One straightforward reason for making a sensor small is that in the application under consideration V is so small that it becomes very difficult to expose the sensor to the sample solution without diluting it. Using the denotation in Figure 1, one can say that the sample volume V must be at least as large as the product Ah in order to establish proper contact. For instance, assume one wants to detect an analyte in one bacterial cell using a quartz crystal microbalance (QCM). We then have a sample volume below V = 10⁻¹⁸ m³, but an ordinary QCM is a very large sensor based on binding to a ∼1 cm² sensor crystal [9]. This sensor test is clearly not feasible since the sample cannot be introduced properly (it requires an unrealistically small h). The situation then obviously requires diluting the sample by many orders of magnitude and C will become much lower, making efficient detection problematic (see further below). Instead, better performance will clearly be reached if A can be made smaller, at least when nothing else changes in the system (yet, it should be kept in mind that handling a volume as small as 1 fL will be problematic regardless of A). Given that miniaturization can be needed when V is extremely small, let us consider how common this situation is. Besides single cell analysis, one can also think of certain biopsy samples, in which case the sample volume would be on the order of V = 1 μL. (This is also a good measure of the smallest volume that can be easily handled with a common pipette.) In this case, one can actually expose a sensor as large as A = 1 cm² using a h = 100 nm nanofluidic channel. Such a liquid cell would be quite extreme in terms of aspect ratio, but it is not necessarily an unrealistic configuration. However, it is probably preferable to have a sensor with slightly smaller A, suggestively on the order of 1 mm².

Besides small V, the sample containing the analyte can in principle also be too “small” in terms of number of molecules available (CV). Even if V is reasonably large, there might not be many molecules present depending on the value of C. This is certainly the case for single cell analysis by QCM (in addition to the problem with too small V). Here the value of A naturally comes into play once more: Even if the sample liquid can be properly introduced to the sensor surface it is possible that Γ_max cannot be reached even if all molecules bind. Let us look at another example, namely the peptide hormone calcitonin, which is present at a mass concentration of CM ≈ 10 pg/mL in blood (the exact concentration indicates the state of medullary thyroid cancer). This is a very low concentration [10], making calcitonin one of the most challenging biomarkers to detect with surface sensitive techniques. The highest surface coverage possible will depend heavily on the receptor arrangement on the surface, but for relatively dense protein layers a value of Γ_max on the order of 100 ng/cm² can be assumed [11]. Thus, for a large sensor with A = 1 cm², a (standard) 10 mL blood sample will only provide calcitonin molecules enough to reach Γ/Γ_max = 0.1%. Does this mean a smaller sensor is needed, so that fewer molecules are needed to reach Γ_max? The answer is actually no, not in practice, because it has implicitly been assumed that one can reach Γ_max, but this holds true only for the very rare case of irreversible binding. Because of the low concentration of calcitonin and the limited affinity of recognition elements one can only expect a small fraction of the binding sites to be occupied even at equilibrium. Reversibility is usually illustrated with the common Langmuir isotherm for reaction limited binding [12]: (1) Γ ( t ) = Γ max k on   C k on   C + k off   ( 1 − e − ( k on   C + k off ) t )

Here k_on and k_off are the rate constants for association and dissociation. The surface coverage at equilibrium (t → ∞) is Γ_eq and the fraction of binding sites occupied is given by: (2) Γ eq Γ max = k on   C k on   C + k off = 1 1 + K D C

Here K_D is the dissociation constant defined as the ratio of the association and dissociation reaction coefficients, i.e., K_D = k_off/k_on. It is a measure of the strength of the interaction (low K_D represents high affinity). Preferably, K_D should be within the dynamic range of the sensor, i.e., the range of analyte concentrations one wishes to detect, but it tends to be higher. Reaching Γ_eq = Γ_max requires K_D = 0, while in reality one can usually only get down to K_D ≈ 1 nM at best for antibodies [13]. Aptamers are no better in this respect [14]. Interestingly, it is also clear from Equation (2) that for nanosensors (e.g., A = 100 × 100 nm) and low C in general, one cannot expect that even a single receptor (on the order of ∼100 in total) has an analyte bound to it. Therefore, if one evaluates the sensor state at a given point in time, nanosensors will often actually give Γ = 0 instead of Γ = Γ_eq simply as a consequence of “molecular discretization” breaking the mathematically continuous model in Equation (2) [8]. One should also consider possible edge effects: do the receptors just at the edge of the active spot really behave the same way as the others? For smaller A the relative number of receptors just at the edge is higher.

Let us now look again at the calcitonin example when reversibility is accounted for. In this example M = 15 kg/mol, which gives C slightly below 1 pM and it is thus clear from Equation (2) that Γ_eq is around 0.1% of Γ_max for a high affinity receptor. As calculated above, this is exactly the highest possible Γ that the number of molecules in the sample allows us to reach when A = 1 cm² and V = 10 mL. Thus, the advantage of sensor miniaturization is questionable: There seems to be enough molecules available even when A = 1 cm². Strictly, this is not true since the Langmuir model assumes that the concentration in the sample is not reduced once some molecules have bound to the surface (molecules must be continuously supplied to the surface). Therefore a slightly smaller A is probably preferable, or more blood is needed, but pushing towards nanoscale sensors is obviously not necessary. Rather, it seems like a sensor on the microscale would represent more than enough miniaturization. I emphasize that this example was for what is arguably a “worst case scenario” biomarker and that there is much more molecular material available for other protein biomarkers in blood [10].

In order to establish equilibrium and approach reaction limited binding kinetics, it is often useful to implement continuous flow during the experiment. The volumetric flow will be approximately Q = vhA^1/2 when the channel is as wide as the sensor (Figure 1). Further, one can let τ denote the time it takes to perform the sensor test. Naturally, the use of continuous flow then requires that neither Q nor τ reaches extreme values and for flow operation one clearly needs Qτ < V. This leads to a potential problem for larger sensors due to depletion of V, while smaller sensors should require a lower Q for the same flow velocity at the sensor (v determines how efficiently molecules are supplied to the sensor). Ignoring some extreme examples such as single cell analysis, do we have sufficiently large V in general also for operation under flow on large sensors? Let us once more look at the 10 mL blood sample together with a relatively large flow channel that has h = 1 mm and a long measurement with a duration of τ = 1 h. It is then clear that even for A = 1 cm², one can implement flow at a rate of v ≈ 300 μm/s without depleting V. This is arguably a velocity that represents efficient delivery of molecules to the sensor [1]. How does the situation look like in other applications? For biosensors, the most common applications are related to medical diagnostics, food safety and environmental monitoring. In terms of medical diagnostics, one can normally acquire bodily fluids in amounts of V > 1 mL. It may be preferable to consume less volume, but A can decrease very many orders of magnitude before one reaches the nanoscale. Practically all samples from food industry or the environment are obviously even larger (people do not eat microscale portions and the environment is a very big place). This suggests that even for A on the mm scale, V is sufficiently large in most cases even for operation under continuous flow.

Looking further at the time of the sensor test, how can one know whether τ will not be so long that our sample is still too small for work under flow to be feasible? In other words, assuming the flow delivers new molecules to the sensor efficiently so that the system follows Equation (1) fairly well, is equilibrium reached? If not, a smaller sensor could be needed since it requires a lower Q to reach the same v. One clearly wants τ to be long enough for equilibrium establishment, so that Γ is close to Γ_eq and the signal maximized. At the same time, τ needs to be quick enough for the sensor test to be of practical use. It can be seen from Equation (1) that the system is close to equilibrium if k_offτ or k_onCτ (not necessarily both) are comparable to one. Therefore, it is sufficient that k_off⁻¹ alone is comparable to τ regardless of C [8]. In other words, equilibrium can be reached as long as the molecular interaction lifetime is comparable to the time of the experiment. Although the desired τ will be quite different depending on the scenario, a typical high affinity k_off⁻¹ value is on the order of 10⁵ s [13] and it seems likely that one will often have k_off⁻¹ comparable to τ. Note that this does not mean that a high k_off is something desirable. A higher k_off will unfortunately always result in a lower Γ_eq. It just happens to be the case that even the best receptors available normally provide interaction lifetimes comparable to the timescale of a typical experiment. It is also noteworthy that the commonly studied biotin-avidin interaction [15–17] is exceptionally strong with k_off⁻¹ > 10⁷ [18] and not at all representative for typical biomolecular interactions (although the k_on value is quite ordinary). One must be careful when using this interaction as a model system to draw generic conclusions about binding kinetics and sensor performance [17].

One can summarize this section by saying that in many real world sensor applications there will be sufficient volume and analyte molecules available, even if A is relatively large. However, the use of nanosensors is clearly motivated in some important cases when the number of analyte molecules available is very low, such as single cell analysis [19]. In some sensor applications, one aims to study a molecular interaction rather than detecting an analyte, such as in drug development and proteomics. In these cases, the researcher prepares the “sample” and can thus influence both V and C. The number of molecules available can then often be reasonably high without too much effort. For instance, drugs must obviously be possible to produce in relatively large quantities or else they would not be suitable candidates. Also, extensive research on protein production (see the Elsevier journal Protein Expression and Purification) has made it possible to produce quite large amounts of many proteins of interest, at least those that are water soluble.

The effect from reduced A on the incident diffusive flux of molecules is perhaps the least discussed yet the most obvious advantage of miniaturized sensors, although it only applies for binding that is strongly influenced by mass transport [20]. For fully diffusion limited binding to a planar infinite surface in contact with a stagnant sample, Γ is a function only of concentration and the diffusion constant D of the analyte [12]: (3) Γ ( t ) = 2 CM Dt π

Equation (3) represents a sensor that captures molecules immediately when they are at the surface (k_on → ∞), which never saturates (infinite Γ_max) and the binding is implicitly considered to be irreversible (k_off = 0). Naturally, this can only ever be valid for the initial binding phase well before equilibrium is reached [12]. Even though reaction kinetics are effectively ignored in Equation (3), it is possible to say something about the upper performance limit of the sensor only from knowing D. This is because a molecule can diffuse to the surface without binding to it, but not bind to the surface without first diffusing to it. Looking again at proteins as model analytes, one has D around 10⁻¹⁰ m²/s [21] and Equation (3) predicts that reaching sufficiently high Γ for detection, typically Γ = 0.1 ng/cm² [1,22], may unfortunately take years if C is below 1 pM (e.g., calcitonin in blood).

Although only approximate analytical solutions are available for diffusive flux to other sensor geometries, it is clear that the infinite surface geometry provides the lowest flux of molecules to the sensor [23]. Therefore, smaller sensor spots will tend to cause an increased flux into A due to the edges [20]. The increased flux associated with smaller sensors is conceptually illustrated in Figure 2 for different architectures. When diffusion determines binding rate, the smaller sensors will result in higher ∂Γ/∂t because their geometry induces increased flux at the borders. One can also think of this phenomenon as an effect of sensor dimensionality: Equation (3) represents the slowest type of binding because the depletion zone only grows in one dimension, but “wires” or “dots” have depletion zones that grow in two or three dimensions [23]. For the one dimensional problem [Equation (3)], the depletion zone at the sensor has a characteristic thickness determined by the independent Brownian motion of the molecules: (4) z ¯ = 2 Dt

It follows that the ideal sensor in this context is infinitely small and preferably also somehow suspended from the supporting surface and fully introduced into the sample. Such a sensor can “suck in” analyte molecules from all directions to a single point, thereby maximizing the flux. Notably, heat transport follows the same principles and if heating is an issue for the sensor under consideration, miniaturization can lead to more efficient cooling.

As illustrated in Figure 2, a smaller sensor may result in a higher Γ, but one must then consider in what situations this conclusion applies. It is quite rare to have purely diffusion limited binding from a stagnant liquid. Indeed, the most obvious way of countering the constraints imposed from diffusion, namely by introducing flow [1], has so far been ignored in this analysis. As discussed in the previous section, many samples of relevance are large enough for measurements under constant flow. Miniaturized sensors actually make it more difficult to enhance the binding through convection. This is simply because for small A, molecules tend to pass by the sensor element without binding to it [7,8]. To understand this conceptually, consider once more our flow channel geometry in Figure 1. For flow to be reasonably efficient in increasing the incident flux to A, each molecule must spend a time above the sensor equal to the time it takes to diffuse down to the surface on average [8]. (The dimensionless Peclet number, which represents the ratio of these times, should be one.) The corresponding flow, denoted Q_capt, can be calculated by using Equation 4 and integrating across the channel. Since the average time a molecule spends above the sensor is Ah/Q one gets [1]: (5) Q capt = Ah 1 h   ∫ 0 h z 2 2 D   d z = 6 AD h

The flow rate calculated from Equation (5) represents a compromise between enhancing mass transport to the sensor while preserving decent capture efficiency, i.e., the probability that a molecule introduced with the flow actually encounters the sensor surface should be around 50%. Of course, if one has practically unlimited amounts of sample volume available, it is still possible to increase the flow (Q > Q_capt) and further enhance ∂Γ/∂t. However, the effect becomes much weaker and Γ approaches a Q^1/3 dependence [8], which is also what happens for large channels [24] (h → ∞). Therefore, Equation (5) is still important because it indicates the absolute flow rate at which it becomes difficult to achieve further enhancement from flow in diffusion limited systems. Figure 3 shows results from using Equation (5) to calculate Q_capt for different A and a few different h, fixing D as 10⁻¹⁰ m²/s. Clearly, a small A is associated with a low Q_capt, although the flow channel height also is important. The larger sensors (up to 1 cm²) have no problems capturing molecules even at relatively high Q. In contrast, the nanosensors (A down to 10 × 10 nm) essentially cannot be used together with flow at all if good capture efficiency should be maintained, regardless of the flow channel height. Admittedly, for small sensors some enhancement is still possible even in the Q^1/3 limit since Q can be increased so many orders of magnitude above Q_capt. Note that it has been assumed that the flow channel width is equal to the width of the sensor spot, which is an arbitrary rectangle. It should also be kept in mind that for binding controlled by diffusion under convection one gets an inhomogeneous Γ in the direction of the flow [24].

In summary, one can say that although sensor miniaturization may help to increase the binding rate from stagnant liquids in the diffusion limited case, it also tends to spoil the enhancement which is possible by introducing flow. When the sensor is larger, it is possible to introduce higher flow rates and still expect the sensor to capture a large fraction of the molecules. A complete optimization requires taking the sensor and flow channel geometry into account, but one must also consider how much sample there is available and to what extent diffusion controls the binding rate. Interestingly, Q_capt is proportional to the number of molecules introduced to the sensor but also proportional to A [Equation (5)]. Therefore, given that Equation 5 well represents the highest flow at which a reasonably high fraction of the introduced molecules bind (for any given A), Γ becomes essentially independent of sensor size in diffusion controlled systems (Γ equals the number of molecules captured divided by A). This suggests that focus should be put on minimizing h instead of A, i.e., one should work towards efficient flow configurations [16,25]. It should then be kept in mind that the high pressure drops associated with narrow microfluidic channels may be problematic.

Many surface sensitive techniques are compatible with detection in arrays, even with parallel readout. Examples include SPR imaging [26], piezoelectric resonator arrays [27] and cantilevers [3]. It is obvious that a higher degree of multiplexing can be reached if each sensing element is smaller. So how small should A (the size of one sensor element) preferably be in order to reach the degree of multiplexing necessary? Although this will clearly vary from case to case, it seems hard to suggest the use of nanosensors just for the sake of multiplexing. In many biosensor applications, such as detection of biomarkers in blood or pathogens in food, it seems like one would want to detect around 100 analytes at the most, which will not require very small sensing spots. But one can anyway look at an extreme example: assume one wants to scan the entire human proteome, including alternate splicing and posttranslational modifications, on a 1 cm² chip. This corresponds to almost ∼10⁶ analytes, but one only needs sensor spots of about 10 × 10 μm. The only practical example where so many interactions need to be measured is most likely antibody library scans [28]. It thus seems rarely necessary to approach the nanoscale just for the sake of sufficient multiplexing unless the total size of the sensor chip must be very small for some reason. Another reason why it is questionable to have small A just for the sake of multiplexing is that depending on the transducer mechanism there are various practical limits on parallel readout of many individual close packed sensor spots. Also, even if readout is possible, handling data from millions of channels operating in real-time is clearly problematic.

In addition to the abovementioned one must also consider the limitations of serial surface modification methods [28]. Multiplexed detection is based on the possibility to functionalize one sensor spot with one type of receptor only, with the affinity for its analyte preserved. For the case of protein receptors, the most efficient patterning strategies operate with microscale resolution [29]. For instance, the fastest and most reliable method for array functionalization is arguably still microdispensing, which has a spatial resolution limited to ∼100 μm for the liquid droplets on the surface (depending on contact angle). Scanning probes techniques [30] can introduce chemical patterns on very small regions but operate slowly and are not compatible with fragile biological molecules. Electrochemical methods [31] and material specific chemistry [17] can be used to pattern nanostructures, but again only in a slow serial manner. It seems clear that with the toolset available, functionalization of a single nanosensor element is feasible, so the functionalization issue is not a valid criticism against nanosensors. However, it is still fair to claim that efficient functionalization of nanometric sensing arrays is highly problematic, which in turn questions the value of miniaturization just for the sake of multiplexing.

Resolving single molecule binding events appears to be a dream for most researchers working with sensor development. When the sensor signal is determined by Γ, reducing A is the most obvious way to get a fewer number of molecules corresponding to the same signal. The smallest sensors to date are probably single plasmonic nanoparticles [32], which seem to be just at the limit of resolving single biomolecular binding events [33,34]. It is clear that as long as A is ∼1 μm² or higher, single molecule resolution cannot be realized because the detection limit in Γ is always so high that the signal to noise ratio is lower than the number of bound molecules. It thus appears reasonable to claim that nanosensors are required for resolving single molecules. In this context it is interesting to note that detection of single molecule binding events were first reported with a relatively large (microscale) optical sensor (Figure 4) utilizing so called whispering gallery modes [35]. Since then the heating mechanism suggested to be responsible for the large signal per molecule has been questioned [36], but development of the sensor continues [6]. Although it represents an impressive engineering achievement worthy of attention, it is worth to ask the question what the single molecule resolution feature contributes with in terms of sensor efficiency.

Unfortunately the literature seems to confuse single molecule resolution with single molecule detection, which is an extreme case of low CV. The single molecule detection problem has already been discussed above and this section deals with the concept of single molecule resolution, which is a prerequisite for (but does not directly enable) the possibility to detect a single molecule. This is because even if single molecule binding events can be resolved, it is still very problematic to guide the single molecule to the (extremely small) active sensor site. Looking at the sensor in Figure 4(A) as an example, it has not been proven that every bound molecule gives rise to a step higher than the noise in the binding curve [Figure 4(B)]. Indeed, in surface sensitive techniques, especially those compatible with extreme miniaturization, the same molecule normally gives a different signal depending on where on A it binds [17]. In other words, variations in the magnitude of the step-like signal from a molecule will not represent useable information because the sensitivity is not homogenously distributed to start with. In addition, surface sensitive techniques rarely make it possible to say anything about which receptor that was responsible for an observed binding event [37]. The question is then what new information one can obtain from monitoring the number of discretized binding/unbinding events during a certain time.

The general argument why studying single molecules (or nanoparticles, quantum dots etc.) is important is that it enables the observation of different molecular states or interaction mechanisms. Such effects are averaged out and difficult to observe in ensemble measurements. However, it is important to distinguish between scientific insights on molecular behavior and the efficiency of surface-based sensors. It is only when one uses sensors to investigate the nature of a molecular interaction that single molecule resolution may be relevant. Even then it is relatively hard to find concrete examples where it is obvious that single molecules must be resolved. One special case is the study of conformational changes during a molecular interaction, which may be interesting for biomacromolecules [38,39]. Another exception is the case when two molecules bind to each other in different ways, leading to different k_on and k_off values [Equation (1)]. For instance, an antigen can have two epitopes that an antibody can recognize. By studying individual binding events, it becomes relatively easy to see that two types of interactions are occurring [37]. On the other hand, these effects are to some extent observable also in ensemble measurements (i.e., larger sensors) by proper analysis of binding/unbinding kinetics [40]. For instance, to make Equation (1) valid, efficient flow is needed during the association to avoid depletion and during the dissociation phase to avoid rebinding. If the curves acquired under such circumstances clearly differ from single exponentials, it shows that the interaction is more complex than simple first order kinetics and other models can be tested instead. Yet, single molecule resolution will naturally help illuminate the mechanism. This is especially so if it is due to heterogeneity in the population of molecules.

Many people consider single molecule resolution to be the obvious ultimate limit of sensor performance [32–35] and miniaturization is clearly needed if one should attempt to reach this goal. I suggest that this dogma should be reconsidered. By pushing towards single molecule resolution one has implicitly assumed that the detection limit in terms of number of bound molecules (ΓA) is more important than the detection limit in terms of Γ. As I have tried to explain, in many real world sensor applications there are plenty of molecules available and it is better to think in terms of concentration. Given that you have obtained single molecule resolution, I suggest asking questions like: What new information have I gained by observing single molecules? In which real-world application will this sensor be more efficient than its larger counterpart? What would my detection limit in Γ be if I had made my sensor larger? If you do get close to single molecule resolution, you will most likely be looking at relatively large analytes [34] or lack real-time data [33]. Indeed, you will likely have a poor detection limit in terms of surface coverage because of the problems associated with miniaturization, which will be considered next.