Analog Electronic Circuits to Model Cooperativity in Hill Process

In the field of computational biology, electronic modeling of bio-cellular processes is in vogue for about a couple of decades. Fast, efficient and scalable electronic mimetics of recurrently found bio-chemical reactions are expected to provide better electronic circuit simulators that can also be used as bio-sensors or implantable bio-devices at cellular levels. This paper presents some possible electronic circuit equivalents to model dynamics of one such bio-chemical reaction commonly involved in many bio-cellular processes, specifically pathways in living cells, known as the Hill process. The distinguishing feature of this process is cooperativity which has been modeled in silicon substrate using a pair of transistors, one transistor driving current in the other the same way ligand binding to one receptor site controls the binding affinity of the other receptor sites. Two possible circuits have been proposed and compared to electronically model cooperativity of a Hill reaction. The main idea is to exploit the natural analogies found between structures and processes of a bio-cell and electronic transistor mechanics, to efficiently model fundamental bio-chemical reactions found recurring in bio-processes. These circuits can then be combined and rearranged quickly to form larger, more complex bio-networks, thus mitigating the intricacies involved in modeling of such systems.


INTRODUCTION
tudy of various life forms has always been a subject of colossal interest to the researchers. It has long been identified as a topic of biology, and biologists as well as bio-scientists have proficiently been able to gather enormous amount of data through observation and experimentation. Advances in biosciences have unleashed powerful approaches to determine various components and underlying biochemistry of living cells too. The biocellular machinery might appear simple in structure at a glance, but is enormously complex in its working and networking abilities [1]. The complexity and size of these bio-cellular networks have made their 1 Department of Computer and Information Systems Engineering, NED University of Engineering and Technology, Karachi, Sindh, Pakistan. Email: a mariaw@neduet.edu.pk (Corresponding Author), b mkhurrum@neduet.edu.pk 2 Centre for Research in Analog and VLSI Microsystems Design (CRAVE), Massey University, Auckland, New Zealand. Email: hasanmic@massey.ac.nz comprehension task arduous by theory and experimentation alone, while opening up new opportunities for mathematicians, computer scientists and even engineers [1][2], resulting in emergence of the interdisciplinary field called Computational Biology. Versatile modeling approaches have been employed in this domain, from mathematical and analytical models to computer-based simulations, to help understand these complicated and sophisticated bio-networks better [1,[3][4][5].
Although mathematical and computational modeling of bio-cellular entities and processes have interested researchers since many decades, electronic modeling of bio-cellular processes is an emerging research trend S 679 in biosciences as well as engineering. Computational models suffer from a potential requirement of big, power hungry machines to run simulations. Simulation of one whole-cell cycle of the smallest bacteria on earth, using an over simplified model, was run on a 128-node Linux cluster in 10 hours [6]. To overcome these issues an alternate direction in vogue nowadays is modeling of bio-cellular processes functionalities in specialized analog [2,[7][8], digital [9][10][11] or mixed IC electronic hardware [12][13]. The research work presented in this paper advocates analog transistor mimetics for bio-cellular processes as the bio-cellular processes and circuits are naturally more analog than digital; they are noisy and act in progressive way rather than stern off/on digital states [14][15]. Many logical similarities are found among parameters and processes of biological and electronic domains. Biological processes largely comprise of feedback processes and oscillators which are also common characteristics of many analog electronic domain devices. Digital electronic circuits employing merely two states, are unable to make use of complete informational capacity of an electronic transistor; which otherwise can work on a very wide range of values [16]. Thus transistor provides more versatile functionality when used in analog domain, fitting to biological circuits much more appropriately.
Although the bio-cellular processes are immensely complex in their interactions, a finite set of biochemical reaction underlie them. A modular approach is to efficiently model these fundamental bio-chemical reactions in analog electronic domain anticipating that this would make modeling of larger networks easy [2,17]. This paper presents an extension to the work published by the author in [17] in which two such processes namely receptor-ligand binding and Michaelis Menten reaction, were modeled using analog transistors in electronic domain. This work presents several approaches to model a third fundamental reaction called the Hill process with argumentative and simulation support. The same process has also been worked upon for equivalent electronic model in [2] but there are some key differences: One key difference is that [2] and its preceding work used MOS transistors operating in sub-threshold region or BiCMOS technology for electronic modeling, whereas electronic modeling present ed in this work uses MOS transistors in analog domain, operating in triode and saturation regions. The second main difference is the approach used to mimic the bio-cellular processes. In [2], the Hill coefficient is represented by amplifying the difference between the voltages representing the respective molecular concentrations, whereas here this constant is matched to the device parameter mechanics of a single MOS transistor, thus simplifying the overall model. A new set of analogies between the two domain has been developed, and the concept has been validated mathematically as well analytically through reasoning and simulations.
Some recent significant contributions in related domains include [18], in which a new solver called Arco for complex cytomorphic and neuromorphic computations has been presented, and most recently [19], which presents a digitally reconfigurable cytomorphic electronic chip to simulate a variety of bio-cellular reaction networks. Same research group has to its credit significant work on noise of such circuits too [20]. Expanding the research to implementation of biosensors, they have also devised a biosensing device for measurement and control of cellular molecules [21].
Mimicking the functionalities of bio-cellular networks in analog and mixed IC hardware can help in producing improved simulators for bio-cellular networks and processes [7][8] that can aid in better understanding of biological behaviors and drug testing. This kind of specialized hardware, due to size compactness and power efficiency, would also be easier to integrate with synthetic biological circuits [2,9,12]; thus can be used as bio-sensors, and as implantable bio-devices as well.

DETERMINISTIC MATHEMATICAL MODEL OF HILL PROCESS
In organic cells, signals are propagated from one part of the cell to other through cell signaling pathways. These pathways comprise of cascaded proteins which interact with each other through various bio-chemical reactions to amplify and transport signals from one point to another [22][23]. One such reaction, receptorligand binding occurs when a ligand protein binds to a receptor protein to form product, the receptor-ligand complex. This reaction involves receptor having only one site where a ligand molecule can bind. An alternate phenomenon is cooperative binding, where the receptor has more than one identical binding sites. The receptor reacting in cooperative binding molecule is a multi-meric (multi-site) protein, where binding of one molecule of ligand to a receptor site affects the affinity of other ligand molecules towards the remaining receptor sites, resulting in non-linear cooperative behavior. Hence for positive cooperativity, a small increase in concentration of ligand molecules rapidly increases reaction rate. This kind of cooperative binding is exhibited when oxygen binds to Hemoglobin protein in blood. Hemoglobin transports oxygen through blood and this cooperative binding behavior makes this oxygen shuttling all the way efficient [2]. Hill function quantifies this cooperative binding and is modeled by the reaction given in Equation (1). In this reaction a receptor having binding sites, binds to ligand molecules, representing each ligand molecule, forming a receptorligand complex . and respectively are the forward and reverse reaction rate constants.

R + nL
RLn (1) The steady state behavior as reported in [22] is given by Equation (2).
where R , L and RL represent the concentrations of receptor, ligand and receptor-ligand complex respectively. K is called dissociation constant and is defined as k /k . k and k have dimensions of concentration -1 sec -1 and sec -1 respectively; while K has units of concentration. [22] defines θ as the fraction of receptor sites bound at equilibrium, as shown in Equation (3).
Equation (3) produces graphs which are sigmoidal in shape. Here n is defined as the Hill's coefficient, and it regulates the steepness of the curve. From Equation (1) is expected to be the number of binding sites on the multimeric receptor, but experimental results show that this is the degree of interaction between the sites instead and never reaches the actual number of binding sites [22]. The more the cooperativity, the higher the value of ; hence steeper the curve. Monomeric receptor-ligand binding is a special case of Hill reaction with 1, showing no cooperativity. Fig. 1 depicts the graphical results as discussed. The results clearly suggest that increasing makes the sigmoidal curve steeper and more switch-like. When value of the Hill coefficient is 1, the plot becomes hyperbolic.

PROPOSED ANALOG ELECTRONIC MODELS FOR HILL PROCESS
This section proposes some possible MOS transistor equivalents to model Hill process. The equivalences and electronic circuit validations stated in [17] suggest that a single MOS transistor operating in analog mode can model a non-linear bio-cellular reaction like receptor-ligand binding or Michaelis Menten kinetics. The drain and gate voltages are analogous to ligand and receptor concentrations respectively. For Hill process model, another transistor has been cascaded, whose gate voltage is proportional to the drain current of the first transistor as shown in Fig. 2 [17], the drain current I " flowing through the short channel transistor M is given as Equation (4).
where E "./ is the critical electrical field at velocity saturation point and L is the length of the transistor. The gate drive voltage of transistor M is proportional to I " , hence I " is limited by I " and is given as Equation (5).
As can be seen from Equation (5), the cascaded arrangement of transistors M and M produces a current I " as a higher order equation, converting hyperbolic curve of mono-meric receptor binding to sigmoidal curve for a bi-meric receptor binding. Equation (5) is comparable to Equation (2) for n 2.
To represent variation in Hill coefficient n, two modeling approaches have been proposed. In the first approach the hill coefficient n is adjusted via resistor R 3 and the channel length L of the transistor M . The basic idea here is to reduce resistance to transistor drain current which is analogous to bio-chemical product formation; thus modeling increased affinity. For n = 1, when there is only one binding site, the drain current through transistor M shows the rate of final product formation. When there are more than one binding sites, the production rate is measured as drain current of transistor M , adjusting values of R 3 and L to model different values of . In the second approach, is modeled as channel width of the transistors. The proposed idea is again to reduce the resistance to drain current by widening the path of current flow. Increase in transistor channel width models increasing n. Fig. 3 shows the suggested electronic model where transistors M , M 4 and M 5 channel widths are in increasing order respectively. Hence drain currents of these transistors respectively represent the production rates for increasing values of n.  Receptor-ligand concentration RL is plotted against ligand ligand concentration L being varied from 0 to 5mM, keeping R equal to 1mM, to 1.2KM -1 sec -1 , and k to 1 sec -1 . These values have been adopted from data and results reported in [22][23]. As can be seen that when n is 1 the resulting curve is hyperbolic. When n is 2 the curve becomes sigmoidal showing the effect of cooperativity. As n increases further the curve becomes steeper or more switch like.   Fig. 2 in Cadence. For value of n equal to 1, voltage proportional to the drain current of M is measured as V is varied from 0 to 3V. As stated earlier, this voltage represents the receptor-ligand complex concentration RL for a receptor having only one binding site. Here V is kept 1.3V, R 3 is 2KΩ, channel width for both M and M is 2μm, and channel length for both the transistors is 750nm. Like Matlab simulation curves, it can be seen that when n is 1, the resulting curve is hyperbolic showing zero cooperativity. For value of n greater than 1, RL is proportional to the drain current of M . It shows that with increasing V , there is a negligible increase in drain current of M initially, which is turned into a rapid increase as the drain current in M grows, generating a sigmoidal curve. When there are more than one binding sites, the production rate is measured at drain current of M . For n being 2, 3 and 4, values taken for R 3 are 1 KΩ, 3 KΩ and 4 KΩ respectively, and values for are 750nm, 550nm and 350nm respectively. The electronic simulation run in Cadence for this dataset produced graphical results as shown in Fig. 6. Graphical curves for four different values of follow the same trend as shown in Fig. 4

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Cadence simulation, when V is varied from 0 to 5V, are shown in Fig. 7. These curves for RL vs L again show similar behavior as depicted in Fig. 4.
Simulation results for both the suggested electronic circuit equivalents for Hill process, as given in Figs. 6-7, show similar progression as depicted in the Matlab graphs of Fig. 4.

CONCLUSION
This research endeavor has put forth some possible electronic models for the bio-cellular reaction called Hill process. To model dynamics of Hill process, two transistors have been used in cascade, one controlling the activation of the other. It has been shown that the drain current of the controlled transistor increases rapidly with increase in the drain current of the controlling transistor; the same way as ligand binding to one receptor site increases the ligand binding affinity of other site on the same receptor. These electronic circuits have been simulated in Cadence Virtuoso IC design software, producing the expected sigmoidal curves. To represent the Hill coefficient which governs the steepness of the sigmoidal curves, reduction in transistor drain current has been proposed which can be achieved by adjusting either length or width of the transistors. Results obtained for different circuit approaches have been compared quantitatively as well as qualitatively. The prospects of computational biology in electronic domain may seem promising in producing better, faster electronic simulators for bio-cellular processes; however, the work done so far in this domain is still in rudimentary stages. Researchers working in this area are trying different options; nevertheless, it is very early to comment on the qualitative order of these approaches. A lot of improvements can be applied to the circuits presented in this paper to gauge veracity of the results, still it puts forth some new dimensions to this research area.