These three parameters, together with the process mechanism and the perturbation efficiency in the process, can describe the phenomenon. One of the main purposes of this paper is to link the parameters ta, Aa, and Ba and the perturbation efficiency to the characteristics of the system. There is another fundamental process for life that has some characteristics similar to allostery: the light-harvesting (LH) process in photosynthesis. Here, as in allostery, two distant macromolecular sites are involved—two sites dozens of Ångströms away. In both processes, it is particularly important that the perturbation is distributed efficiently without dissipating in the infinite degrees of freedom within the macromolecule. The importance of considering quantum effects in the LH process is well documented in literature, and the quantum coherences are experimentally proven by time-dependent spectroscopic techniques. Given the existing similarities between these two processes in macromolecules, in this work, we suggest using Quantum Mechanics (QM) to study allostery.The allosteric effect is one of the most important processes in regulating the function of proteins, and the elucidation of this phenomenon plays a significant role in understanding emergent behaviors in biological regulation. In this process, a perturbation, generated by a ligand in a part of the macromolecule , moves along this system and reaches a specific (active) site, dozens of Ångströms away, with a great efficiency. The dynamics of this perturbation in the macromolecule can model precisely the allosteric process. In this article, we will be studying the general characteristics of allostery, using a time-dependent quantum approach to obtain rules that apply to this kind of process. Considering the perturbation as a wave that moves within the molecular system, we will characterize the allosteric process with three of the properties of this wave in the active site: (1) t The emergence of life in a system includes so wide a range of aspects and meanings to prevent us from giving reductive and univocal definitions of “life.” There are, of course, many definitions of the concept of life; each of them accentuates important aspects of life and highlights processes, and/or cycles of processes, that are indispensable to life, at least in the forms and ways we know it on Earth. Here we will not deal with these general issues. This paper focuses on the emergence of life in the cell. We believe that the importance of regulating the fundamental processes of life is an essential component, right from its definition. The allosteric effect is one of the most important processes for regulating the function of macromolecules . The elucidation of the general characteristics of this phenomenon plays a major role in understanding the emergence of cellular life.Allostery has been defined as the phenomenon in which a ligand, bound to one site  of a protein, alters the binding interaction of the protein with a second ligand in another site (active site). Here, we prefer to use a more general definition of this phenomenon. Nowadays, allostery is generally defined as any process were an event in a macromolecule site affects functions, dynamics, or distribution of the conformations of another site.Allostery is a remote regulation through the transmission of information from one site to another, and this process is strictly related to the intramolecular dynamics in proteins. Allosteric effectors communicate with the active site by altering protein dynamics either through large-scale structural changes or through more subtle changes in correlated residue motions. A proper understanding of the allosteric mechanism is impossible unless the underlying dynamic conformational flexibility and energetic landscape are fully appreciated.Schematizing a protein with an internal network of interactions of the components, the study of the allosteric process allows us to understand how an environmental perturbation, exerted in a part of the macromolecule, can move inside it and reach a specific distant site. In this sense, allostery becomes a characteristic of all macromolecules that are considered part of a complex interaction network among the component elements. The interactions in these networks allow the formation of “channels” within these systems, and, along these channels, a perturbation, generated in a specific site, can move. At this point, due to the presence of countless interactions and channels in the macromolecule, the problem is not to explain why the perturbation moves in the network but to explain how it is possible to avoid the complete dissolution of the perturbation. Moreover, the problem of the quantitative efficiency of this phenomenon arises because allostery is an action-at-a-distance force, and the active site can be many Ångströms away from the sensor site    with Pi(ti ± c4) = Pi(ti)/100 ≅ 0.in the range  and in a, <Pa (t)> becomes independent of time:After a time much greater than tOur modeling of allostery is based on the main characteristics of the largely used models of literature , but with some important differences. Firstly, our model is the only one applying QM to study the dynamics of the allosteric signal among sites. Secondly, another relevant difference between our model and the current models studying this phenomenon is the attention to time-dependent aspects of allostery. Until now, in fact, all scientists have affirmed that the perturbation moves in the network, but no one has related the characteristics of allosteric systems to the arrival time in the active site and to the propagation efficiency until arrival at that site. Finally, for the first time, in our model some properties of allostery have been related to the characteristics of the systems  by simple rules.An important common feature shared by most of photosynthetic organisms is that they capture photons in pigment molecules embedded in the protein environments of light-harvesting complexes (LHCs). The excitons created in such LHCs remain well protected despite being waged by environmental fluctuations mainly due to protein degrees of freedom, and they are delivered successfully to their far away destinations in about hundred picosecond time scales.A fundamental feature shared by many LHCs, which has been clarified during the past decade, is the intermediate nature of the terms constituting the Hamiltonian. In other words, for LHCs, it is common that the difference of energies between the states and the coupling terms constituting the Hamiltonian are of comparable magnitudes. Moreover, typically, these parameters are in the same order as those of the room temperature thermal energy and also of the major portion of the energy spectrum that comprises the macromolecule. The energetic convergence of these multiple terms provides LHCs with a rich repertoire of pathways and mechanisms for exciton dynamics and energy harvesting. At the same time, the lack of apparent small parameters in these systems render simple perturbation theories rather unreliable quantitatively. Therefore, advanced levels of quantum dynamics theories and computational approaches have become necessary to achieve accurate descriptions of exciton dynamics and relevant spectroscopic observables. In this sense, the excitons in LHCs have served as prominent testing cases for modern quantum dynamics, electronic structure calculation approaches, and spectroscopic methods.All these characteristics are common to allosteric processes. In allosteric processes the energetic terms are often similar to each other and similar to those characterizing the interaction between the system and the environment, both in the case of degrees of freedom of proteins different from those of the two sites involved in the process, and for the environment surrounding the protein. However, in the allosteric process, although all those similarities are present, it is necessary that the perturbation leads the reaction site effectively. These similarities make it possible to apply the same methods to both processes.There have been studies into three major LHCs: Fenna-Matthews-Olson (FMO) complexes of green sulfur bacteria, light-harvesting 2 (LH2) complexes of purple bacteria, and phycobiliproteins (PBPs) of cryptophyte algae . Here weThe FMO complex is of great historical significance because it was the first LHC for which X-ray crystallography structural data became available  and it hAs for the FMO complex, a real time coherent beating signal was observed in 2DES data of LH2 complex. Angle resolved coherent four wave mixing spectroscopy showed evidence of coherent dynamics that can be separated from the relaxation signal due to the energy transfer . Single In a theoretical study on these systems, it is possible to employ the standard quantum mechanical procedure to derive an exciton-bath Hamiltonian based on the assumption that the ground electronic state and the site excitation state can be defined in terms of direct products of adiabatic electronic states of independent chromophores defined at reference nuclear coordinates. The default choice for these reference nuclear coordinates is that of the optimized ground electronic states of chromophores. The common assumption, implicit in most theoretical models developed so far, is that the environment induced non-adiabatic coupling between the ground and the exciton state, is very small and hence negligible.Quantum models used to study physical systems involved in light-harvesting consist of the same components needed to study the allosteric process. First of all, in both processes the complex system (the macromolecule) must be reduced to a small number of sites. This is possible using molecular mechanics studies which identify parts of molecules that are highly sensitive to the arrival of the perturbation into the allosteric site. Each site can be described electronically by one or more molecular states and the vibrational part can be added on each site specifically if relevant experiment information is known. The same applies to site couplings at both electronic and vibronic level. The only apparent difference between allostery and light-harvesting in photosynthesis is quantum coherences, which are not evident in allostery while they are present and studied in the second process. According to us, this does not mean that quantum aspects are not important in allostery, rather that it is not easy to prove them experimentally. In both processes, in fact, it is hard to believe that a classical approach could be sufficient to describe the complex dynamic.When we reduce a macromolecule to a set of efficient sites, all the sites will be coupled. Although this is generally true, some consideration must be taken for sequential site coupling models. Sequentiality is necessary in every process from an initial site until you get to a specific final site. If all sites are equally paired, in fact, a perturbation generated in a site redistributes equally.In allostery, the perturbation selective passage from the allosteric site to its specific active site is necessary. Obviously, nothing can assure that there is only one path to connect the allosteric site to the active one. This is the reason why, in our work, the case studies illustrate more than one connection path.Moreover, each system can be approximated by a sequential coupled system and the dynamics of the original system is well described by the sequential system dynamics in a limited time range. This is the principle behind techniques like Lanczos’s that, by repeatedly applying a Hamiltonian matrix on a start state (door-state), constructs a tridiagonal matrix in a perturbative way. Of course, in Lanczos’s approach, the diagonal and extra diagonal elements of this matrix are all different. However, the model case with equal diagonal elements, equal to zero energy, and with equal extra diagonal elements among adjacent sites, is interesting. In this case, in fact, among equal diagonal elements the conservation of energy only selects degenerate sites vibronic states at the door-state and only a medium coupling is considered.i = cost and the coupling between the sites Vj,k of Eq. (3) equal: Vj,k = hi,i+1 = h  for any pair of neighboring sites. Since all sites are degenerate, we can assume the energy of the generic site to be |i> like the zero energy. The sign of the site coupling (h) is irrelevant in the case of N identical sequentially coupled sites. In QM, two degenerate sites |1>e |2> (E1 = E2 = 0) with couplings ± h, for example, produce two eigenstates of energy −h, + h, with a difference in energy Δe = 2 h that generate the same periodic dynamics (function of Exp [−i*Δe*t]) between the states |1>e |2>.Let us consider a system of N identical sequentially coupled sites  with Ti This system is Markovian because the probability of transitioning between discrete sites is independent of previous transitions. This system can be studied in a CM approach by the help of its master equation of the process  = 1Pi(0) = 0i = 2,Nwith Pi is the population of site i and the amount of perturbation in the site i. The application of this master equation to this system gives, after a certain time, a uniform distribution of the perturbation in all sites since, following this classical approach, we are assuming that a perturbation spread is ongoing and it is redistributing the perturbation in all sites equally.and where t = 0 in the initial allosteric site is |i≥|1> and with active site |a≥|5>. The time unit in this paper, unless otherwise noted, is the period 2*ta of a degenerate states couple with coupling h = 1. In Let us consider a simple case of five sites, where the perturbation at Over the past few years, the application of a master equation has increasingly been used after the transformation of a protein in a set of sites and edges by MD simulations, suggesting that it is a promising tools to study allostery in a drug-discovery context . A detaisite = 36 give a general result: the perturbation completely disperses in all residues with a population of 1/Nsite (in this case 1/36 = 0.028) for each site. We have found a similar behavior in Recently, A different analysis has been done in a second example. In a recent study, A different behavior can be highlighted comparing this last case with Leitner’s. In this last case, a complete dispersion of the perturbation is not present. The difference in the behavior of these two systems is due to the transition probability matrix among the states. In this last case, this matrix is not symmetrical [see Supplementary Table S1 of Going back to the comparison between the CM and QM approaches on our five-state SC system, we want to highlight a different behavior in the propagation of the perturbation in the quantum approach :(1)In the quantum treatment, we see that the signal propagates like a wave of perturbation that moves along the sites, reaching the active site We are generally interested in the characteristics of the perturbation wave in the active site. It is possible, therefore, to find characteristics of the perturbation wave in any site. Our model, hence, allows us to determine the perturbation path in the network.(3)Although there is a sequential movement of the perturbation wave, at any instant, the wave is never completely in a site .(4)The perturbation wave has a simple form (like a Gaussian in time), but the process cannot be described with a single Gaussian moving among the sites. In a Gaussian moving in a space coordinate; if the amplitude decreases, the width must increase in order to preserve the normalization of the wave. This is not the case of the perturbation wave in time in any site and, in particular, in the active site. Moreover, since this site is the last one considered, and therefore it is not possible for the wave to proceed further, the perturbation wave is accumulated in this site before being reflected. This explains the difference between the perturbation wave in the active site compared to the other sites. This behavior, however, can be a good representation of the physical phenomenon that the signal at the active site is displaying.(5)The perturbation wave moves efficiently in the network  that describe the allosteric process in SC systems are related only to the number of sites present (N) and to the value of couplings between the sites (h). In the analytical case of two sites (expressing everything in the atomic unit), we will haveThe three parameters of the active site , we will find an almost linear relationship, as a function of the number of sites  with hi,i+1 = 1, we will haveIf we calculate tof sites . Fittinga and Ba which vary with the coupling as explained above , while the amplitude of the perturbation wave in the active site, Aa, is independent from the value of the coupling .with tN) only as a function of the distance from the active site and the sensor one. From the analysis of In this model system, we have obtained a result that we would like to highlight: the amplitude of the perturbation wave in the active site decreases  site |i> has two or more alternative paths to get to the active site |a> are important and schematized in A, B, as in A from site |1> three different paths leave through the coupling h1,2, h1,3, and h1,4. In A . In B . In B  with h1,|i> to site |a>. Case I represents two paths with the same number of intermediate sites; cases II and III are two paths of different lengths (with 1 and 2 sites more in a path). In particular, case III represents the approximation of the sequential coupling of the light-harvesting FMO complex.Let us consider now three cases  that rep1 = |1> → |2> → |3> → |4> as principal and study the modifications to this path adding the second path, Pt2, characterized by fewer couplings. If we decide to indicate the coupling between two sites with h (path Pt1) and c*h (path Pt2), same length (case I) or longer (case II and III), with c as a numerical factor between 0 and 1, for example in case III (one state per site) we will obtain a Hamiltonian:Let us choose the path Pth = 1 for the case with c = 0 and for the three systems I, II, and III with c = 0.5.In a) until about 80, 60, and 50% in the three cases. Obviously, the active site population, once reaching the maximum, decreases until 0, and then starts increasing again so that it can reach another maximum. Using as a reference the first arrival time of the perturbation to the active site, in the case of Pt1 (case with c = 0) the adding of a new path Pt2 alters slightly the arrival time of the perturbation to the active site and, generally, reduces it. Therefore, the ta of the principal connection path between the allosteric site and the active site can be considered the maximum limit for multipath systems. This can also be applied to the enlargement of the Gaussian Ba. Identifying the principal connection path between the allosteric site and the active site assumed a constant (medium) coupling between its sites, using Eq. (8) it is possible to estimate the three parameters ta, Aa, and Ba of the complex system. This is an important achievement that deserves to be highlighted.The examination of a(t)> in time, compared to the average population in other sites.In order to evaluate the transfer efficiency from the initial site  to the active one, we have to consider the values <P1(t)> (state |i≥|1>), shown in blue <Pa(t)> (state |a≥|4>), shown in green <Pj(t)>  + P3(t))/2 refers to the sites with coupling h), and shown in brown <Pk(t)> (refers to the sites with coupling c*h) with Pk(t) = Pj(t) = (P5(t) + P6(t))/2 , Pk(t) = (P5(t) + P6(t) + P7(t))/3 (case II) and Pk(t) = (P5(t) + P6(t) + P7(t) + P8(t))/4 (case III).In a>, compared to the first passage time in the site |a>) to the active site is between 20 and 30%. In particular, not all the perturbation has left the allosteric site  in long time, and the remaining perturbation, situated in different sites from the initial and the active ones (green and brown curves) is smaller than the one situated in the active site in all cases.In I (assumed constant c = 1) this is evident. In this case, in fact, the two perturbation waves started from |i>, following the same paths, arrive at the same time and in phase in the state |a>. In j, maximum amplitude in the site |j> at first passage and <Pj>, average population in the site |j> for long time compared to the first passage time have been reported.The prevalence of the active site compared to other sites depends on the quantum effect of coherent superposition between the perturbation wave arriving from a path and the wave arriving from the other path. In case a = 0.750.In this case, the maximum population at first passage in the allosteric site is Asites = 6) that enter the time evolution in the form:Despite the high symmetry of this system and its apparent simplicity there is no evidence that the perturbation moves completely from site |i> to the active site |a>. Also in this case, in fact, there are many different times  or small (and related to the entropy term). In the previous paragraph, the perturbation does not change either the sites (energy) nor the edges (coupling) of the network. Here, we would like to test the idea that, during the movement of the perturbation along the network, the fraction of the perturbation wave present on the site changes the coupling between the sites. Of course, this is a non-Markovian process.i,i+1 = h between all pairs of sites adjacent to time t = 0. As a function of time, in this model we will have that:In iP (t) the perturbation fraction at site i and with the assumption of x = ±0.2, ±0.8 to study four different types of variations of the coupling between sites as a function of the perturbation fraction. We will use the following notation: su  for x = 0.2; sd  with x = −0.2; lu (large-up) with x = 0.8, and ld (large-down) with x = −0.8 for the four cases studied. The sign of the variation of the coupling between the sites describes a positive or a negative cooperation in the propagation of the perturbation wave. This means that the coupling between the two sites has been changed as a function of the fraction of perturbation wave at the starting site of the i → i + 1 transition, increasing or decreasing it because of what we want to describe.with Analyzing a, Aa, and Ba to the characteristics of the system, also for these non-Markovian cases, the fitting of these parameters was made according to the number of sites . In these cases, for the increase of coupling as a function of the perturbation, the precision of the fitting is similar to that of su, sd, lu, ld) are reported in In order to connect the three parameters ta, Aa, Ba.and F = ta, Aa, Ba vary in an almost linear way with the number of sites. Therefore, these three parameters can also be used to estimate those of a more complex system.We want to underline that, in the case of non-Markovian sequential coupling with a transition probability between two adjacent sites depending on perturbation (Eq. 10), the three parameters tSequential coupling systems can be complicated in two ways to get closer to real situations. In the first case, we can add and/or remove randomly during the perturbation dynamics smaller couplings that bond spatially distant sites. This represents, for example, the case of a protein where some hydrogen bridges connecting distant sites have formed and then been destroyed. In the second case, it is possible to simulate a random interaction of the system with the environment that occurs while the perturbation wave moves inside the protein. This can randomly generate new non-sequential couplings (likewise the first case considered) or randomly modify the sequential couplings between the sites.Physically, these modifications reflect the effect of the protein environment that dynamically modulates the site characteristics. All information about the system-bath interaction of each site is contained in its corresponding spectral density and in our model can be related to the reorganization energies of sites and/or the changes of couplings among them.In From the analysis of 1.The system is largely resistant to a large random perturbation (40%) of the couplings. In fact, a perturbation of 20% of the amount of the sequential coupling in this case changes only a little both the arrival time and the broadening (result not reported) of the wave in the active site . It is n2.A large random perturbation of the amount of the couplings leads to a reduction of the amplitude and to a larger broadening of the signal arriving to the active site. These effects increase when the random perturbation is more frequent.3.The time to reach the active site is not very sensitive even to large random perturbations.This model of the allosteric effect brings together the states of the system in different groups and then the groups of states are coupled. A set of states can be grouped in different ways, but in QM no specific difficulties arise if, with a model in mind, we compute the coupling of the states inside the groups and among the groups. For example, we can form a group of states with the vibrational states of each electronic state (vibronic states). In this case, it is possible to consider the electronic and the vibrational characteristics of allostery and also discriminate between energetic (E) and entropic (S) driven phenomenon in our approach. In a quantum oscillator, in fact, these two terms can be related to the partition function as follow:BT) and Q(β) the partition function and kB and T, Boltzmann constant and temperature, respectively. When the vibrational component of the vibronic states are equal to those of the harmonic oscillator of frequency ω, the partition function is:with β = 1/ of 25 k Dalton, the energetic term generates an allosteric phenomenon with a time smaller than approximately two orders of magnitude compared with the entropic term.The reason for considering groups of states and following their population over time is that, in this way, we can approximate complex systems with simpler ones, and we can study the dynamics of the former in an approximate way. Let us now consider the systems of These figures show a 16-site system and a 22-site system with non-sequential couplings, respectively. In these figures, the splitting and joining of states in the passage from a region to another does not take place in the energy field, but in the network field. In fact, they represent alternative paths in the network, but all the states considered have been taken at the same energy. Our goal is to study the dynamics of these two complex systems with that of two sequentially coupled systems of 5 and 7 degenerate states, those indicated in red in the figures (together with the states |i> and |a>).The systems of Recently, some authors  have useIn order to compare the dynamics of the 16-site system and the 22-site system with those of the 5-site and the 7-site systems where,for all pairs of states |i> and |j>, the effective couplings between states of two adjacent regions in the 16-site system are assumed to be:and in the 22-site system:In From h = 1, while among the groups, we have considered two cases (for both the coupling is 0.1*h): (a) each state of a group is coupled to all states of the neighboring groups; (b) only the last state of a group is coupled to the first state of the neighboring group. These two different types of coupling among the groups can be related to different models. In the first case, the ideal model is that of weakly interconnected groups; in the second, a sequential coupling in the groups is continued, from the last state of each group to the first state of the next group, with a weak coupling. To study a model system following this approach, here we have considered the case of a 20-state system divided into four groups of five states. Within a group, the coupling is sequential with In In In this article, we have considered the allosteric phenomenon with a time-dependent quantum approach. To our knowledge, it has been the first time that such an approach has been applied to the study of the perturbation movement in the allosteric process. Quantum approach was used by Davidov to study the transport of energy by moving his soliton along the protein chains . DavidovIn particular, we have described the allosteric process, considered as a propagation of the perturbation wave from the sensor to the active site, through three parameters of the first arrival in the active site: the arrival time, the amplitude and the broadening of this wave of perturbation in the active site, and through the transfer efficiency of the perturbation at a much large time than the time of first arrival.Quantum mechanics has been widely applied to another fundamental macromolecular process: light-harvesting process in photosynthesis. The similarities between these two processes are evident: both require a site where the perturbation arrives and a second site (dozens of Ångströms away) where it produces an effect.The long distance between the two sites and the presence of infinite degrees of freedom in a macromolecule make it necessary to explain the specificity of the propagation in both processes. It cannot be explained, in fact, by a master equation in terms of its spreading to all sites because, in this case, the process efficiency will be small. On the other hand, it is well known that the light-harvesting process is very efficient in transferring energy from the sensor site to the reaction site. Also in allostery we must disregard lower efficiency if we want to explain the specifics of a macromolecule site and understand that plays an allosteric function with regard to the active site.Firstly, in this article we have compared our quantum approach to a stochastic one that uses a master equation for studying the evolution of the perturbation in the network of coupled states. In this case, we have analyzed the time-dependent evolution of the perturbation and we have shown that a quantum approach is suitable to follow the movement of the perturbation among the sites. The master equation, on the other hand, may have difficulties in reproducing the movement of the perturbation site-by-site or in identifying its possible paths in the network. Moreover, a master equation approach redistributes the perturbation in all sites equally, breaking the preferential relationship between the allosteric site and the active site. We then applied our approach to different types of model systems. In particular, we have studied four types of models:(a)A case in which allosteric perturbation does not modify the couplings between the sites  during its movement in the macromolecule. This is a Markovian model.(b)A second case in which the arrival of the perturbation modifies the couplings between the sites slightly or significantly, both increasing the couplings and reducing them. This is a non-Markovian model.(c)Another case in which the environment changes the couplings between the sites randomly.(d)A last case in which we grouped the states and then coupled these groups of states to each other.In our study we attempted to draw general conclusions:1.For a Markovian SC system, the perturbation wave moving from the start site to the arrival site is of a Gaussian type and the three Gaussian parameters  are sufficient to describe the allosteric process of the first arrival to the active site. This is true both for linear SC systems and for multipath SC systems, estimated with a simpler case of the principal path. If the chemical process in the active site (not included in this description) is much slower compared to the first arrival time of the perturbation wave in this site, we must consider the average population of the active site, related to the efficiency of this allosteric process. We can link all these parameters to characteristics of the system .2.When the coupling between the sites is time-dependent and related to the fraction of the perturbation wave that reaches a site, in this non-Markovian SC system the variation of the allosteric process depends both on the amount of the variation of the couplings and of its sign.3.The system is resistant to a small random perturbation  during the propagation of the signal from the sensor to the active site. This means that the dynamics of this kind of system are mainly determined from the inside characteristics of the system itself than from the environment. A large random perturbation of the system alters the allosteric dynamics according to its frequency. The main characteristics changed are the amplitude and the broadening of the perturbation wave in the active site, while the arrival time is less sensitive. This random variation of coupling deserves further consideration. In our paper we have used the sequential coupling among sites with a constant value as a useful model to study more complex cases. The fact that a random perturbation, which modifies the coupling and/or creates new non-sequential couplings, does little to change the system dynamics tells us that our simple model can be used to predict the dynamics of more complex systems.4.The allosteric effect treated with a model of sets of coupled sites states. In particular, we have considered two model systems, a 16-site system and a 22-site system, with the aim of studying their dynamics with the help of two sequentially coupled systems of five and seven states. With regard to the time and amplitude of the perturbation wave in the active site, we find that the simplified analyzes are very good approximations.Furthermore, we studied two other model cases. If each state of a group is slightly coupled to all states of the neighboring group, the perturbation is distributed on all states of the system and the amplitude that arrives to the active site is small. On the other hand, if it is possible to think the neighboring groups of states coupled only by their final/initial state, the perturbation wave arrives into the active site with considerable amplitude. In both cases, the arrival times to the active site are significantly increased, compared to the SC case, and there are two characteristic periods of oscillation.From this theoretical study of the main characteristics of the allosteric process, it is possible to draw some general remarks to deal with a real system that includes this process. First of all, it is necessary to identify the macromolecular parts which are involved in this process and that can be schematized as coupled sites. This can be done, as it is now, with molecular mechanics simulations. Once the sites are identified, their electronic and vibrational states and their couplings should be calculated with an accurate quantum calculation using a Born-Oppenheimer approximation and through spectroscopic techniques. We should then model the system with a set of coupled sites and find the preferential paths which lead from the allosteric site to the active one; finally, we should follow the dynamics using a quantum approach as shown in our work. We also point out that the quantum approach allows us to take into account the constructive interference among the different paths that connect the allosteric site to the active one. These quantum coherences can play a decisive role in complex systems with many paths. It is also not possible to consider the interference between the paths.A last general question can be posed. If there is such a perturbation that moves along proteins in the allosteric process, is it possible to measure its speed of propagation experimentally? The answer seems to be positive. On the basis of recent pump-probe-type experiments, a conformational transition, similar to the one occurring upon ligand binding, can be found on a picosecond timescale by a laser pulse . What isFinally, we want to return to the more general problem of a complex network of sites with interactions of different types  among the components. These systems can form both macromolecules and less aggregated entities called supra-systems. In our opinion, the dynamic of these supra-systems should be studied with QM. Only a quantum approach allows us to understand the movements within this kind of weakly bonded network and allows us to study the preferential passage of a perturbation among their different sites. The idea that some biological processes, normally intended to function in a classical thermodynamic limit, may be able to use states and interferences of mechanical-quantum superposition is intriguing and challenging.All datasets generated for this study are included in the article/supplementary material.The author confirms being the sole contributor of this work and has approved it for publication.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Inhibitory control, or inhibition, is one of the core executive functions of humans. It contributes to our attention, performance, and physical and mental well-being. Our inhibitory control is modulated by various factors and therefore fluctuates over time. Being able to continuously and unobtrusively assess our inhibitory control and understand the mediating factors may allow us to design intelligent systems that help manage our inhibitory control and ultimately our well-being.The aim of this study is to investigate whether we can assess individuals’ inhibitory control using an unobtrusive and scalable approach to identify digital markers that are predictive of changes in inhibitory control.We developed InhibiSense, an app that passively collects the following information: users’ behaviors based on their phone use and sensor data, the ground truths of their inhibition control measured with stop-signal tasks (SSTs) and ecological momentary assessments (EMAs), and heart rate information transmitted from a wearable heart rate monitor (Polar H10). We conducted a 4-week in-the-wild study, where participants were asked to install InhibiSense on their phone and wear a Polar H10. We used generalized estimating equation (GEE) and gradient boosting tree models fitted with features extracted from participants’ phone use and sensor data to predict their stop-signal reaction time (SSRT), an objective metric used to measure an individual’s inhibitory control, and identify the predictive digital markers.P=.005), the mean duration between 2 consecutive phone use sessions (P=.02), the change rate of battery level when the phone was not charged (P=.04), and the frequency of incoming calls (P=.03). The top digital markers negatively associated with SSRT include the standard deviation of acceleration (P<.001), the frequency of short phone use sessions (P<.001), the mean duration of incoming calls (P<.001), the mean decibel level of ambient noise (P=.007), and the percentage of time in which the phone was connected to the internet through a mobile network (P=.001). No significant correlation between the participants’ objective and subjective measurement of inhibitory control was found.A total of 12 participants completed the study, and 2189 EMAs and SST responses were collected. The results from the GEE models suggest that the top digital markers positively associated with an individual’s SSRT include phone use burstiness (We identified phone-based digital markers that were predictive of changes in inhibitory control and how they were positively or negatively associated with a person’s inhibitory control. The results of this study corroborate the findings of previous studies, which suggest that inhibitory control can be assessed continuously and unobtrusively in the wild. We discussed some potential applications of the system and how technological interventions can be designed to help manage inhibitory control. Inhibitory control, or inhibition, is the ability to inhibit prepotent responses to goal-irrelevant stimuli. It is one of our executive functions and is essential for sustained attention , workingAlthough some individuals have lower inhibitory control than others, our inhibitory control is not a stable trait. Instead, it fluctuates over time based on internal and environmental factors, such as our moods, activities, and the context or the surrounding environment we are in. For example, it has been shown that physical activities  and goodAs the ebbs and flows of inhibitory control consequently influence an individual’s daily behaviors, such as attentional ability , alcoholSome of the aforementioned factors that modulate inhibitory control, such as locations visited and physical activities, can be captured with smartphones or wearables. Behavioral patterns that are associated with changes in inhibitory control may also manifest themselves in people's phone sensor data, such as their phone use patterns. Some previous work suggested that phone sensor data can be used to predict an individual's cognitive performance . HoweverTo this end, the goal of this study is to investigate measuring inhibitory control using a scalable approach within a broader population in addition to people who have problems with inhibitory control and markers indicative of changes in inhibitory control. We developed an iOS app and conducted a 4-week in-the-wild study to collect users’ phone sensor data, HRV, and subjective and objective measurements of their inhibitory control. We used the data to infer the changes in participants’ inhibitory control and to examine the factors that were associated with the changes. The contributions of this paper are 3-fold:To the best of our knowledge, this is the first in-the-wild study that collected participants’ inhibitory control with objective and subjective measurements along with their phone sensor data and HRV. We also made our code for our InhibiSense publicly available on Github  so that We identified how different behaviors and contexts influenced people’s inhibitory control in the wild based on more than 1100 measurements collected from this study. We also used these predictors as features to train machine learning models to determine whether we could predict high and low inhibitory control.Finally, we discussed the implications of the study and several applications of the system that can help us manage our behaviors and well-being in real-world scenarios.We developed InhibiSense, an iOS study app that collects participants’ phone sensor data, HRV, ecological momentary assessment (EMA), and performance of SSTs . The colInhibiSense collected users’ phone sensor data continuously using the AWARE framework . The AWAHRV calculation is based on a person’s interbeat intervals, or RR intervals. We collected participants’ RR intervals using Polar H10 . The RR ”  to 6 (Very much so) using a slider.EMA is a method to repeatedly sample participants’ experiences in real time during a study . We empl” . ParticiSSTs are valid and widely used objective measurements for assessing inhibitory control . They tyL or R was shown with equal probability, and a participant had to tap the left or the right side of the screen, respectively. In a stop trial, the go signal was replaced with a stop signal X after an SSD, and a participant had to inhibit their response immediately. To estimate a participant's stop-signal reaction time (SSRT) more accurately, we used the tracking method (the one-up-one-down method). The initial SSD was set to 250 milliseconds and then was increased by 25 milliseconds in the next stop trial after a successful stopping and was decreased by 25 milliseconds after an unsuccessful stopping.Our implementation of SST  consisteThe study lasted for 4 weeks. During the study, participants received push notifications for answering the EMA questions and completing the SST every hour, from 7 AM to 11 PM. Each task would expire in an hour. They were encouraged to wear the heart rate monitor for the entire day and were able to proceed with the task only if they were wearing it.The study was approved by the Institutional Review Board at Cornell University. We used convenience sampling for our recruitment. During the onboarding meeting, participants were asked to fill out a demographic questionnaire, including their age range and sex, and a Barratt Impulsiveness Scale (BIS 11) questionnaire . Then, tOur goal was to assess an individual’s inhibitory control at a more granular level, specifically every hour. As such, we needed to use the information on the context and physiological signals during the past hour to infer a participant’s inhibitory control at the end of the 1-hour window. We extracted passive sensing features from the phone sensor data and HRV features from their RR intervals during the 1-hour window before a participant started an SST. Specifically, the features from phone sensor data capture information on factors that are associated with changes in inhibitory control, including activities, phone use, surrounding environments, and sleep. We will describe our data preprocessing and feature extraction below.self-control, also known as trait inhibitory control. We added up the scores for the 6 questions and used it as the baseline inhibitory control.The questions asked in BIS correspond to different order factors (dimensions). Specifically, 6 out of the total 30 questions correspond to order factor 2).Activity can be broken down into the following categories: total number of steps, distance traveled, activity type, and acceleration. We computed the percentage of time an individual was stationary, walking, running, automotive, cycling, and nonstationary  during each 1-hour window based on the inferred activities provided by Apple’s activity recognition API . For accWe computed phone use–related features as follows:Phone use sessions: On the basis of screen unlock and lock events, we computed the burstiness (the number of phone use sessions), the number of short sessions (sessions less than 30 seconds) , the totCall: The number and the mean duration of incoming and outgoing calls.Battery: The frequency and percentage of time a user’s phone was being charged and the rate of battery level change when the phone was and was not being charged, respectively.We computed environment-related features as follows:Ambient noise: We computed the mean, median, and standard deviation of the frequency, loudness (dB), and power, namely root mean square, of ambient noise.Location: We used the GPS coordinates and Foursquare API  to retriWi-Fi: We computed the similarity between the Wi-Fi device names (SSID) in 2 consecutive windows using the Jaccard, Dice, Second_Kulczynski, and Ochiai distance metrics. The presence of the individual SSIDs during a window was encoded as a binary vector, and the similarity metrics were then applied to the vectors.Network: The percentage of time a user's phone was connected to the internet and the percentage of time the phone was connected to the network through a mobile and Wi-Fi network, respectively.Sleep duration, sleep onset, and sleep offset were inferred based on participants’ phone use . Sleep dInformation on the day of the week, weekday or weekend, and the hour of the day was obtained based on the timestamps.The features extracted for each type of sensor data are summarized in Physical activityMean and standard deviation of accelerationNumber of stepsDistance traveledPercentage of time a participant was stationary, walking, running, automotive, cyclingPhone usePhone use burstinessFrequency of short phone use sessionsMean and standard deviation of phone use durations (seconds) and intervals between consecutive phone use sessionsCallFrequency and duration (seconds) of incoming and outgoing callsBatteryFrequency and duration (seconds) of chargingMean change rate of battery level (units/second) when the phone was charged and not chargedGPS locationLocation categoryJaccard, Dice, Second_Kulczynski, and Ochiai coefficients of location similarityWi-FiJaccard, Dice, Second_Kulczynski, and Ochiai coefficients for Wi-Fi service set identifier (SSID) similarityAmbient noiseMean and standard deviation of frequency (Hz), loudness (dB), and sound root mean square (RMS)NetworkPercentage of time the phone was connected to the internet through a Wi-Fi or mobile network or was not connectedTimeWeekend or weekdayDay of the weekHour of the daySleepSleep onset and offsetHours of sleepInterbeat intervalMean and standard deviation of heart rateStandard deviation of NN intervalsStandard deviation of 5-min average NN intervalsRoot-mean-squared NN interval differencesStandard deviation of NN interval differencesTriangular indexRelative power of low (0.04-0.15 Hz) and high (0.15-0.4 Hz) frequency band computed using Fast Fourier transform (FFT), Lomb-Scargle periodogram, and autoregressive methodTo ensure the quality of the HRV features, HRV features were extracted if and only if an interval contained at least 5 min of continuous RR interval recordings. For windows that met the criteria, we first removed outlier and ectopic beats  and replnth RT (the RT at the nth percentile) among the RT distribution in that session, where n is equal to ×100). Then, we used linear interpolation to find the SSD where p equals 0.5. Finally, the central SSRT in that session was obtained by subtracting the SSD from the 50th RT.SSRT is the amount of time it takes for a participant to inhibit their response, which is inversely correlated with one’s inhibitory control. A longer SSRT means lower inhibitory control and vice versa. We used the integration method based on the horse race theorem  to compuWe excluded data points in which the stop unsuccessful rate was below 0.25 or above 0.75 and the estimated SSRT was below 50 milliseconds or above 1500 milliseconds [To investigate how the different constructs of inhibitory control in the EMA are related to each other and how people perceive their own inhibitory control in comparison with their inhibitory control measured using SST, we computed the correlation coefficients between the different dependent variables using repeated measures correlation (RMCORR) ,52. RepeTo examine the predictive features for inhibitory control, namely, features that have significant main effects for estimating SSRT, we fitted 2 generalized estimating equation (GEE)  models wIn our analysis, the independent variables were the sensor features and their inhibitory control baseline derived from their BIS responses; the dependent variable was SSRT. We chose SSRT to be the dependent variable because we assumed that an individual's inhibitory control is a continuum. Before fitting the models, we first removed features whose collinearity was above the variance inflation factor threshold  and thenAfter determining which features are more predictive of changes in inhibitory control, our next research question was whether we can automatically infer the state of a person’s inhibitory control using these features. To this end, we trained gradient boosting tree (GBT)  classifiA total of 22 participants signed up for the study; 10 of them dropped out in the middle of the study due to their school workloads. The remaining participants . Conversely, none of the EMA questions had significant RMCORR with SSRT, except that the responses for the EMA question “I am having trouble pulling myself together” had a marginal positive correlation with SSRT . No statistically significant RMCORR between the aggregated EMA score (the scores for the negatively worded questions were first inverted before being added together) [The distribution of participants’ responses to the individual EMA questions and their SSRT are shown in ogether)  and SSRTP=.001), that is, lower trait inhibitory control (higher self-reported score in BIS) was associated with higher SSRT (lower inhibitory control). Regarding activity, the standard deviation of acceleration had a significant negative effect ; larger changes in one’s movement were associated with higher inhibitory control. As for phone use, phone use burstiness , the mean duration between 2 consecutive phone use sessions , the change rate of battery level when the phone was not charged , and the frequency of incoming calls  had significant positive effects on predicting SSRT (associated with lower inhibitory control), whereas the frequency of short phone use sessions  and the mean duration of incoming calls  had significant negative effects (associated with higher inhibitory control).Specifically, in terms of internal factors, individual inhibitory control baseline had a significant positive effect on predicting SSRT , the mean decibel level of ambient noise , the percentage of time in which the phone was connected to the internet through a mobile network , the Jaccard similarity coefficient for the locations visited during the past 2 hours , and whether an outdoor recreational place was visited during the past hour  had significant negative effects on predicting SSRT (associated with higher inhibitory control), whereas whether a participant was in a residential building in the past hour  and the hour before  had significant positive effects (associated with lower inhibitory control). The GEE model had an R2 score of 0.22.With regard to surrounding environments, the standard deviation of the frequency of ambient noise . After data cleaning, we had a total of 577 data points. For comparison, we applied the same set of analyses as we did for phone sensor features. After removing features that had high collinearity, the features that were included in the analysis were the standard deviation of heart rate, the standard deviation of the average NN intervals during each 5-minute segment (SDANN), the root mean square of the successive differences (RMSSD), and the relative power of high frequency (HF) bands.To include HRV features for our analysis, we first removed data points that contained missing or incomplete HRV features (details about data preprocessing are given in the P=.01), that is, lower trait inhibitory control was associated with lower inhibitory control. Regarding activity, the standard deviation of acceleration had a significant negative effect on predicting SSRT , whereas the percentage of time being stationary had a significant positive effect . In other words, larger changes in one’s movements were associated with higher inhibitory control. With regard to phone use, the mean phone use duration  and the mean change rate of battery level when the phone was not charged  had significant positive effects on predicting SSRT (associated with lower inhibitory control), whereas the mean duration of incoming calls  and the number of short sessions  had significant negative effects (associated with higher inhibitory control). With regard to surrounding environments , the standard deviation of the frequency of ambient noise , the percentage of time the phone was connected to the internet through a mobile network , and the Jaccard coefficient of location similarity between 2 consecutive hours  had significant negative effects (associated with higher inhibitory control). In addition, we found that hours of sleep  and the standard deviation of heart rate  also had significant effects on predicting SSRT. The estimated coefficients for the individual features are summarized in R2 score of 0.25. Similarly, we trained GBT classifiers to examine whether adding additional HRV features could improve the performance of the models. With leave-one-subject-out cross-validation, the mean AUC-ROC slightly improved to 0.62.The predictive features after HRV features were taken into account are similar to the predictive features given by the GEE model fitted with phone sensor features. Specifically, individual inhibitory control baseline had a significant positive effect on predicting SSRT  had a positive main effect on predicting an individual’s inhibitory control, which suggests that short phone use sessions might be less detrimental than longer ones. We suspect that the short and longer phone uses might be associated with different types of phone interruptions: endogenous interruptions  and exogenous interruptions , respectively . An examThe estimated coefficients for the different types of environments show that environments also play an essential role in predicting an individual’s inhibitory control. Specifically, outdoor environments, state parks for instance, had a positive association with an individual’s inhibitory control, whereas being in a residential building, such as being at home or in a dorm room, had a negative association with their inhibitory control. Moreover, the mean battery change rate when the phone was not charged and the percentage of time a user’s phone was connected to a mobile network, both of which are indicators of whether a user was outdoors, also had positive associations with their inhibitory control. Taken together, the results suggest that higher inhibitory control might be associated with people spending more time outdoors, which corroborates the findings of previous studies ,61,62. FAccording to the literature, heart rate and HRV features, particularly the relative HF power, have been found to be related to different levels of inhibitory control. We collected participants’ interbeat intervals over the course of the study. However, the standard deviation of heart rate was the only feature that was found to have a significant main effect on predicting the participants’ inhibitory control. We suspected that this may be due to some confounding factors, which were difficult to control during in-the-wild studies, such as the environment, posture, and physical activity . For exaStudies have shown that sleep plays an essential role in inhibitory control . HoweverP=.02) in College and University locations. Also, in professional places , the responses for the EMA question “I am full of willpower” also had a slight negative correlation with SSRT . In other words, on campus or in the workplace, people might feel that it was harder for them to pull themselves together or had less willpower when their inhibitory control was lower. On the contrary, in outdoor or recreational places, the responses for the EMA question “I have no trouble bringing myself to do disagreeable things” had a positive correlation with SSRT . In nightlife spots , the responses to the EMA question “I am having trouble paying attention” had a marginal negative correlation with SSRT .In this study, we employed both subjective and objective measurements to assess and compare participants’ responses for both types of measurements. We did not find any significant correlation between their subjective and objective responses. The distributions of different participants’ EMA responses were very distinct, whereas the distributions of their responses for objective measurements, SST, were relatively similar. When we looked at the correlations by location, we found that the responses for the EMA question “I am having trouble pulling myself together” had a mild positive correlation with SSRT  there might be some publication bias in the earlier literature that overestimated the correlation between self-reports and inhibitory control tasks [The little correlation between subjective and objective measurements was also observed in some recent studies. Some plausible explanations for the inconsistency suggested by the researchers include (1) questions in self-report scales may capture multiple constructs rather than inhibitory control alone, whereas inhibitory control tasks typically measure a narrower construct; (2) behavioral inhibition tasks, including SSTs, were designed to minimize between-subject variability to achieve high reliability (which corroborates our findings that the distributions of participants’ SSRTs were fairly similar), a phenomenon called reliability paradox, which limits the tasks’ ability to capture individual differences; (3) what inhibitory control tasks measure might be participants’ ol tasks -71. In aol tasks . The gapIn summary, we identified several markers that can be used to infer inhibitory control in the wild. However, these are only a subset of markers that manifest the effects of the factors modulating an individual's inhibitory control. There is information on other factors, such as diet, stimulant intake, and exposure to natural light, which is difficult to directly and accurately measure using smartphone sensors, which in turn limits the performance of our prediction models. That said, being able to track patterns that are associated with inhibitory control is still of great use in many real-world applications, such as productivity management technologies. Showing statistics about users’ phone use patterns, physical activities, location patterns, and inferred levels of inhibitory control during the past few weeks can provide insights into how their inhibitory control fluctuates over time. As such, users can adjust their behaviors to change the pattern of their inhibitory control. Alternatively, they can schedule their daily tasks according to their inhibitory control patterns so that they work on tasks requiring a great level of attention when they are in states of high inhibitory control.Previous studies have shown that there are contexts in the workplace where people are more susceptible to distractions. For example, people are more likely to cyberloaf after returning from a break . TherefoThere has been growing awareness of digital well-being and phone addiction. Android and iOS have included screen time to help users restrain the amount of time they spend on their phones and prevent excessive phone use. However, prolonged phone use time is just one of the behaviors  that indIn addition to excessive phone use, there are other types of addictive behaviors, such as food addiction and substance dependence, which also result from reduced inhibitory control. Detecting the downward trends in those individuals’ inhibitory control and the signatures and contexts that are associated with the decrease in their inhibitory control can prevent them from falling off the wagon by suggesting that they take some preemptive steps. For example, a system can track whether a user is in an environment that is likely to trigger alcohol craving based on the type of location and the loudness in the surrounding environment  and suggThe level of inhibitory control is also related to which episode an individual with mental disorders is in. For example, studies have shown that patients with bipolar disorder had impaired inhibitory control when they were in their manic and depressive episodes . As suchThe size of our study was relatively small and homogeneous (the participants were mainly recruited from a college campus and were limited to iOS users). Follow-up studies with a large sample size and more diverse demographics will need to be conducted to examine how different demographic factors impact inhibitory control. Other information, such as light and app use, which we hypothesized would be useful for inferring inhibitory control, could not be collected on iOS devices (because the majority of the participants on campus were iOS users). The phone use patterns of users who installed productivity apps might be different from the phone use patterns of those who did not. Besides, outliers (of SST performance and inferred sleep) were removed from our analysis. Further research is needed to investigate whether such extreme values will affect the assessment of inhibitory control. For future work, in addition to phone sensor and heart rate data, we are planning to collect data using other types of wearables, such as fitness or sleep trackers, to obtain more granular information, such as physical activity, sleep quality, and chronotype, to account for the influences of these confounding factors. Being able to account for these pieces of information can potentially improve models’ performance for inferring different states of inhibitory control.This paper presents a preliminary study using mobile and wearable sensor data to assess individuals’ inhibitory control in the wild. To the best of our knowledge, this is the first in-the-wild study aimed to develop a new approach to accessing inhibitory control unobtrusively and continually. On the basis of the data from 12 participants during a 4-week study, we found that factors including an individual’s inhibitory control baseline, the number of phone use sessions, the change rate of battery level, the mean duration of incoming calls, physical activity, the mean frequency and loudness of ambient noise, whether a user was outdoor, and the types of locations a user visited had significant main effects on predicting an individual’s inhibitory control. We also trained a machine learning model to predict whether an individual had high or low inhibitory control at different moments using those features, which achieved an AUC-ROC of 60%. The findings provide some insights into how we can potentially design personalized technologies that can help support and manage productivity in school or the workplace and intervention tools that can prevent individuals with mental disorders such as substance abuse from relapsing or engaging in problematic behaviors.


We present a global ice sheet reconstruction for the past 80 000 years, called PaleoMIST 1.0, constructed independently of far-field sea level and δ18O proxy records. Our reconstruction is compatible with LGM far-field sea-level records without requiring extra ice volume, thus solving the missing ice problem. However, for Marine Isotope Stage 3 (57 000-29 000 years before present) - a pre-LGM period - our reconstruction does not match proxy-based sea level reconstructions, indicating the relationship between marine δ18O and sea level may be more complex than assumed.The evolution of past global ice sheets is highly uncertain. One example is the missing ice problem during the Last Glacial Maximum  – an apparent 8-28 m discrepancy between far-field sea level indicators and modelled sea level from ice sheet reconstructions. In the absence of ice sheet reconstructions, researchers often use marine  The configuration of past ice sheets, and therefore sea level, is highly uncertain. Here, the authors provide a global reconstruction of ice sheets for the past 80,000 years that allows to test proxy based sea level reconstructions and helps to reconcile disagreements with sea level changes inferred from models. All of these reconstructions use assumptions on far-field sea level as part of their tuning strategy. In the period prior to the LGM, proxy records based on oceanic δ18O fluctuations are often used to tune reconstructions when there is an absence of paleo sea-level data4.Glacial isostatic adjustment (GIA)-based ice sheet reconstructions are commonly used for paleoclimate modeling and assessing present-day Earth deformation and sea-level change. As a result, it is essential that the modeled history of the ice sheets adheres to local geological and geophysical constraints as much as possible. Prior reconstructions of ice sheets have been created by using methods that try to fit GIA constraints without regards to ice physics7 shows that during part of MIS 3, ice-free conditions may have existed. Under this interpretation, the climatic conditions in this area were favorable to allow the growth of forests, with a climate that was potentially analogous to present8. This would indicate that not only was the Laurentide Ice Sheet reduced in size but it also had to be far enough removed from southern Hudson Bay to not strongly affect the climate there. Pico et al.9, through GIA modeling, provided additional support for a reduced extent Laurentide Ice Sheet to explain high relative MIS 3 sea-level indicators along the eastern coast of the United States. The dating methods used for inferring reduced ice sheet extent are near the limit of their reliability during mid-MIS 310. If regarded as minimum ages, then these deposits could be from an earlier ice-free period, such as the last interglacial.Recent work in the Hudson Bay region in North Americaδ18O values based on empirical relationships between ice volume (and therefore sea level) and δ18O but is consistent with some of the sea-level indicators and prior GIA studies12. From our results, we propose that these relationships of δ18O proxy records to sea level and ice volume are not valid.Our reconstruction, called PaleoMIST 1.0 , is created independently of indirect proxy records and far-field sea-level records. This allows us to investigate two of the most contentious problems when assessing past ice sheet configuration and sea level. First, our reconstruction can achieve the sea-level lowstand observed in many far-field locations at the LGM. Since our model adheres to ice physics, geological observations, and local relative sea-level change, we consider it to be a plausible depiction of the ice sheet configuration at the LGM. Therefore, the origin of the long-debated missing ice problem was likely from the starting assumptions on where ice was distributed and the Earth rheology model, while achieving the far-field sea-level lowstand is a nonunique problem. Second, the ice volume in our reconstruction is unable to match the pre-LGM 14, we tune the reconstruction in a way that adheres to ice sheet physics, albeit in a very basic way, and obeys constraints on ice margin location and flow direction indicators. Our aim is to avoid using far-field proxy and sea-level records, instead of relying on direct geological evidence of ice sheet evolution and near-field sea-level data. This allows us to assess whether or not there truly is missing ice at the LGM lowstand15, which happened 26,000–19,000 yr BP (years before present), or if this is an artifact of the assumptions (and circularity) used in creating the previous ice sheet reconstructions. Our method also allows us to independently verify if the low δ18O values found in deep-sea sediment proxy records prior to the LGM indeed correspond to large ice sheet volume.In our approach to ice sheet reconstruction16, and preceding the MIS 4  ice advance. Since our reconstruction focuses on general ice sheet configuration changes over the past 80,000 years, we use a relatively coarse time step of 2500 yr BP. This time resolution is insufficient to model near-field Holocene sea-level changes that are sensitive to rapid deglaciation rates but is sufficient for inferring ice sheet configuration prior to the LGM, which has much poorer age control. It is also sufficient to compare with far-field sea-level records that are less sensitive to the exact history of the ice sheets17.Since the GIA signal is dependent on the history of the ice sheet and ocean loading, it is necessary to include a time period prior to our periods of interest, MIS 3  and the LGM. For these purposes, we start our reconstruction at 80,000 yr BP, when there was relatively high global sea level18 . The main philosophy, especially when reconstructing the North American ice sheets, was to maximize the extent of the ice sheet in order to maximize the possible ice volume. As a result, our North American MIS 3 margin extent is generally larger than the recent assessment by Batchelor et al.20. The timing of subsequent events is targeted to coincide with the timing of detrital carbonate deposits in the Atlantic Ocean that are correlated with Hudson Strait sourced Heinrich Events21. The extent of the ice margin in Hudson Strait is set to reach a maximum at the timing of the event, and subsequent time steps showing a temporally local minimum extent. In the lead-up to the MIS 4 maximum, which we have set to coincide with Heinrich Event 6 , Quebec-centered ice advanced over Hudson Bay, culminating with a merger of the proto-Keewatin dome. The Keewatin Dome was unlikely to have been large prior to the MIS 4 maximum, as the early ice flow directions in southwestern Hudson Bay are oriented directly westward22. After Heinrich Event 6, we depict a two-dome structure for the Laurentide Ice Sheet, due to the activation of an ice stream in Hudson Strait, and it is set to remain this way throughout most of MIS 3. In the minimal scenario  . We found the best compromise of maximizing ice volume and near-field sea-level observations32 was with a high lower mantle viscosity, which we set to 4 × 1022 Pa s. The Eurasian ice sheets reconstruction was initially tuned to provide a relatively good fit to postglacial sea-level indicators. Specifically, areas near the edge of the ice sheet33 required relatively thin ice at the LGM. The GIA response for areas covered by the Eurasian ice sheets also fit better with our chosen mantle viscosity (4 × 1022 Pa s) than with lower values. Since the GIA response of the Eurasian ice sheets is relatively insensitive to the lower mantle34, this response is likely influenced by the GIA signal from the North American ice sheets35. The Antarctic ice sheets were not tuned to the same extent, as the increase of ice thickness was likely limited at the LGM36. The shear stress values on the Antarctic shelf were kept to a low value in order to prevent the excessive ice thickness on continental Antarctica. Ice volume estimates for the Eurasian and Antarctic ice sheets are consistent with estimates from other ice sheet reconstructions and models15, while the combined Laurentide and Greenland volume is at the lower end of recent estimates.Changes in the geometry of the Laurentide Ice Sheet dominate the contributions to global sea level in the reconstruction Fig. . Due to 38, fluctuations in global sea level were controlled almost exclusively by the Laurentide Ice Sheet .Since the Eurasian ice sheets were generally restricted to mountainous areas during the middle of MIS 3eet Fig. . Previou6 km3) are both substantially less than reported by Lambeck et al.31 . We, therefore, find no basis for the missing ice problem15, as our LGM reconstruction is compatible with existing sea-level constraints, even with a relatively small Antarctic contribution (10 m SLE). Given that our reconstruction does not include components like thermal expansion, groundwater storage, and contributions from smaller glaciers and ice caps (estimated to be 3–4 m SLE15), our estimated ice volume is likely slightly overestimated. The cause of the discrepancy between our results and those of Lambeck et al.31 probably lies in the difference in the starting ice volume that they used in Antarctica. Lambeck et al.31 started their analysis with an assumed ice volume of 28 m ESL, which is much greater than the 10 m SLE value in our reconstruction. Due to gravitational effects, more ice will need to be added to Northern Hemisphere to counteract the extra ice in Antarctica to achieve a good fit to far-field sea level, leading to an overall larger ice sheet volume.The calculated sea level from our reconstruction is able to match the far-field constraints at Barbados, Sunda Shelf, Bonaparte Gulf, and Great Barrier Reef Fig. . Our max42. In this case, it is not so much that our model fixes the missing ice problem, but that the definition of global sea level is model dependent. We recommend that GIA-based ice sheet reconstruction work should not target a specific ice volume or global sea-level lowstand value, but consider each location as independent observations of relative sea level within a framework of sites around the world.An alternative explanation is that the far-field records that show LGM sea-level values below −120 m are not representative of the global average sea level. In our model at the LGM Fig. , much ofδ18O-based sea-level reconstructions during MIS 3 are incompatible with the currently available constraints on ice sheet configuration. Whether this discrepancy can be solved by determining a new relationship between δ18O and sea level, perhaps with the aid of water isotope modeling, or by systematically eliminating current ice sheet extent indicators for MIS 3 should be the focus of future studies. Our reconstruction also demonstrates that there is no consensus on Late Pleistocene ice volume, and we anticipate future refinements, for instance with different Earth rheology assumptions and ice margin histories, will produce different configurations.Our ice sheet reconstruction was created independent of far-field records of sea level. This reconstruction has solved the LGM missing ice problem, and ice volume estimates are consistent with the far-field sea-level drop. Our reconstruction has however created a new missing ice problem, as the 45. This program uses a 1D, spherically symmetric Earth structure. We use a three-layered Earth model with a 120 km elastic lithosphere, an upper mantle with a viscosity of 4 × 1020 Pa s, and a lower mantle with a viscosity of 4 × 1022 Pa s. This set of Earth model parameters is within the range of values used in Eurasian GIA studies46, while the lithosphere thickness and upper mantle viscosity are the same as used in the NAICE North America reconstruction13. The lower mantle viscosity is compatible with a best-fitting solution in far-field regions31, but is higher than values used in prior North American GIA-based ice sheet reconstructions47. The GIA was calculated to a spherical harmonic degree 256, with three iterations to adjust the ocean function due to changes in the ice load. The sea-level change and Earth deformation were calculated on a 1° grid directly from the output Stokes coefficients to be input back into the ice sheet reconstruction. Since the ice sheet calculation was done on a 5 km grid, the 1° grid was projected and interpolated using Delaunay triangulation49. The development of the reconstruction was done in a progressive manner, where the deformation calculated from a previous ice sheet reconstruction was used to construct an adjusted reconstruction. The final ice sheet reconstruction was iterated twice to ensure that there is close correspondence to the ice load and the calculated sea level and deformation. For a data-model comparison of the sea-level indicators, the sea level was calculated at the location of the indicator, rather than grouping all of the indicators into a region. This is required since there can be large regional variations in calculated sea level, even in far-field locations , so for these areas, the sea-level indicators were only used to give a general sense of how well the model performed. The base topography used in this study is RTopo-257. The ice sheets were calculated on a projected coordinate system at 5 km resolution. For input into SELEN, the output ice thickness was converted into a hexagonal grid58 with a resolution of roughly 66 km, which is sufficient for calculating to spherical harmonic degree 256. The workflow is essentially the same as in a previous study13 but is briefly summarized as follows:To reconstruct the ice sheets, we use the software ICESHEET59. (2) The GIA (Earth deformation and sea level) was calculated and the result was added to modern topography to produce the underlying paleotopography. (3) The ice sheet reconstruction was again calculated using the adjusted paleotopography. This step is necessary to do before analyzing sea level because the deformed topography significantly increases ice sheet volume even with identical shear stress parameters60. (4) Calculate the GIA again, and calculate sea level at locations where there are sea-level indicators. (5) Using the comparison of the calculated sea level with the sea-level indicators, we adjusted the basal shear stress up or down, including time dependence , in an attempt to reduce the misfit, while taking into consideration broad-scale ice flow direction features. (6) Using the previously calculated paleotopography, a new ice sheet reconstruction was determined. Steps 4 and 5 were repeated many times until a generally satisfactory fit was achieved. (7) Since prior to the LGM there are few sea-level indicators, we relied more on matching the inferred changes in ice flow direction through a time when creating the reconstruction. The shear stress and margin locations were adjusted to conform to specific geomorphic and geologic information, which is detailed below. (8) After the targetted geological and sea-level constraints were satisfied, we did a second calculation of the reconstruction without changing the shear stress parameters to ensure that there was a minimal mismatch between the underlying deformed topography, GIA response, and the ice load.(1) First, the ice sheet reconstruction was calculated using modern topography using a shear stress model with values based on surface topography and surficial geology. For Antarctica and Greenland, these values were adjusted to recreate the modern ice sheet surface topography, while for North America and Europe, it was based on the previous studies62. Thus, by examining geological indicators of ice flow direction, it is possible to evaluate if the ice sheet reconstruction is realistic. Specifically, for the pre-LGM Laurentide Ice Sheet, we made adjustments to the ice margin and basal shear stress to ensure that the ice sheet surface gradient followed known flow direction shifts22. While this does not directly constrain ice thickness or volume (and is therefore nonunique), it ensures that for our chosen combination of ice margin and shear stress, that geometry remains internally consistent with geological constraints.For the flow direction indicators, we assume that the ice sheet flow will be in the direction of the greatest ice sheet surface gradient66. The margins prior to 20,000 for the Eurasian ice sheets were also based on existing reconstructions67. The main characteristics of the Eurasian ice sheets were that they were limited in extent during MIS 337, and not a significant contributor to sea level during this time. Compared to a previous reconstruction68, the ice extent is less at 40,000–50,000 yr BP, but similar to the minimal extent at 35,000 yr BP. The margins for the pre-LGM period for Antarctica and North America were drawn from scratch for this study, details which are expanded on below. We have also included the Patagonian Ice Sheet66, but it contributes less than 1 m SLE ice volume, even at the LGM. Chronology of Patagonian Ice Sheet fluctuations is based on dated landforms in southern Patagonia69. For time slices that are intermediate of the timing of existing margin reconstructions, we set the margin to be intermediate of them.The ice sheet margins were developed for the past 80,000 years at the 2500 year time steps. The LGM to present interval utilized existing margin reconstructions70. This was likely attained about 30,000 yr BP, which is the time slice we set the maximum extent. In the Ross Sea sector, the margin may have gradually advanced to the shelf edge from some minimum limit in MIS 5e71. This is used as a template for the entire West Antarctic Ice Sheet, which we depict as gradually advancing from an extent that was similar to present at 80,000 yr BP, to a maximum extent at 30,000 yr BP. The East Antarctic Ice Sheet remained close to its present-day extent through MIS 373, so we have used the present-day margin throughout MIS 4 and 3.The pre-LGM constraints on the Antarctic ice sheet are sparse. In parts of the West Antarctic Ice Sheet, the maximum extent at the continental shelf edge may have been reached prior to the LGM21. For this reconstruction, the timing of the Heinrich Event represents a local maximum ice-sheet configuration, and the subsequent time step shows a retreat from that position. The 2500 year time step used in our reconstruction fits conveniently with the roughly 7000-year intervals of Heinrich Events. It must be acknowledged that the margin chronology we propose has uncertainty on the order of several thousand years. We utilized the recent compilation of pre-LGM indicators of ice-free conditions6 to guide the ice extent history for MIS 3. In the minimal scenario, the data that indicate ice-free conditions in Hudson Bay existed are used, while in the maximal scenario, these are ignored. The calculated Holocene sea level within Hudson Bay is not sensitive to the choice of scenario. The maximum extent of the MIS 4 glaciation is set to be 60,000 yr BP, which coincides with Heinrich Event 6, after which there is a retreat of the ice sheet. Retreat of the margin is also set to happen at 50,000, 42,500, and 37,500 yr BP, which are after Heinrich Events 5a, 5, and 4. The MIS 3 deglaciation of Hudson Bay in the minimal scenario is set to be at Heinrich Event 5 rather than 5a, due to the fact that the radiocarbon ages are finite (which would not be the case during 5a), and that this event was likely short lived enough that it did not produce recognizable erosional features in far-field regions. In the maximal scenario, the ice margin south of Hudson Bay is set to remain in northern Ontario and southern Manitoba, with fluctuations that are coincident with the Heinrich Events. In this scenario, it is assumed that the presence of the Laurentide Ice Sheet saddle between the Keewatin and Quebec–Labrador centers remained near the southern end of Hudson Bay throughout MIS 3, which may explain why the pre-MIS 2 sediments were preserved. The western margin of the Laurentide Ice Sheet is set to remain in the vicinity of Great Slave Lake during MIS 3, based on recent work that presents evidence of ice-free conditions west of the lake prior to the LGM74. The Laurentide Ice Sheet did not reach Banks Island during MIS 4 and 375, so the limit is set to be east of there on Victoria Island. The ice sheet did not advance over Lake Superior and into Lake Michigan until near the end of MIS 376, and so this is set to be the southern limit during MIS 4. The ice margin advanced and retreated into southern Ontario and the St. Lawrence Valley several times between 80,000 and 30,000 yr BP. This reflects the relative stability of the Quebec–Labrador dome of the ice sheet. Our margin reconstructions in this region are based on records of these fluctuations28. The MIS 4 extent in the Eastern Canada region is set to reach the shelf edge off the coast of Nova Scotia77. There is some evidence of a possibly brief retreat of the ice sheet in eastern Baffin Island during MIS 378, but we have kept the margin near the edge of Baffin Island throughout MIS 3 and 4. The evolution of the margins in the central parts of the ice sheet was based on the relative chronology of major changes in flow direction81. The margin reconstructions between 30,000 and 20,000 yr BP are described in detail in Gowan60.Fluctuations in sea level for the past 80,000 years are dominated by changes in the Laurentide Ice Sheet, so it is most sensitive to ice sheet margin location uncertainties. Since the error ranges of many of the chronological constraints become large the further back in time they go, the chronology of advance and retreat events was set to coincide with the timing of Heinrich Events82. Prior to the beginning of advance after about 35,000 yr BP, the ice extent may not have been much different than the present, and non-glacial sediments are preserved throughout British Columbia85. In many areas of the Cordillera, the MIS 4 maximum extent was the same or even somewhat larger than the LGM limit91. The MIS 4 glacial maximum for the Cordilleran Ice Sheet happened at about 55,000 yr BP88, which is somewhat later than what we set the Laurentide Ice Sheet (which was based on the timing of Heinrich Event 621).The Cordilleran Ice Sheet history is defined as having major glacial episodes during MIS 4 and MIS 2, with little ice cover during MIS 3. The Late MIS 3 margins are based on the previous reconstructions94. The retreat during MIS 3 was in some places substantial, such as in northeastern Greenland, where the extent is set to be less than present95. The northwestern Greenland Ice sheet also may see a similar retreat during MIS 396. The Holocene margin is based partially on the modern-day extent57, with adjustments in western Greenland to take into account the Holocene thermal maximum94.The Greenland Ice Sheet in MIS 4 was similar in extent as the MIS 2 maximum98. The buildup of this ice sheet may rely on the Greenland Ice Sheet advancing over the Nares Strait, which may not have happened prior to the LGM during the Late Pleistocene99. As a result, we have kept the modern extent ice caps as the margin limit for the pre-LGM period.For the Innuitian Ice Sheet in northern Canada, evidence for pre-LGM late Pleistocene glaciation beyond the modern ice cap limits is lackingAfter finalizing the margins, the basal shear stress was adjusted to increase or decrease the thickness of the ice sheet for modeling sea level and to match large scale indicators of ice flow direction. By increasing the basal shear stress, the ice sheet surface slope is steeper and results in thicker ice. As mentioned above, the main goal was to be able to maximize global ice volume at the LGM, while balancing near-field Holocene data. The initial tuning was purely to achieve the originally inferred global 130-m sea-level drop by increasing the thickness of the core of the Laurentide and Eurasian ice sheets, but achieving this produced unrealistically thick ice sheets . The first sea-level datasets added to tune the basal shear stress, and therefore ice thickness, were within the core regions of the ice sheets, and it became necessary to reduce the shear stress from the initial values that we used.14 and Antarctic ice sheets to reproduce the modern ice-sheet configuration. Secondary control is that areas with continuous sediment cover have lower shear stress than places with discontinuous sediment cover100. This can be seen in the fact that the peripheral regions of the ice sheets were relatively thinner than the core areas to accommodate pro-glacial lake strandline tilts13. The third control on the shear stress was that regions with major ice streams have lower shear stress than surrounding areas. This is particularly true for the Hudson Strait, where the shear stress had to be significantly lower than the adjacent Quebec–Labrador region in order to produce a much thinner ice load within Hudson Bay. During deglaciation, the shear stress values generally needed to be reduced in order to get closer correspondence to the GIA response. This lowering of shear stress is likely a result of increased basal meltwater and warming of the base during deglaciation and increased interactions with pro-glacial lakes and the ocean at the margins. Since topographic variability is the main factor controlling the shear stress, the geographical domains were divided on this basis, to separate mountainous areas from areas with flatter topography. After this step, the regions were further adjusted to account for variability in surficial geology. After some initial tuning, we tested the impact of varying the lower mantle viscosity on postglacial sea level in Hudson Bay. The initial Earth model that was utilized had a viscosity of 4 × 1021 Pa s, but the modeled sea level was much higher than the observations. Tests using different lower mantle viscosity values revealed that values approaching 1023 Pa s reduced the postglacial sea level. Using a higher viscosity value made it possible to have a larger ice load in the center of the Laurentide Ice Sheet, and therefore maximize ice volume, so a value of 4 × 1022 Pa s was chosen for subsequent tuning.Although in most places there are few physical constraints on the thickness of the Northern Hemisphere ice sheets at the LGM, there are some general patterns that can be deduced by looking at the pattern of Holocene sea-level change. First, mountainous areas have much higher shear stress than areas with relatively flat terrain. This was directly observed when we tuned the shear stress values for the Greenland13, which was only tuned on the basis of GIA constraints in the western Laurentide Ice Sheet and using a lower value for lower mantle viscosity, the core region (Hudson Bay and the Quebec–Labrador sector) of the ice sheet is ~1000–1200 m thicker, while some of the peripheral regions, such as the Great Lakes, northeastern United States, and Boothia Peninsula are thinner by 300–500 m. The general two-dome structure with lower elevation within Hudson Bay remains.Compared with our earlier reconstruction42. We have included the plots from Churchill (located near the center of the Laurentide Ice Sheet) and Ångermanland (located near the center of the Eurasian ice sheets) in Supplementary Figs. 21 and 1023 Pa s. This demonstrates that using a lower mantle viscosity toward 1023 Pa s improves the fit of the calculated sea level in ice-covered areas. The second report shows the results from the standard version of PaleoMIST, a 500-year time-step version where the ice load was linearly interpolated, the variant with Hudson Bay remaining ice-covered through MIS 3, plus three commonly used Earth models101. The high temporal resolution version provides a better fit to sea-level indicators in formerly glaciated areas, with lower sea level since the ice load is reduced in a more gradual, realistic way. The available sea-level indicators during MIS 3 cannot distinguish which Laurentide Ice Sheet scenario is more likely (see Supplementary Figs. We have created two reports comparing calculated sea level to sea-level indicators from a global databaseSupplementary InformationPeer Review File