in (12).

The equations to determine D _eff in tight junctions are from (8) and (14):

$$ D_{{eff}} = \frac{{\varepsilon DF}} {\tau }{\left( {f_{0} + {\sum\limits_\kappa {f_{\kappa } {\left( {\frac{\kappa } {{1 - e^{{ - \kappa }} }}} \right)}} }} \right)},\;{\text{where}}\;F = {\left( {1 - \frac{r} {R}} \right)}^{2} {\left( {1 - 2.104\frac{r} {R} + 2.09{\left( {\frac{r} {R}} \right)}^{3} - 0.95{\left( {\frac{r} {R}} \right)}^{5} } \right)},\;{\text{and}}\;\kappa = \raise0.7ex\hbox{${ez{\left| {\Delta \psi } \right|}}$} \!\mathord{\left/ {\vphantom {{ez{\left| {\Delta \psi } \right|}} {KT}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${KT}$} $$

D _eff is the effective diffusion coefficient through the tight junctions. ɛ is the proportion of surface area covered by the tight junctions. D is the aqueous diffusion coefficient. F is the Renkin factor to describe the sieving effect of tight junctions on molecules of different sizes. τ is the tortuousity of the tight junction passage, which is estimated as 2.5–3.0 times the height of tight junctions. κ is the electrochemical energy. f ₀ is the fraction of unionized molecules and $f_{\kappa }$ is the fraction of charged molecules. R is the estimated average tight junction pore size. e is the unit charge of an electron. z is the valence of a molecule. Δψ is the electrical potential difference across the membrane. K is the thermal constant, and T is the Kelvin temperature converted from 37°C.

The mathematical equations derived from (8) to compute P _total are:

$$ \frac{{M_{\operatorname{R} } }} {{M_{{\operatorname{D} {\left( 0 \right)}}} }} = {\left( {\frac{{H_{\operatorname{R} } }} {{H_{\operatorname{R} } + H_{\operatorname{D} } }}} \right)}{\left( {1 - e^{{ - \alpha \beta T}} } \right)},\;where\;\alpha = \raise0.7ex\hbox{${P_{{\operatorname{tota} l}} }$} \!\mathord{\left/ {\vphantom {{P_{{\operatorname{tota} l}} } {H_{D} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${H_{D} }$},\;{\text{and}}\;\beta = \raise0.7ex\hbox{${{\left( {H_{\operatorname{R} } + H_{D} } \right)}}$} \!\mathord{\left/ {\vphantom {{{\left( {H_{\operatorname{R} } + H_{D} } \right)}} {H_{\operatorname{R} } }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${H_{\operatorname{R} } }$} $$

M _R is the mass in the acceptor space; M _D(0) is the initial mass in the donor space; H _R (2.6/4.71 cm) and H _D (1.5/4.71 cm) are the thicknesses of the receiver and donor spaces, respectively; T is time.

The analytical solution to the substrate initial reaction rate ν in the presence of inhibitors for the two-site enzyme is based on the kinetic model in Fig. 11, and is (17):

$$ v = \frac{{V_{{\max }} {\left( {\frac{1} {{K_{\operatorname{s} } }} + \frac{{\beta \prime {\left[ S \right]}}} {{\alpha \prime K_{i} K_{s} }}} \right)}}} {{\frac{1} {{{\left[ S \right]}}} + \frac{1} {{K_{s} }} + \frac{1} {{K_{i} }} + \frac{{{\left[ S \right]}}} {{\alpha \prime K_{\operatorname{i} } K_{s} }}}} $$

V _max is the maximal reaction rate of the substrate without inhibitors; K _S is the Michaelis–Menten constant of the substrate; K _i is the Michaelis–Menten constant of the inhibitor; α′ is the enhancing factor for the second site binding when one and only one site is bound with a molecule; $\beta \prime {\left( {0 \leqslant \beta \prime \leqslant 1} \right)}$ is the inhibition factor of generating metabolites. The parameters in Table III used the equation above. Dissociation constants for all species in Fig. 11 are defined by $K_{S} = {k_{{ - 1}} } \mathord{\left/ {\vphantom {{k_{{ - 1}} } {k_{1} }}} \right. \kern-\nulldelimiterspace} {k_{1} }$, $K_{i} = {k_{{ - 2}} } \mathord{\left/ {\vphantom {{k_{{ - 2}} } {k_{2} }}} \right. \kern-\nulldelimiterspace} {k_{2} }$, $\alpha \prime K_{i} = {k_{{ - 3}} } \mathord{\left/ {\vphantom {{k_{{ - 3}} } {k_{3} }}} \right. \kern-\nulldelimiterspace} {k_{3} }$ and $\alpha \prime K_{\operatorname{S} } = {k_{{ - 4}} } \mathord{\left/ {\vphantom {{k_{{ - 4}} } {k_{4} }}} \right. \kern-\nulldelimiterspace} {k_{4} }$.

Fig. 11

A kinetic model for substrate inhibition from (17). When S binds to the catalytic site of the enzyme (ES), the rate and V _max of product formation (ES –> M) are determined by k _p[ES] and k _p[E _total], respectively. When S binds to the inhibitory site (SE) that is non-productive, the rate of SES → M is determined by factor β (0 ≤ β ≤ 1). Dissociation constants for all species are defined by $K_{\operatorname{S} } = {k_{{ - 1}} } \mathord{\left/ {\vphantom {{k_{{ - 1}} } {k_{1} }}} \right. \kern-\nulldelimiterspace} {k_{1} }$, $K_{i} = {k_{{ - 2}} } \mathord{\left/ {\vphantom {{k_{{ - 2}} } {k_{2} }}} \right. \kern-\nulldelimiterspace} {k_{2} }$, $\alpha \prime K_{i} = {k_{{ - 3}} } \mathord{\left/ {\vphantom {{k_{{ - 3}} } {k_{3} }}} \right. \kern-\nulldelimiterspace} {k_{3} }$ and $\alpha \prime K_{\operatorname{S} } = {k_{{ - 4}} } \mathord{\left/ {\vphantom {{k_{{ - 4}} } {k_{4} }}} \right. \kern-\nulldelimiterspace} {k_{4} }$. All parameters are shown in Table III using the above equation.