Option pricing of geometric Asian options in a subdiffusive Brownian motion regime

Nowadays, the miniaturization of bioelectronic devices in order to improve their performance, the design of more efficient compounds to exert their function accurately, the fabrication of durable and green-friendly packaging items minimizing the content of scarce based-fossil fuel materials or the precise knowledge of the entire mechanisms which govern biomolecules and cellular systems require the progress of high-throughput techniques. From its discovery [1], atomic force microscopy (AFM) has emerged as promising technique to interrogate multitude of properties from samples with different nature. AFM presents some advantages in comparison to other nanoscale microscopies. AFM does not request the addition of external contrast agents to measure carbon-sourced biological samples like scanning electron microscopy (SEM) or conduct the experiments at ultra-low temperatures as cryo-transmission electron microscopy (cryo-TEM). Furthermore, AFM is capable to devote the measurements at liquid media mimicking the near-physiological conditions inner the living cell. AFM consists of a flexible cantilever where a laser beam is reflected towards a photodetector sensor device. The flexible cantilever ends with a sharp tip which interacts with the external sample surface. The close-loop feedback maintain the setpoint parameter under controlled by modifying the proportional-integral-derivative (PID) settings [2]. Finally, the sample is mounted on a piezoelectric tube made by ceramic materials which are deformed under applied voltages. This fact enables to deliver excellent vertical resolution. AFM treasures multitude of operational modes specialized to acquire one specific property of the sample of interest.

AFM imaging consists to raster the sample with an AFM tip in order to collect topology maps. Today it exists many operational modes to gather the topography information of soft matter systems. Contact mode (CM-AFM) sets the cantilever bending as setpoint and the user can define the grade of interaction between the AFM tip and the scanned sample. Tip-sample interaction is settled by the constant applied force between both surfaces. Alternatively, tapping mode (TM-AFM) controls the oscillation amplitude, the frequency shift or the phase variance as feedback. Here, the tip-sample interaction is based on the AFM cantilever oscillation damping. TM-AFM is considered a semi-intermittent mode that minimizes the non-desirable frictional forces when the AFM probe oscillates with the sufficient amplitude. The main drawbacks experienced for CM-AFM and TM-AFM are the possibility to drag the imaged features during data acquisition in combination with potential sample damage due to the high-lateral forces exerted by the AFM tip and the lack of control of absolute vertical forces that could drive the soft samples compression, respectively. Suppliers have developed other operational modes in order to overcome the aforementioned limitations like peak-force tapping (Bruker) or QI mode (JPK-Bruker). The first mentioned technology monitors the applied AFM tip load forces on the sample surface. This fact enables to preserve the integrity of soft matter systems recording high-quality topography information. Peak force tapping mode has pave the way to evolve peak-force quantitative nanomechanics (PF-QNM) to address the nanomechanical properties of soft samples (See section titled nanoindentation). On the other hand, QI-mode makes negligible the sample invasiveness by efficient algorithms that allows automated tip-sample interaction rendering thus, nondestructive imaging straightforward. All the above described modes display low temporal resolutions (from tens of seconds to minutes scale). High-speed AFM (HS-AFM) through the design of ultra-short cantilevers [3] achieved to work at high-resonance frequencies that leads microseconds acquisition times [4]. Independently of the selected operational mode, imaging always is advisable to choose AFM cantilevers with low spring constants (range of 0.01–0.1 N/m) to gather topology information of soft matter systems.

Table 1 recaps the capabilities of all mentioned AFM imaging modes. AFM imaging succeed to study the biomolecular morphological changes upon ligand catalysis [5]–[8] which is relevant to know those conditions where the biomolecule of interest renders the best catalytic performance. Next generation of bioreactors strongly depends on the insights achieved on this regard [9]. In addition, AFM imaging is capable to discern the number of units involved in hepatotoxic strand formation under the presence of certain metal ions [10] or the effect of coniferyl alcohol polymerization by Fenton reaction on bioinspired lignin films [11]. This latest study is relevant to create novel antibacterial films based on plant sourced polymers [12]. The record of biomolecular surface charges [13] or conformational dynamics of ATP synthases [14] and rhodopsin dimers from the G-protein-coupled photoreceptor family [15] under the presence of protons and light, respectively are additional relevant cases that showcase AFM as core technique in this field. The final example of AFM imaging studies is the assessment of the interplay between proteins and DNA chains [16], [17] which can be substantial to devise biomolecular scaffolds [18]. Hence, AFM is a powerful tool to elicit the morphological properties of soft matter systems [19].

Table 1.  Comparison between available AFM imaging modes according to the feedback and the main advantages and weaknesses depicted.

Operational mode	Setpoint	Main advantage	Main weakness
CM-AFM	Cantilever bending	High stability	High lateral forces
TM-AFM	Resonance amplitude Frequency shift Phase difference	Low lateral forces	No control vertical force
PF-TM	Applied force (>50 pN)	Constant feedback	Low temp. resolution (~ms)
QI-mode	Deflection of cantilever down to the height of zero force	Constant feedback	Low temp. resolution (~ms)
HS-AFM	Resonance amplitude	High temp. resolution (~µs)	Low scan range

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Then, AFM based force spectroscopy (AFM-FS) determines the intra- or intermolecular interactions between biomolecules [20],[21]. The classical example of multidomain protein unfolding is the case of titin [22]. The role of redox conditions can lead to prevent and better diagnose cardiac diseases [23]. On the other hand, big efforts have been made to decipher the intermolecular interactions of DNA strands [24], protein-protein [25], cell-cell [26], small molecules like rotaxanes [27], and lignocellulosic polymers with cellulose nanocrystals functionalized AFM levers [28]. The potential industrial applications of the latest described scientific work is the optimization of composite materials by introducing sustainable plant fibers as fillers inside matrices [29]. AFM-FS experiments require proper functionalized strategies [30] by orienting the partner towards the soft matter sample of interest [31]. AFM-FS can become to dynamic force spectroscopy (DFS) by varying the retraction velocity of the AFM cantilever.

Through the Ritchie-Evans equation the dissociation parameters of the system studied can be obtained [32] (1):

where, F* is the most probable rupture force of one individual analyzed complex, k_B is Boltzmann's contant, T is the absolute temperature, x_β is the distance between the bound state and the energetic maximum, R is the loading rate and k_off is the dissociation rate at equilibrium. Thus, x_β and k_off dissociation parameters are obtained when the F* is plotted respect to the neperian logarithm of R. The dissociation parameters have been obtained by DFS for flavoenzimes [33], enzyme-cofactor [34], proteins [35] or peptide-cell membrane receptors [36] complexes, among others. The Ritchie-Evans equation is rooted in the Bell model [37] which in turn is based on the Arrhenius equation or Kramers rate theory (2) [38]:

where, ΔU and k₀ parameters are the variation of the single reaction coordinate and the intrinsic rate, respectively. Kramers rate theory states the free energy barrier (ΔG^†) proportionally decreases when external load forces are exerted. Subsequently, these load forces also cause the increase of the bond rupture rate. Conventional AFM setups (except HS-AFM) only provides small range of loading rates which are low. At this regime of loading rates, the mean values of F* obtained in (1) result from Arrhenius equation of kinetic rate [39]. Thus, it is demonstrated the loading rate dependence and stiffness of the AFM apparatus on the single-dimensional energy landscapes from the chemical bonds of interest [40],[41]. Experimental measurements have been carried out to demonstrate this hypothesis. The rupture force of native bonds formed by β1 and β5 polypeptide chains from ubiquitin protein increases with the loading device stiffness when varies from 10 to 1000 pN/m at the same tested AFM lever loading rate [42]. Therefore, Ritchie-Evans equation only is accepted for low regimes of loading rate. When the plot renders one single linear trend means that only exists one energetic barrier when the complex decouples, whereas two linear regimes are related to two different energy landscapes between the bound and the dissociate complexes. Free energies can be also obtained by physical simulations [43]. Recently, some applications have been found in this field like the exploration of the energy landscape of DNA respect to antiviral drugs [44] which will significantly aid to advance in the creation of more effective therapies against human diseases. In addition to binary transient biomolecular complexes, DFS can also be devoted to reconstruct energy landscapes for intramolecular folding events observed in multidomain proteins or nucleic acids [45].

Thereupon, nanoindentation measurements consist in penetrate some nanometers the external sample surface by the AFM tip through applying external force on the AFM cantilever. After recording the force-distance curve, the mechanical deformation underwent by the sample is obtained by the slope of the force profile. Higher slopes are related to stiffer samples, whereas smoother slopes correspond to soft materials [46]. Nanomechanical parameters like the Young's modulus or energy dissipation are obtained by nanoindentation studies. Many theoretical frameworks have been built up to address the Young's modulus of soft matter systems using the AFM tip as nanoindenter. The approaches most used in this field are rooted in Hertz and Derjaguin-Müller-Toporov (DMT) works.

Hertz model hypothesized the nanoindentator geometry is an ideal sphere exerting perpendicular penetration on the studied surface [47]. Hertz model neglects the adhesion forces established between the AFM tip and the sample surface (3).

here, d is the indentation depth, F is the indentation force, Eeff is the effective Young's modulus and R_eff is the effective curvature radius of the nanoindenter.

The elastic modulus of the soft matter sample of interest can be obtained by (4):

being, υ the Poisson's ratio and the subscripts T and S the AFM tip and sample, respectively. Then, DMT model takes into consideration long-range AFM tip-sample adhesion forces [48]. The Young's modulus of the soft matter system can be achieved by (5):

where, a is the contact radius between the surfaces of the AFM tip and the indented sample, F is the load force applied by the nanoindenter and w is the required energy to separate the unit area of both surfaces. Commonly, it is recommended the use of Hertz model when the radius of the AFM tip employed is large, whereas DMT model is broadly conducted for sharp AFM tips. Young's modulus of cells [49], virus capsids [50], cancer cells [51], lignocellulosic films at different relative humidities [52] or hydrogels [53] have been revealed. It has been reported that changes on cellular membrane rigidity affects mechanotransduction processes [54] and proliferation [55] which impacts on the growth of human diseases [56]. On the other hand, revealing the nanomechanical properties of biopolymer films, composites, hydrogels or biotissues serve to acquaint their wettability, rigidity and elastic performance which is crucial to use them for many industrial applications [57].

AFM-nanoscale infrared spectroscopy (AFM-nanoIR) is based on a pulsed, tunable IR laser light aligned to the same scanned area by the commercial AFM tip. The photothermal expansion induced by sample absorption causes the vibration of the chemical bonds involved in the scanned sample surface area being thus, chemically characterized [58]. Two different modes to acquire chemical information of soft matter systems by AFM-nanoIR are available. First, AFM-nanoIR can perform chemical mapping at one defined wavenumber value. This aspect is of enormous importance when the user aims to discriminate between surfaces with different chemistry and it exists previous knowledge about the expected chemical bonds that serve as target. One illustrative example is the characterization of plant cell sections embedded by resins with benchmark wavenumber of 1710 cm⁻¹ typical of carboxylic acids coming from the tested resin agents [59]. Chemical mapping offers excellent lateral resolution below 10 nm. The second approach is the “point and shoot” method based on obtaining conventional full IR spectral at specific soft matter sample regions. It is not unusual to observe slightly shifts of the wavenumbers found on the maximum of the IR spectra peaks between nanoscale and bulk measurements. This fact is attributed to the changes underwent by chemical bond lengths that eventually can take place when differences in electronegativity of neighboring atoms are experienced [60]. AFM nanoIR technique has been exploited to locate cell membrane receptors [61], the interrogation of chemical changes observed by the interaction of amyloidogenic proteins involved in neurodegenerative disorders with peptide inhibitors [62] or polymeric blends [63] for the manufacturing of sustainable food packaging or other enveloping items.

Finally, magnetic force microscopy (MFM) measures the magnetic response provided by the sample upon AFM tips coated with cobalt-chromium. Distance between AFM tip and the external sample surface area is fixed at several hundred of nanometers in order to prevent the non-desirable short-range Van der Waals interactions which can interfere during the data acquisition. Repulsive and attractive magnetic forces coming from the sample cause a bend of the flexible cantilever which will be recorded by the AFM electronic system. Thus, MFM detects local magnetic fields with the classical AFM spatial resolution [64]. Magnetic properties of magnetosome nanoparticles coming from magnetotactic bacteria [65] which could be exploited as microswimmer robots under induced flow fields [66], proteins functionalized with magnetic nanoparticles [67] or magnetic nanoparticles embedded in polymer matrices [68] have been measured by MFM. The main applications of the most relevant outcomes found by MFM are based on hyperthermia [69], drug delivery [70] treatments, development of energy storage devices [71], or emulsion separation in petroleum industry [72], among others. To conclude, AFM is presented as suitable alternative to assess the physico-chemical properties of soft materials. Five different operational modes have been presented to illustrate the potential of this technique (Figure 1) and their potential applications (Figure 2).

Figure 1.  Schematic illustration of atomic force microscopy (AFM) technique highlighted in black circle. AFM imaging (TM-AFM), AFM force spectroscopy (AFM-FS), nanoindentation, AFM-nanoscale infrared spectroscopy (AFM-nanoIR) and magnetic force microscopy (MFM) are remarked in blue, purple, brown, red and orange rectangles, respectively. The adhesion force and young's modulus values are obtained from the regions depicted in red color from their respective force-distance curves.

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Figure 2.  Some relevant applications of the five addressed AFM operational modes (AFM imaging, AFM-FS, nanoindentation, AFM-nanoIR and MFM, respectively) for Research & Development and Industry fields.

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Promising future perspectives are expected in the use of AFM in this field. Volumetric analyses have been developed by defining a mask threshold on topography images obtained by AFM imaging [73]. This aspect facilitate the statistical study of the observed features providing less scattering results. The main limitation to conduct imaging with conventional AFM apparatus is the poor temporal resolution achieved. HS-AFM allows to acquire ultra-fast images that allows to monitor in real-time the change biomolecular dynamic conformations in relevant conditions. The design and fabrication of smaller cantilevers in combination to high-response Z-scanners with the subsequent optimization of hardware and software components will be the basis of the next-generation of HS-AFMs. In this framework, the “only trace imaging” (OTI) mode has been evolved by discharging the record of the backward scan period [74]. OTI mode enables scan rates ranging from 8 frames per second (fps) to 30 fps improving nearby 2.5 times the data acquisition from previous HS-AFMs. Moreover, HS-AFM has recently used in the field of AFM-FS to determine the dissociation parameters of streptavidin:biotin [75]. Streptavidin:biotin system serves as proof-of-concept being the strongest non-covalent complex known in nature. HS-AFM succeed to obtain the information related to the complex rupture at loading rates more similar than the taken place in the living life organisms. Alternately, the optimization of AFM-FS experiments drives molecular recognition imaging (MRI) studies [76]. MRI simultaneously acquires the topography map and the maximum AFM-tip adhesion force for each pixel of the scanned area being possible to quantitatively correlate the visualized features with their adhesion properties. Moreover, the combination of AFM-FS with Kelvin probe force microscopy can open new avenues in the recognition of soft matter systems by combining specific intermolecular interactions with surface potentials [77]. These quantitative MRI methods significantly improve previous qualitative MRI approaches like AFM-based simultaneous topography and recognition imaging (TREC) [78] where the functional principle is based on the contrast image created between the lower part of the AFM cantilever amplitude oscillation respect to the upper part. The lower and upper parts of the amplitude oscillation yield topography information and molecular recognition images, respectively. Quantitative MRI technologies may open a door in the development of ultrasensitive detection technologies [79]. Furthermore, MRI experiments have shown to map biomolecular alterations like DNA methylation [80] which is relevant to monitor DNA mechanism effects. This technology could be expandable to study the redox functions involved in aging and disease [81]. It is also expected the employment of alternative techniques like computational modeling in combination with AFM can shed light to the physico-chemical properties of soft matter systems [82]. Nevertheless, AFM-FS also needs to deal with challenges, such as the noise threshold limit during sensing soft matter inter-/intramolecular forces which is nearby 15–20 pN. Optical tweezers (OT) [83] and magnetic tweezers (MT) [84] tools have raised to solve this limitation. OT and MT can trap the soft matter sample (typically biomolecules or living cells) by a laser beam source or through magnetic microbeads, respectively. Other advantage presented by OT and MT is the lack of sample immobilization on solid surfaces eliminating detrimental lateral forces during data acquisition. We expect the crosstalk of the five presented AFM operational modes in combination with other complementary single molecule techniques will significantly aid to better understand the underlying mechanisms of soft matter systems at the nanoscale level. This aspect is relevant to assess information that can be hidden in bulk studies like transient phenomena, rare events, conformational changes or crowding effects or local heterogeneities, among others.