Hill Equation Research Articles

A Novel Approach to Analyzing Binding Data from Na+ Driven Transporters: Beyond Non-Integer Hill Coefficients

Membrane transport proteins play a central role in many biological processes by mediating the transport of many different solutes, such as ions, nutrients, and neurotransmitters across biological membranes. Numerous pathological conditions result from impairments in the function of molecules of this class.Many prokaryotic and eukaryotic Na+-driven transporters couple the movement of one or more Na+ ions down their electrochemical gradient to the active transport of a variety of solutes. When more than one Na+ is involved, Na+ binding data are usually analyzed using the Hill equation with a non-integer exponent n. The results of this analysis are an overall Kd-like constant and n, a measure of cooperativity. This information is usually insufficient to provide the basis for mechanistic models. In the case of transport utilizing two Na+ ions n < 2 indicates that molecules with only one of the two sites occupied is present at low saturation. Here we propose a new way of analyzing Na+ binding data for the case of two Na+ ions that, by taking into account binding to individual sites, yields bounded ranges of pairs of possible values for the Na+ affinities of the individual sites. These values provide far more information than can be obtained by using the Hill equation with a non-integer coefficient.To illustrate the advantages of the new analysis we used experimental data on Na+ binding to the Na+/I- symporter (NIS)—the key protein that mediates the first step for the biosynthesis of the thyroid hormones—obtained by scintillation proximity assay (SPA). The same formalism can be used to analyze data of other proteins that bind the same substrate at two binding sites.

Long-Term Modulation of Ion Channels by Aldosterone in Adult Rat Atrial Myocytes

In recent years, both aldosterone and mineralocorticoid receptor (MR) have drawn attention as important factors that promote structural remodeling of the atrium. Here, we investigated effects of chronic aldosterone treatment on both intracellular Ca2+ and ion channels. Atrial myocytes were cultured in either the absence or presence of aldosterone, and then the activity of ion channels was studied, under whole-cell patch-clamp conditions. Aldosterone increased both the cell membrane capacitance (Cm) and the maximal conductance of ion channels that give rise to IKsus, INa, ICaT, and ICaL (30-100%). Except for inactivation curves of INa, which were shifted by −10 mV, aldosterone produced no major alterations in the biophysical properties of the channels. Interestingly, at resting membrane potentials the increase in INa was cancelled by a greater fraction of inactivation. The onset and recovery of the changes -in ICaT, ICaL, and Cm- were also assessed. In general, they required 2d to be noticeable, reached their maximal value in 6d, and returned to basal values after 1-3d of aldosterone removal. The effects on both Cm and ICaL were further studied -to explore both a potential dose-response relationship and a possible implication of the MR. In fact, co-incubating with 10 µM of spironolactone (an MR antagonist) abolished both effects. Furthermore, their corresponding magnitudes fitted well with the Hill equation, being the EC50 values for Cm and ICaL 20 and 130 (nM), respectively. Interestingly, aldosterone did not alter expression levels of Cav1.2, suggesting that the action on ICaL arises from a stimulus in open probability. The hormone also produced a 40% increase in the amplitude of Ca2+ transients along with a higher proportion of arrhythmic cells (2.5-fold increase). These results contribute to understanding the role of the MR and aldosterone in atrial electrophysiology.

Chromosome Reorganization by the Highly Cooperative Dps Protein

All living creatures organize their genome into a dynamic three-dimensional nucleoprotein complex. This organization has a vital importance for cellular processes such as gene expression, DNA replication, and DNA repair. In order to understand the nature of chromosome architecture we need to gain insight into the biophysical parameters that drive compaction. However, critical details of this process remain unknown. In this study we investigate a DNA-binding enzyme from E. coli that rearranges the bacterial genome and protects the chromosome against cellular stresses ranging from starvation to antibiotic exposure. The enzyme responsible for this transformation is called Dps (DNA-binding protein from starved cells). The interaction between Dps and DNA results in the rapid formation of a tightly packed three-dimensional crystal lattice termed a biocrystal. Although static structures of biocrystals have been documented, little is known about how the structures form. We have developed a fluorescent assay for tracking the physical interaction between individual DNA molecules and Dps under TIRF microscopy. We found that Dps induces collapse extremely rapidly after a slow nucleation event. In addition, we have used magnetic tweezers to obtain biophysical parameters of the conformational transition of stretched DNA initiated by Dps protein. We found that cooperative Dps binding to DNA cannot be fitted with Hill equation but displays the qualitative features of an Ising system. Consequently we have developed a mean field model that captures important features of the collapse. These in vitro experiments provide the crucial details about the highly cooperative mechanisms of DNA-Dps biocrystal formation.

On the Robust Stability of the Hill Equation with a Delay Term: A Frequency-Domain Approach

The paper considers the problem of robust stability of the Hill equation with a damping and a time-delay term. Analytical results are presented in terms of the coefficients of the equation. The derivation is based on the frequency-domain approach, specifically, the geometric interpretation of the Yakubovich stability criterion for linear time-periodic systems. There are two approaches to this interpretation: One involves analysis of the so-called Lipatov plots and the other is based directly on the Nyquist hodograph.

Muscle Shortening Velocity is Modulated by Alternative Myosin Converters

The myosin converter has been proposed to be critical for setting muscle shortening velocity by influencing the amplification of small conformational changes at the ATPase site into the larger lever arm swing. To test this hypothesis, we exploited the natural variation of the converter region found in Drosophila muscle types. In Drosophila, a single myosin heavy chain gene is alternatively spliced to generate myosin isoforms. The converter region is encoded by five alternative versions of exon 11 that are expressed in different muscle types. Through genetic modification we forced the expression of the alternative versions in the Drosophila jump muscle and found that two of the three versions tested to date, 11a (native to indirect flight muscles) and 11e (embryonic muscles), caused faster maximum shortening velocities of skinned jump muscle fibers relative to the native control 11c. Increases in velocity were primarily responsible for 70.9% and 79.0% higher power outputs for 11a and 11e fibers, respectively, because maximum force generation was not significantly different from control fibers. The fibers expressing 11a and 11e also exhibited a straighter force-velocity curve, as shown by increased Hill equation parameters a (95.4% and 84.5% higher, respectively) and b (72.6% and 65.5% higher, respectively) relative to control fibers, indicating that the converter modulates load dependent cross-bridge kinetics. The higher power and shortening velocity of 11a and 11e fibers enabled increased jumping ability, 76.0% and 48.8% farther, respectively. Converter homology models, using the new Drosophila myosin S1 crystal structure (pdb 4QBD), revealed minimal tertiary structure differences suggesting residue specific interactions with the relay, N-terminus, or essential light chain are critical. Overall, our data support the hypothesis that the converter helps set muscle shortening velocity under loaded and unloaded conditions and thus contributes to muscle fiber type diversity.

In vitro Reconstitution of Skeletal Muscle Contraction using Native Thin Filaments

Intracellular Ca2+ concentration regulate the muscle contraction by changing the conformation of Tropomyosin molecule over Actin filament. In this work we have purified native thin filaments (NTF) from Chicken Pectoralis muscle and reconstituted in in vitromotility assay (IVMA) with skeletal muscle myosin II. NTF prepared were 1.67±0.14 µm in length and showed good regulation under different Ca2+ concentrations. At pCa 4, more than 90% of NTF were continuously sliding and when solution changed to pCa 9 all the filaments were rigidly stuck to surface. Sliding velocity of NTF was VF=5.63±1.21 µm/s at 30 oC and 17% higher compared to unregulated actin filaments. Fraction of sliding filament showed a direct correlation with pCa of buffer and pCa50= 6.69 was obtained using Hills equation fit. However the sliding filament velocity was not affected under different pCa concentration. Under varying substrate [MgATP] conditions in IVMA we found k+ATP is 4.3± 0.56 x 106 M−1s−1 and k-ADP is 404.33±2.7 s−1 at 30 oC. These kinetic parameters are comparable to values obtained using unregulated actin filaments, suggesting the detachment kinetics of myosin II is not affected due to Ca2+ regulation. We studied the effect of myosin density on thin filaments activation in IVMA. Even with fully activated thin filament (pCa 4.0), decrease in myosin density leads to increase in fraction of stuck filaments. Decrease in fraction of motile filaments with myosin density in IVMA suggests there is minimum number of myosin head binding required to completely activate the NTF.Supported by Department of Science and Technology, Government of India.

Computational tools for fitting the Hill equation to dose–response curves

Computational tools for fitting the Hill equation to dose–response curves

Contribution of CYP3A isoforms to dealkylation of PDE5 inhibitors: a comparison between sildenafil N-demethylation and tadalafil demethylenation.

The aim of this study was to characterize the kinetics of metabolite formation of the phosphodiesterase type-5 (PDE5) inhibitors sildenafil and tadalafil by CYP3A4, CYP3A5, and CYP3A7 isoforms. The formations of N-desmethyl sildenafil and desmethylene tadalafil were examined using CYP3A supersomes co-expressing human P450 oxidoreductase and cytochrome b5. Both sildenafil N-demethylation and tadalafil demethylenation were catalyzed by CYP3A4, CYP3A5, and to a lesser extent by CYP3A7. The kinetics of desalkyl metabolite formation of the two drugs were well fitted to the Hill equation; however, the Hill coefficients (n) suggested CYP3A-mediated negative cooperativity. Next, we analyzed the kinetics with a two binding sites model assuming two reaction steps: reaction 1 with high-affinity and low-capacity metabolism and reaction 2 with low-affinity and high-capacity metabolism. The kinetics of desalkyl metabolite formation were also fitted to the two binding sites model. The intrinsic clearance (CLint) values of reactions 1 and 2 for sildenafil N-demethylation were 0.733 and 0.033 µL/min/pmol P450 for CYP3A4, 0.788 and 0.019 µL/min/pmol P450 for CYP3A5, and 0.079 and 0.004 µL/min/pmol P450 for CYP3A7, respectively. The CLint values of reactions 1 and 2 for tadalafil demethylenation were 0.187 and 0.014 µL/min/pmol P450 for CYP3A4, 0.050 and <0.001 µL/min/pmol P450 for CYP3A5, and 0.004 and <0.001 µL/min/pmol P450 for CYP3A7, respectively. These results may provide the basis not only for understanding the metabolic properties of the two PDE5 inhibitors, but also for one possible explanation of the mechanisms of CYP3A-mediated negative cooperativity.

Effect of potential on the early stages of nucleation and properties of the electrochemically synthesized ZnO nanorods

The zinc oxide nanostructured were synthesized on Indium doped tin oxide substrate using cathodic reduction of H2O2 and ZnCl2 from chloride aqueous electrolyte under differents applied potentials. The effects of this potential on the nucleation of ZnO seeds were investigated by performing transient current measurements and using model based on Scharifker–Hills equations. The results suggest that the nucleation mechanism of ZnO is progressive with a three-dimensional growth of the hemispherical nuclei. The XRD patterns show that the ZnO crystallizes in a hexagonal Würtzite-type structure with a phase preferentially orientated along c-axis. The structural analysis evidences a strong relationship between the directional growth along (002) crystallographic plane and the applied potentials. ZnO crystalizes in hexagonal nanorods (NRs) with diameters in the ranges of 160–250nm. Photoelectrochemical study indicates that the obtained films have n-type semiconducting behaviour, and generate high photocurrents up to 40μA/cm2 at −1.0 and −1.1V. The transmittance spectra indicate that the films exhibit a good optical quality with low defects density with an averaged band gap of ~3.31eV.

MODELING OF THE SPONTANEOUS POLYPROPYLENE SORBENTS IMBIBITION WITH EMULSIONS

Oil penetration into aqueous media results in various chemical and physical processes including formation of “oil-in-water” emulsions, that have devastating effects on the environment. This research was conducted to evaluate the kinetics of permeation of “oil-in-water” emulsions into synthetic porous material. It gives the ability to develop more effective sorbents, decreasing risks of negative consequences and environmental contamination. The following publication focuses on assessment of the possibility to describe and predict a process of porous medium imbibition with emulsions by means of the Hill equation. Its adequacy is compared with the classical Lucas–Washburn equation and its modified version. The advanced approach is to substitute the commonly used parameter of the height of an imbibed emulsion (h(t)) by the parameter mh(t), which is the mass of an imbibed emulsion in the case of the classical and modified Washburn equations. According to the obtained results, the Hill equation provides the most appropriate and precise description of the porous material imbibition with “oil-in-water” emulsions and oil in comparison with others, and shows the highest correlation values (Rav = 0.995±0.001) and the lowest normalized root mean square error (NRMSEav 1.95±0.138).

Effects of guided random sampling of TCCs on blood flow values in CT perfusion studies of lung tumors.

Tissue perfusion is commonly used to evaluate lung tumor lesions through dynamic contrast-enhanced computed tomography (DCE-CT). The aim of this study was to improve the reliability of the blood flow (BF) maps by means of a guided sampling of the tissue time-concentration curves (TCCs). Fourteen selected CT perfusion (CTp) examinations from different patients with lung lesions were considered, according to different degrees of motion compensation. For each examination, two regions of interest (ROIs) referring to the target lesion and the arterial input were manually segmented. To obtain the perfusion parameters, we computed the maximum slope of the Hill equation, describing the pharmacokinetics of the contrast agent, and the TCC was fitted for each voxel. A guided iterative approach based on the Random Sample Consensus method was used to detect and exclude samples arising from motion artifacts through the assessment of the confidence level of each single temporal sample of the TCC compared to the model. Removing these samples permits to refine the model fitting, thus exploiting more reliable data. Goodness-of-fit measures of the fitted TCCs to the original data (eg, root mean square error and correlation distance) were used to assess the reliability of the BF values, so as to preserve the functional structure of the resulting perfusion map. We devised a quantitative index, the local coefficient of variation (lCV), to measure the spatial coherence of perfusion maps, from local to regional and global resolution. The effectiveness of the algorithm was tested under three different degrees of motion yielded by as many alignment procedures. At pixel level, the proposed approach improved the reliability of BF values, quantitatively assessed through the correlation index. At ROI level, a comparative analysis emphasized how our approach "replaced" the noisy pixels, providing smoother parametric maps while preserving the main functional structure. Moreover, the implemented algorithm provides a more meaningful effect in correspondence of a higher motion degree. This was confirmed both quantitatively, using the lCV, and qualitatively, through visual inspection by expert radiologists. Perfusion maps achieved with the proposed approach can now be used as a valid tool supporting radiologists in DCE-CTp studies. This represents a step forward to clinical utilization of these studies for staging, prognosis, and monitoring values of therapeutic regimens.

Modeling and Simulation of a Parametrically Resonant Micromirror With Duty-Cycled Excitation.

High frequency large scanning angle electrostatically actuated microelectromechanical systems (MEMS) mirrors are used in a variety of applications involving fast optical scanning. A 1-D parametrically resonant torsional micromirror for use in biomedical imaging is analyzed here with respect to operation by duty-cycled square waves. Duty-cycled square wave excitation can have significant advantages for practical mirror regulation and/or control. The mirror's nonlinear dynamics under such excitation is analyzed in a Hill's equation form. This form is used to predict stability regions (the voltage-frequency relationship) of parametric resonance behavior over large scanning angles using iterative approximations for nonlinear capacitance behavior of the mirror. Numerical simulations are also performed to obtain the mirror's frequency response over several voltages for various duty cycles. Frequency sweeps, stability results, and duty cycle trends from both analytical and simulation methods are compared with experimental results. Both analytical models and simulations show good agreement with experimental results over the range of duty cycled excitations tested. This paper discusses the implications of changing amplitude and phase with duty cycle for robust open-loop operation and future closed-loop operating strategies.

ASSA14-03-44 CaMKII inhibition influences IKATPchannel's function through SR Ca2+feedback regulation

Objectives We thought to investigate whether SR Ca 2+ content can regulate I KATP channels function using the cross-breeding mice of CaMKII inhibition (Inh) and phospholamban (PLN) gene knockout mice (PLN -/- ). Methods Mice were sent from Dr. Anderson’s Lab in USA to Animal Centre of Tonji Medical College in China. I KATP from ventricular myocytes was recorded using inside-out patch-clamp configuration of the patch-clamp technique. Steady-state dependence of membrane current on [ATP] was obtained by calculating the relative current indexed to zero ATP ( I rel). The data were fitted using the Hill equation: I rel = 100/(1+([ATP]/K 1/2 ) H , where I rel is the relative current, K 1/2 is the concentration causing half-maximum blockade, and H is the Hill coefficient. Single channel I KATP recordings were obtained from inside out patches using fire-polished pipettes (resistance ~9–10 Ohm). The threshold for judging the open state of I KATP channels was set at half of the single channel amplitude. The nP o, where n represents the number of channels in the patch and P o the probability of each channel to open, was assessed using Clampfit-10 software. Using qRTPCR, we measured mRNA expression levels of genes encoding SUR1 (ABCC8), SUR2 (ABCC9), Kir6.1 (KCNJ8) and Kir6.2 (KCNJ11), the subunits reconstituting functional I KATP channels. Results Consistent with our previous study, I KATP was significantly increased, while the negative regulatory dose-dependence of ATP was unchanged in Inh mice compared to wild type (WT) and CaMKII inhibition control (Con) mice, suggesting that CaMKII inhibition has feedback regulation of the functional I KATP channels. Furthermore, I KATP channel opening probability was equivalent in cell membrane patches from INH, WT and Con ventricular myocytes. We found that CaMKII inhibition does not affect mRNA levels for I KATP encoding genes, indicating that CaMKII mediated increases in I KATP are independent of augmented transcription of I KATP channel subunit genes. Breeding Inh mice with PLN -/- mice returned I KATP and I KATP channel opening probability to control levels and equalised the APD and QT intervals in Inh mice to Con and WT levels. Dialysis of CaMKII inhibitory peptide into WT cells did not result in increased IKATP, suggesting that enhanced I KATP in Inh mice is an adaptive response to chronic CaMKII inhibition rather than an acute effect of reduced CaMKII activity. Conclusions These findings provide novel evidence that CaMKII links intracellular Ca 2+ to cardiac I KATP and suggest that PLN is a critical CaMKII target for feedback regulation of I KATP in ventricular myocytes.

ASSA14-02-13 CaMKII inhibition influences IKATPchannel's function through SR Ca2+feedback regulation

<h3>Objectives</h3> We thought to investigate whether SR Ca²⁺ content can regulate I_KATP channels function using the cross-breeding mice of CaMKII inhibition (Inh) and phospholamban (PLN) gene knockout mice (PLN^-/-). <h3>Methods</h3> Mice were sent from Dr. Anderson’s Lab in USA to Animal Centre of Tonji Medical College in China. I_KATP from ventricular myocytes was recorded using inside-out patch-clamp configuration of the patch-clamp technique. Steady-state dependence of membrane current on [ATP] was obtained by calculating the relative current indexed to zero ATP (<i>I</i>rel). The data were fitted using the Hill equation: <i>I</i>rel = 100/(1 + ([ATP]/K_1/2)^H, where <i>I</i>rel is the relative current, K_1/2 is the concentration causing half-maximum blockade, and H is the Hill coefficient. Single channel I_KATP recordings were obtained from inside out patches using fire-polished pipettes (resistance ∼9–10 Ohm). The threshold for judging the open state of I_KATP channels was set at half of the single channel amplitude. The <i>nP</i>o, where <i>n </i>represents the number of channels in the patch and <i>P</i>o the probability of each channel to open, was assessed using Clampfit-10 software. Using qRTPCR, we measured mRNA expression levels of genes encoding SUR1 (ABCC8), SUR2 (ABCC9), Kir6.1 (KCNJ8) and Kir6.2 (KCNJ11), the subunits reconstituting functional I_KATP channels. <h3>Results</h3> Consistent with our previous study, I_KATP was significantly increased, while the negative regulatory dose-dependence of ATP was unchanged in Inh mice compared to wild type (WT) and CaMKII inhibition control (Con) mice, suggesting that CaMKII inhibition has feedback regulation of the functional I_KATP channels. Furthermore, I_KATP channel opening probability was equivalent in cell membrane patches from Inh, WT and Con ventricular myocytes. We found that CaMKII inhibition does not affect mRNA levels for I_KATP encoding genes, indicating that CaMKII mediated increases in I_KATP are independent of augmented transcription of I_KATP channel subunit genes. Breeding Inh mice with PLN^-/- mice returned I_KATP and I_KATP channel opening probability to control levels and equalised the APD and QT intervals in Inh mice to Con and WT levels. Dialysis of CaMKII inhibitory peptide into WT cells did not result in increased IKATP, suggesting that enhanced I_KATP in Inh mice is an adaptive response to chronic CaMKII inhibition rather than an acute effect of reduced CaMKII activity. <h3>Conclusions</h3> These findings provide novel evidence that CaMKII links intracellular Ca²⁺ to cardiac I_KATP and suggest that PLN is a critical CaMKII target for feedback regulation of I_KATP in ventricular myocytes.

A qualitative explanation of the origin of torsional instability in suspension bridges

We consider a mathematical model for the study of the dynamical behavior of suspension bridges. We show that internal resonances, which depend on the bridge structure only, are the origin of torsional instability. We obtain both theoretical and numerical estimates of the thresholds of instability. Our method is based on a finite dimensional projection of the phase space which reduces the stability analysis of the model to the stability of suitable Hill equations. This gives an answer to a long-standing question about the origin of torsional instability in suspension bridges.

Moment Lyapunov exponents of the Parametrical Hill's equation under the excitation of two correlated wideband noises

The Lyapunov exponent and moment Lyapunov exponents of Hill\' s equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged Ito differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.

Quantitative high-throughput screening data analysis: challenges and recent advances

In vitro HTS holds much potential to advance drug discovery and provide cell-based alternatives for toxicity testing. In quantitative HTS, concentration-response data can be generated simultaneously for thousands of different compounds and mixtures. However, nonlinear modeling in these multiple-concentration assays presents important statistical challenges that are not problematic for linear models. The uncertainty of parameter estimates obtained from the widely used Hill equation model can be extremely large when using standard designs. Failure to properly consider standard errors of these parameter estimates would greatly hinder chemical genomics and toxicity testing efforts. In this light, optimal study designs should be developed to improve nonlinear parameter estimation; or alternative approaches with reliable performance characteristics should be used to describe concentration-response profiles.

UDP-glucuronosyltransferase (UGT) 1A1 mainly contributes to the glucuronidation of trovafloxacin

Identification of drug-metabolizing enzyme(s) responsible for the metabolism of drugs is an important step to understand not only interindividual variability in pharmacokinetics but also molecular mechanisms of metabolite-related toxicity. While it was reported that the major metabolic pathway of trovafloxacin, which is an antibiotic, was glucuronidation, the UDP-glucuronosyltransferase (UGT) isoform(s) responsible for the trovafloxacin glucuronidation has not been identified yet. In the present study, among the functional human UGT members, UGT1A1, UGT1A3, and UGT1A9 exhibited higher trovafloxacin acyl-glucuronidation activities. While other UGT members such as UGT1A8, UGT2B7, and UGT2B15 showed glucuronidation activity toward trovafloxacin, the metabolic velocity was extremely low. In human liver microsomes, trovafloxacin acyl-glucuronidation followed the Hill equation with S50 value of 95 μM, Vmax value of 243 pmol/min per mg, and a Hill coefficient of 2.0, while the UGT1A1-expressing system displayed Michaelis–Menten kinetics with a substrate inhibition, with Km value of 759 μM and Vmax value of 1160 pmol/min per mg. In human liver microsomes prepared from poor metabolizers (UGT1A1*28/*28), significantly reduced trovafloxacin acyl-glucuronide formation activity was observed, indicating that UGT1A1 mainly, while other UGT members such as UGT1A3 and UGT1A9 partially, contributes to the glucuronidation of trovafloxacin.

An exact solution of the Currie–Hill equations in 1+1 dimensional Minkowski space

We present an exact two-particle solution of the Currie–Hill equations of Predictive Relativistic Mechanics in 1 + 1 dimensional Minkowski space. The instantaneous accelerations are given in terms of elementary functions depending on the relative particle position and velocities. The general solution of the equations of motion is given and by studying the global phase space of this system it is shown that this is a subspace of the full kinematic phase space. • Explicit solution of the 1 + 1 dimensional Currie–Hill equations. • Explicit solution of the equations of motion. • Explicit construction of the Poincaré generators. • Detailed description of the global structure of the phase space.

Stable heterogeneity for the production of diffusible factors in cell populations.

The production of diffusible molecules that promote survival and growth is common in bacterial and eukaryotic cell populations, and can be considered a form of cooperation between cells. While evolutionary game theory shows that producers and non-producers can coexist in well-mixed populations, there is no consensus on the possibility of a stable polymorphism in spatially structured populations where the effect of the diffusible molecule extends beyond one-step neighbours. I study the dynamics of biological public goods using an evolutionary game on a lattice, taking into account two assumptions that have not been considered simultaneously in existing models: that the benefit of the diffusible molecule is a non-linear function of its concentration, and that the molecule diffuses according to a decreasing gradient. Stable coexistence of producers and non-producers is observed when the benefit of the molecule is a sigmoid function of its concentration, while strictly diminishing returns lead to coexistence only for very specific parameters and linear benefits never lead to coexistence. The shape of the diffusion gradient is largely irrelevant and can be approximated by a step function. Since the effect of a biological molecule is generally a sigmoid function of its concentration (as described by the Hill equation), linear benefits or strictly diminishing returns are not an appropriate approximations for the study of biological public goods. A stable polymorphism of producers and non-producers is in line with the predictions of evolutionary game theory and likely to be common in cell populations.