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Rubem P. Mondaini Editor

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics

Rubem P. Mondaini Editor

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018

123

Editor Rubem P. Mondaini President, BIOMAT Consortium – International Institute for Interdisciplinary Sciences Rio de Janeiro, Brazil Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISBN 978-3-030-23432-4 ISBN 978-3-030-23433-1 (eBook) https://doi.org/10.1007/978-3-030-23433-1 Mathematics Subject Classification (2010): 92B05, 97M10, 97M60, 49J20, 93A30 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The present book is a collection of papers which have been accepted for publication after a peer-review evaluation by the Editorial Board of the BIOMAT Consortium (http://www.biomat.org) and international referees ad hoc. These papers have been presented on technical sessions of the BIOMAT 2018 International Symposium, the 18th symposium of the BIOMAT series which was held at the Faculty of Sciences and Technology, University Hassan II, Mohammedia, Morocco, from 29 October to 2 November 2018. On behalf of the BIOMAT Consortium, we thank the members of the BIOMAT 2018 Local Organizing Committee, Karam Allali (chair), Khalid Hattaf, Ahmed Taik, Noura Yousfi, Noureddine Moussaid, Saida Amine, Mustapha Kabil, Chakib Abchir, and Driss Karim, for their professional expertise at following the guidelines and fine tradition of the BIOMAT Consortium and preserving the excellency of the BIOMAT Symposium series on this first BIOMAT Conference in Africa. We are so much indebted to all these colleagues as well as to research collaborators and Ph.D. students of the faculty for their invaluable help since the opening session on Monday morning to the closing session on Friday evening. The financial support in terms of lunches, coffee breaks for all the participants, and accommodation for the invited keynote speakers have been provided by the Faculty of Sciences and Technology. The BIOMAT Consortium has succeeded once more in its fundamental mission of enhancing the interdisciplinary scientific activities of mathematical and biological sciences on developing countries with the organization of the BIOMAT 2018 International Symposium. The authors of papers from Western and Eastern Europe, Africa, and South America had the usual opportunity of exchanging scientific feedback of their research fields with their colleagues from Morocco and other 16 countries: Algeria, Brazil, Cameroon, Canada, Chad, Colombia, France, Germany, Hungary, Italy, Mali, Nigeria, Poland, Portugal, Russia, and Senegal. The editor of the book and president of the BIOMAT Consortium is very glad for the continuous collaboration and critical support of his wife, Carmem Lucia, on the editorial work, from the reception of submitted papers for the peer-review procedure of BIOMAT Consortium Editorial Board to the ultimate publication of the

v

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Preface

scientific programme. He also thanks his former research student Dr. Simão C. de Albuquerque Neto from Federal University of Rio de Janeiro for his computational skills and enlightening discussions during the preparation of our presentations to the BIOMAT 2018. Mohammedia, Morocco November 2018

Rubem P. Mondaini

Editorial Board of the BIOMAT Consortium

Rubem Mondaini (Chair) Adelia Sequeira Alain Goriely Alan Perelson Alexander Grosberg Alexei Finkelstein Ana Georgina Flesia Anna Tramontano Avner Friedman Carlos Condat Charles Pearce Denise Kirschner David Landau De Witt Sumners Ding-Zhu Du Dorothy Wallace Eduardo Massad Eytan Domany Ezio Venturino Fernando Cordova-Lepe Fernando R. Momo Fred Brauer Frederick Cummings Gergely Röst Guy Perriére Gustavo Sibona Helen Byrne Jacek Miekisz Jack Tuszynski Jaime Mena-Lorca Jane Heffernan

Federal University of Rio de Janeiro, Brazil Instituto Superior Técnico, Lisbon, Portugal University of Arizona, USA Los Alamos National Laboratory, New Mexico, USA New York University, USA Institute of Protein Research, Russia Universidad Nacional de Cordoba, Argentina University of Rome, La Sapienza, Italy Ohio State University, USA Universidad Nacional de Cordoba, Argentina University of Adelaide, Australia University of Michigan, USA University of Georgia, USA Florida State University, USA University of Texas, Dallas, USA Dartmouth College, USA Faculty of Medicine, University of S. Paulo, Brazil Weizmann Institute of Science, Israel University of Torino, Italy Catholic University del Maule, Chile Universidad Nacional de Gen. Sarmiento, Argentina University of British Columbia, Vancouver, Canada University of California, Riverside, USA University of Szeged, Hungary Université Claude Bernard, Lyon, France Universidad Nacional de Cordoba, Argentina University of Nottingham, UK University of Warsaw, Poland University of Alberta, Canada Pontifical Catholic University of Valparaíso, Chile York University, Canada vii

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Jean Marc Victor Jerzy Tiuryn John Harte John Jungck José Fontanari Karam Allali Kazeem Okosun Kristin Swanson Kerson Huang Lisa Sattenspiel Louis Gross Ludek Berec Michael Meyer-Hermann Nicholas Britton Panos Pardalos Peter Stadler Pedro Gajardo Philip Maini Pierre Baldi Rafael Barrio Ramit Mehr Raymond Mejía Rebecca Tyson Reidun Twarock Richard Kerner Ryszard Rudnicki Robijn Bruinsma Rui Dilão Sandip Banerjee Seyed Moghadas Siv Sivaloganathan Somdatta Sinha Suzanne Lenhart Vitaly Volpert William Taylor Yuri Vassilevski Zhijun Wu

Editorial Board of the BIOMAT Consortium

Université Pierre et Marie Curie, Paris, France University of Warsaw, Poland University of California, Berkeley, USA University of Delaware, Delaware, USA University of Sao ˜ Paulo, Brazil University Hassan II, Mohammedia, Morocco Vaal University of Technology, South Africa University of Washington, USA Massachusetts Institute of Technology, MIT, USA University of Missouri-Columbia, USA University of Tennessee, USA Biology Centre, ASCR, Czech Republic Frankfurt Inst. for Adv. Studies, Germany University of Bath, UK University of Florida, Gainesville, USA University of Leipzig, Germany Federico Santa Maria Technical University, Valparaíso, Chile University of Oxford, UK University of California, Irvine, USA Universidad Autonoma de Mexico, Mexico Bar-Ilan University, Ramat-Gan, Israel National Institutes of Health, USA University of British Columbia, Okanagan, Canada University of York, UK Université Pierre et Marie Curie, Paris, France Polish Academy of Sciences, Warsaw, Poland University of California, Los Angeles, USA Instituto Superior Técnico, Lisbon, Portugal Indian Institute of Technology Roorkee, India York University, Canada Centre for Mathematical Medicine, Fields Institute, Canada Indian Institute of Science Education and Research, India University of Tennessee, USA Université de Lyon 1, France National Institute for Medical Research, UK Institute of Numerical Mathematics, RAS, Russia Iowa State University, USA

Contents

Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards . . . . . . . . . . . Alexey Tokarev, Nicolas Ratto, and Vitaly Volpert

1

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-Containing Capsids and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adil Meskaf and Karam Allali

23

Dynamics of a Generalized Model for Ebola Virus Disease . . . . . . . . . . . . . . . . . Zineb El Rhoubari, Hajar Besbassi, Khalid Hattaf, and Noura Yousfi

35

Bifurcations in a Mathematical Model for Study of the Human Population and Natural Resource Exploitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. M. Cholo Camargo, G. Olivar Tost and I. Dikariev

47

Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune Response and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sanaa Harroudi and Karam Allali

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Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell Population in the Light of Suns and Stars . . . . . . . . . . . . . . . Y. Elalaoui and L. Alaoui

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The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Driss Kiouach and Yassine Sabbar

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Global Dynamics of a Generalized Chikungunya Virus . . . . . . . . . . . . . . . . . . . . . 107 Hajar Besbassi, Zineb El Rhoubari, Khalid Hattaf, and Noura Yousfi Differential Game Model for Sustainability Multi-Fishery. . . . . . . . . . . . . . . . . . 119 Nadia Raissi, Chata Sanogo, and Mustapha Serhani

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Contents

Towards a Thermostatistics of the Evolution of Protein Domains Through the Formation of Families and Clans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Rubem P. Mondaini and Simão C. de Albuquerque Neto Analysis of Tumor/Effector Cell Dynamics and Decision Support in Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 S. Sabir and N. Raissi Optimal Control of an HIV Infection Model with Logistic Growth, CTL Immune Response and Infected Cells in Eclipse Phase . . . . . . . . . . . . . . . 165 Jaouad Danane and Karam Allali Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Rubem P. Mondaini and Simão C. de Albuquerque Neto Application of the Markov Chains in the Prediction of the Mortality Rates in the Generalized Stochastic Milevsky–Promislov Model . . . . . . . . . . . 191 ` Piotr Sliwka Modelling the Role of Vector Transmission of Aphid Bacterial Endosymbionts and the Protection Against Parasitoid Wasps . . . . . . . . . . . . . . 209 Sharon Zytynska and Ezio Venturino The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic Drugs with Nonlinear Concentration-Time Profiles . . . 231 Tahmina Akhter and Sivabal Sivaloganathan Mathematical Modeling of Inflammatory Processes . . . . . . . . . . . . . . . . . . . . . . . . . 255 O. Kafi and A. Sequeira Modeling the Memory and Adaptive Immunity in Viral Infection . . . . . . . . . 271 Adnane Boukhouima, Khalid Hattaf, and Noura Yousfi Optimal Temporary Vaccination Strategies for Epidemic Outbreaks . . . . . 299 K. Muqbel, A. Dénes, and G. Röst On the Reproduction Number of Epidemics with Sub-exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 D. Champredon and Seyed M. Moghadas Numerical Simulations for Cardiac Electrophysiology Problems . . . . . . . . . . 321 Alexey Y. Chernyshenko, A. A. Danilov, and Y. V. Vassilevski Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory Optimization Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Abdoulaye Mendy, Mountaga Lam, and Jean Jules Tewa A Fractional Order Model for HBV Infection with Capsids and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Moussa Bachraou, Khalid Hattaf, and Noura Yousfi

Contents

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Modeling Anaerobic Digestion Using Stochastic Approaches . . . . . . . . . . . . . . . 373 Oussama Hadj Abdelkader and A. Hadj Abdelkader Alzheimer Disease: Convergence Result from a Discrete Model Towards a Continuous One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 M. Caléro, I. S. Ciuperca, L. Pujo-Menjouet, and L. M. Tine Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards Alexey Tokarev, Nicolas Ratto, and Vitaly Volpert

1 Introduction Blood coagulation is one of the most important defense systems of the organism. This system controls the formation of a polymeric fibrin clot at the sites of vessel wall damage, thus protecting us from blood loss. From the medical point of view, the disruptions of coagulation system can lead to many dangerous states—from bleedings (insufficient hemostatic reaction) to thromboses (redundant hemostatic reaction). Thromboses and bleedings are the leading cause of death in many diseases and conditions such as atherosclerosis, myocardial infarction, stroke, sepsis, cancer, snakebites, frostbites, burns, surgery, as well as hemophilias and other inherited and acquired diseases. That dictates the need in understanding of how this system can be diagnosed and influenced in medical care.

The author “A. Tokarev” was working in the institute “Dmitry Rogachev” at the time of presentation of this work in a session of the BIOMAT 2018 in Morocco. A. Tokarev Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russian Federation N. Ratto · V. Volpert () Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France INRIA, Universit de Lyon, Universit Lyon 1, Institute Camille Jordan, Villeurbanne, France e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_1

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Extensive biochemical, biophysical, and biomedical research is undertaken in this direction during last decades. Any progress on this way needs in-depth knowledge of biochemistry of coagulation and neighbor systems, which are really extremely complex and still have reactions to be discovered and mechanisms to be investigated. Functioning of coagulation on the interface of flowing blood and vessel wall (or inside the growing platelet thrombus) needs application of modern and developing new biophysical approaches. And necessity to understand and consider the patient specificity of coagulation, and final task to push all results into medical practice make all this work biomedical. Thrombin generation is the central process of coagulation as thrombin is a central enzyme of this system (see the Sect. 2 below). Thus, the terms “thrombin generation,” “coagulation,” and “clotting” are often used as synonyms. However, one of the most informative tests of coagulation system in which thrombin concentration is monitored in time in homogeneous conditions is also called thrombin generation [1, 2]. Another approach is to monitor thrombin and fibrin formation in an essentially non-homogeneous, spatially distributed, non-stirred conditions, referred as trombodynamics [3–5]. Finally, a number of experimental assays exists to study coagulation (by monitoring the fibrin or/and thrombin formation) in blood or plasma flow conditions [6, 7]. Needless to say that developing and using of these and all other experimental methods of studying coagulation demand corresponding mathematical models. Mathematical modeling is a well-known and indispensable tool to understand complex biophysical and biological phenomena. Models of various detalization and even completely new computational approaches were developed in this area. Advances in modeling the coagulation alone have been recently reviewed in [8–13], and complex multiscale models of thrombosis which include integrated models of coagulation, platelet aggregation, and blood flow have been reviewed in [14–17]. The main task of the present survey is to show how coagulation models having different background, complexity, and properties can be linked together into one general picture. In most cases, the reaction-diffusion-convection equations are used to describe time- and space-varying concentrations of coagulation substances—zymogens, activated factors, inhibitors, and their complexes: ∂c + (v.∇)c = R(c, k) − ∇(−D∇c) ∂t

(1)

with corresponding initial and boundary conditions. Here, vector notation is used for concentrations, c; rates of chemical reactions, R(c, k); kinetic constants, k; diffusion coefficients, D; and fluid flow velocity field, v; the latter should be specified analytically or be the solution of fluid-phase (for example, Navier–Stokes) equations. In non-stirred conditions v = 0, and Eq. (1) turns into the reactiondiffusion problem. In the fully stirred (i.e., homogeneous) conditions, the diffusion terms vanish, too, so the system (Eq. (1)) turns into ODE system. The level of model detalization determines the dimension and components of the vector c (number

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

3

and list of substances) and the actual form of kinetic functions Ri . The latter are based either on the knowledge/assumptions of detailed mechanisms of biochemical reactions and measured/estimated value of kinetic parameters (in detailed models and in reduced models obtained from detailed ones), on the general reaction scheme and some fitted values of kinetic parameters (in simulation models), or may even be in some unusual, hypothetical form if some system property is proposed and being tested (in phenomenological models).

2 Biochemical Structure of the Coagulation System Coagulation system (see Fig. 1) is a cascade of proteolytic enzymatic reactions with each level consisting of two processes: zymogen (coagulation factor, F) activation to the active enzyme (activated coagulation factor, Fa) followed by its rapid irreversible trapping by inhibitors always circulating in blood. On each level, the short-living coagulation factor catalyzes the reaction of activation on the next cascade level. The final product of coagulation cascade is fibrin (Fn) which rapidly polymerizes into

Fig. 1 General scheme of the coagulation cascade [18]

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3-dimensional mesh (gel); Fn mesh in vivo slows blood flow and glues aggregating platelets and other blood cells into the clot. There are two pathways of coagulation activation [19, 20]. Activation by the extrinsic pathway begins when blood comes in contact with tissue factor (TF): this transmembrane protein is expressed by the majority of cells except those normally being in contact with blood. TF binds with FVIIa, which circulates in tiny amounts (1% of total FVII), making FVIIa able to cleave FIX and FX to their activated forms. Intrinsic activation pathway [21] is initiated by the contact of blood with any “foreign” surface. Upon adsorption on this surface, FXII becomes activated due to conformational changes and then stimulates its own formation both autocatalytically and by activating prekallikrein to kallikrein: kallikrein activates its cofactor high molecular weight kininogen and FXII. Generated FXIIa activates FXI, FXIa activates FIX, and FIXa activates FX. Both pathways unite at the activation of FX to FXa. FXa cleaves prothrombin (FII) to thrombin (FIIa)—the central coagulation enzyme. In addition to cleavage of fibrinogen (Fg) to fibrin (Fn), thrombin controls at least three positive feedback loops activating FV, FVIII, and FXI which are located above in the cascade. Two of these loops lead to the activation of cofactors FVa and FVIIIa which bind with FXa and FIXa, respectively, forming prothrombinase and intrinsic tenase complexes having activities 104 –105 times larger than free enzymes have. Therefore, upon the initial activation by any pathway, local thrombin concentration increases in a dramatically non-linear manner leading to full Fg conversion to Fn. Thrombin also controls negative feedback loop of protein C (pC) activation. This reaction needs thrombin reversible binding to thrombomodulin (Tm) expressed by intact endothelium cells. Formed pCa is an inhibitor of FVa (and, possibly, of FVIIIa, which is unstable by itself). And yet another negative feedback of coagulation is extrinsic tenase (TF-VIIa) complex inhibition by tissue factor pathway inhibitor (TFPI) which depends on the TFPI-Xa binding into the final TFPI-Xa-VIIa-TF complex [22].

3 Phenomenological Modeling of Blood Coagulation: Understanding the General Principles 3.1 Cascade Backbone with Inhibition Control on Every Step: Triggering General agreement on Roman nomenclature of blood coagulation factors [23] and understanding the architecture of coagulation system as a cascade of proenzymeenzyme transformations [24] opened the era of mathematical modeling of blood coagulation. First considerations [24, 25] attempted to understand coagulation as a biochemical amplifier, as it was noted the increase of quantity of enzymes produced on each consequent stage of coagulation cascade. Thus, Levine [25] obtained the

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

S1

E1

E2 E3

S2

E4

S3

E2

S1

5

I2 E3

S2 S3

etc.

I3 E4

I4

etc.

Fig. 2 Schemes of the ‘open’ and ‘damped’ cascades which explain burst-like and trigger-like system response to any activation [26]

analytical solution for an amplifier gain after a limited stimulus time as a function of parameters (kinetic constants and zymogens’ initial concentrations) and pointed out to its sensitivity to perturbations, i.e. to the significant overall effect which can be caused by even small but simultaneous shifts in the rate constants and initial conditions. Hemker [26] revised this concept and argued that the cascade should be viewed as a trigger from zero (no clotting) to upper (full clotting) state. In particular, in the model of an “open” (without inhibition, see Fig. 2) n-stages enzyme cascade the solution for the nth stage had the form Pn (t) ∼ t n /n! (actual formula depends on particular biochemical mechanisms, etc.). This means that increasing n makes P (t) dependence more steep, with more evident lag-time, after which the burst of product formation happens. In the model of a “damped” cascade (all intermediate enzymes are rapidly inhibited, see Fig. 2) this burst is limited, and the equation describing the transition rate and upper level of Pn was also derived [25, 26]. Moro and Martorana [27] extended this approach by including “negative feedbacks” of thrombin inhibition due to thrombin adsorption on generated fibrin strands and by products of fibrin degradation. They obtained the explicit solution for thrombin, fibrin, and products of fibrin degradation, and concluded that the latter cannot influence the stability of the initial state.

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3.2 Positive Feedback Loops: Thresholds Linearity of the above-mentioned models suggests they have single—clotted— steady state, and initial (non-clotted) state is unstable. In fact, blood is liquid, which means that non-clotted state is also stable, and thus coagulation is a threshold process. This issue was addressed by Semenov and Khanin who considered two nonlinear qualitative models of coagulation cascade capable of explaining its threshold properties [28, 29]. The first model [29] was based on the scheme of extrinsic pathway of coagulation and included one positive feedback loop of FV activation by thrombin (see Fig. 3). This scheme resulted in non-linearity in the equation for thrombin: ⎧ d[VIIa] ⎪ = αK1 − H1 [VIIa], ⎪ dt ⎪ ⎪ ⎪ ⎨ d[Xa] = K2 [VIIa] − H2 [Xa], dt (2) d[Va] ⎪ = K3 [IIa] − H3 [Va], ⎪ dt ⎪ ⎪ ⎪ ⎩ d[IIa] = K [Xa] [Va] − H [IIa]. 4 4 dt Ka +[Va]

Extrinsic stimulation

1

VII

VIIs

2

X

Xs Vs

4

II

IIs

5

I

Is

V

3

Fig. 3 Scheme having threshold properties due to accounting of the positive feedback loops [29]

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

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Here, α is a stimulation intensity (triggering signal) assumed to be constant as only initial time was considered. This system was reduced using the Tikhonov’s theorem, yielding two equations for concentration of FVa and FIIa:  dX dτ dY dτ

= Y − X,

(3)

X = α 1+X − bY.

H4 [IIa] Here, X = [Va] Ka ; Y = K3 H3 Ka ; τ = H3 t; b = H3 . Phase-plane analysis of this system shows that, depending on α, it can have one or two steady states in the (H1 H2 H3 H4 ) area of positive concentrations (X, Y ≥ 0). At α < αthres = Ka (K , the 1 K2 K3 K4 ) only steady state is zero, and it is a stable node; after any disturbance, the system returns to this state: blood remains liquid. At α > αthres , there are two steady states, unstable zero (saddle) and stable upper (node), and any disturbance leads to the transition of the system from zero to the upper state: blood clots. Extending their analysis, Semenov and Khanin considered the scheme of intrinsic coagulation pathway including two positive feedback loops—FVIII and FV activation by thrombin [28]:

⎧ dx1 ⎪ dt = k1 α − K1 x1 , ⎪ ⎪ ⎪ dx ⎪ 2 ⎪ ⎪ dt = k2 x1 − K2 x2 , ⎪ ⎪ ⎪ 3 ⎪ ⎪ dx = k3 x2 − K3 x3 , ⎪ ⎨ dt dx4 dt = k4 x7 − K4 x4 , ⎪ ⎪ dx ⎪ ⎪ 5 = k5 x3 x4 − K5 x5 , ⎪ dt Ka +x4 ⎪ ⎪ ⎪ dx6 ⎪ ⎪ = k x 6 7 − K6 x6 , ⎪ dt ⎪ ⎪ ⎩ dx7 x6 dt = k7 x5 K"a +x6 − K7 x7 ,

XIIa, XIa IXa VIIIa

(4)

Xa Va IIa

Here, xi is the concentration of an active factor formed on the ith cascade stage (specified opposite to the equation), and again α is a stimulation intensity. Using the Tikhonov’s theorem, this system was reduced to two equations for concentration of FXa and FIIa:  y dx dτ = 1+y − x, (5) dy y dτ = ax b+y − cy, x5 K1 K2 K3 K5 x7 K4 α k1 k2 k3 k4 k5 k7 α k1 k2 k3 k5 ; y = Ka k4 ; τ = K5 t; a = Ka K1 K2 K3 K4 Ks2 ; b = 7 c= K K5 . Again, phase portrait of the system depends on α.    Ka K6 k4 1/2 2 K3 K4 K5 Ka K7 1 + , only one—zero— < αthres = K1kK  k7 K k6 K4 1 k2 k3 k4 k5

where x = Ka K6 k4 Ka k6 K4 ;

At α

a

stable point exists. At α > αthres , there are three steady states: two stable nodes separated by the saddle and its separatrix. In order for the system to move to an

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upper stable steady state, the initial conditions must be in the region of attraction of this state, which is separated from the region of attraction of the zero steady state by a separatrix. As the activating intensity increases, the saddle approaches zero steady state, and the conditions for the coagulation activation become alleviated. In reality, tiny amounts of all active factors always circulate in blood [30–34]— coagulation “idles,” so that slow steady conversion of fibrinogen into fibrin occurs. Therefore, the bottom stable stationary point at which the unactivated system is located should be slightly shifted from the zero position. This can be achieved by accounting for the reactions of activation of prothrombin and factor X by the free factors Xa and IXa, respectively [35]. The position of this point on the phase plane is determined by the level of the activation signal. Upon sufficiently large stimulation intensity, due to the instantaneous displacement of the isoclines, the system located in the bottom stable steady state turns into the region of attraction of the upper steady state, and blood coagulates. Jesty and Beltrami [36, 37] studied the threshold properties of four modeling enzymatic systems progressively approaching the structure of the coagulation cascade, from 1-enzyme autocatalytic formation to 4-enzyme system with three feedback loops (representing FV, VIII, and XI activation by thrombin). First, they obtained analytical conditions for steady states lose their stability, particularly depending on the feedback loops activities. Then, they extended their analysis using direct numerical simulations. Their systems have many common properties with the actual coagulation system: the threshold behavior, the dependence of the lagperiod duration on the concentration of zymogens, the weak dependence of the system behavior on the activator level (if the latter exceeds the threshold); longrange feedback loop (reaction of FXI activation by thrombin) slightly reduces the activation threshold and slightly enhances the response. In line with this approach, later they supposed that major role of inhibitors of coagulation (native and clinically used) is to control the stability of the bottom “idling” steady state of coagulation system rather than to gradually reduce its response to activation [32].

3.3 Long-Range Feedback in Spatial Case: Blood as an Active Media Understanding the threshold and bistable properties of coagulation system and considering fibrin clot formation as a principally spatial task led Ataullakhanov and Guria to the hypothesis that coagulation should be some new example of autowave systems: thrombin should be able to autocatalytically sustain the spreading of its own formation in space, i.e. in blood plasma [38]. Biochemical background of this hypothesis were (raw at that time) data on the existence of the reaction of FXI activation by thrombin—long-range feedback, looping the coagulation cascade from the very bottom to the very top (see Fig. 1). In the same article, Ataullakhanov and Guria hypothesized that spreading of thrombin autowave is limited by the

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second autowave, which is the wave of protein C activation (see Fig. 1); the latter hypothesis wasn’t supported in future studies. Nevertheless, the first hypothesis [38] and corresponding phenomenological 2-autowave mathematical model [39]: ⎧ ⎪ ⎨ ∂u ∂t = ⎪ ⎩

∂v ∂t

K1 u2 u+K2

− K3 uv + Du,   2  − K8 uv + Dv, 1 + Kv7 = K5 u 1 − Kv6

(6)

started extensive studies of spatial aspects of blood coagulation. Besides influence on the experimental studies, they stimulated the appearance of a long list of mathematical models of various kind and complexity: detailed and reduced, homogeneous and spatial, which we review below in the Sects. 4 and 5 together with the insights obtained with their help.

4 Detailed Modeling of Blood Coagulation: Solving the Biomedical Questions 4.1 Simulating the Thrombin Generation: Many-Parameter Fitting Is Useless and Is Not Really Necessary Extensive search for an adequate coagulation test that can reflect thrombotic/hemorrhagic shifts of the coagulation system has led prof. Hemker’s group to the development of thrombin generation test (TGT), in which the non-linear, explosive kinetics of thrombin formation in plasma followed by thrombin decline due to depletion of prothrombin and inhibition of all coagulation factors—thrombin generation (TG) curve—is detected [1, 2]. This research posed a question of how TG can be simulated mathematically. Such “simulation” model was proposed in 1991 in the same group [40]. Broadly speaking, this was the first quantitative blood coagulation model, as it was based on real (in general) reaction scheme, used experimentally measured kinetic constants (Michaelis–Menten mechanism for all enzymatic reactions was supposed), and was verified by comparison with experimental kinetic curves for three basic coagulation factors. However, model trigger was FXa time-decaying influx and experimental trigger was TF; kinetic constants used were measured in purified system and experiments were done in plasma. That is why this model was indeed simulation rather than detailed. Model study showed that coagulation system has a “threshold” for total triggering FXa concentration, i.e. the prothrombinase activity is central for coagulation to be complete, and the effects of other reactions on this “threshold” were studied. Later analysis resulted in the conclusion that uncertainty in real reaction constants and reaction mechanisms (and also the possibility of shifting to diffusional control in the result of fibrin jellification) make modeling of “simple” TG curve with “complex” mathematical model impractical or even non-scientific [41]. This results from the

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A. Tokarev et al. Factor V 3. Km, kcat

Factor X 1. Km, kcat Factor Vlla (trigger)

Factor Xa + Factor Va

Kd Prothrombinase

2. Km, kcat

Prothrombin

4. Km, kcat Thrombin + Anthithrombin 5. kdec

TAT

Fig. 4 ‘Wagenvoord model’—a minimal, ‘mock’ model which parameters can be fitted to describe any thrombin generation curve [41]

fact that any experimental TG curve is 4-parametric (may be fitted by 4-parametric function, or the sum of two such functions in PRP case), while even the simplest possible coagulation model is based on six reactions and have 14 free parameters, so multiple sets of parameters exist that result in perfect agreement with experiment, even in the case of simulating the influence of substance which model actually lacks. This simple 6-reaction model [41] is sometimes called the Wagenvoord model (Fig. 4). The authors underline that Wagenvoord model is a “mock-model,” i.e. it is improbable simple and shouldn’t ever be used as a realistic predictor of thrombin generation [42]. They stress that for such complex system as coagulation even successful simulation of experimental data does not validate the underlying model assumptions, and propose the strategy of real successful use of simulation techniques [11, 41]. To study the threshold and stability properties of coagulation intrinsic pathway, prof. Ataullakhanov’s group in 1995 built similar simulation mathematical model [43, 44]. They suggested constant [FXIa] as a trigger to mimic the intrinsic pathway and S-shaped [Ca2+]-dependence of tenase and prothrombinase complexes formation rates to describe the threshold-like coagulation response to [Ca2+] [45]; they varied some unknown constants to fit the experimental AMC kinetics with the model (AMC is a fluorogenic substance produced by thrombin from the specially added synthetic substrate), and ended with two sets of best-fit model parameters and corresponded threshold [FXIa] values. Later this model underwent at least two variants of reduction considered below in Sect. 5.1. To study the processes of coagulation in standard laboratory clotting tests— prothrombin time, PT, and activated partial thromboplastin time, APPT, Khanin

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and coworkers constructed qualitative models describing coagulation triggering by extrinsic and intrinsic pathways, respectively, in these tests [46, 47]. Models’ outputs were PT and APPT, and no kinetic parameter fitting was made. The effect of coagulation factors’ deficiencies on clotting times was studied; low sensitivity to moderate deficiencies was demonstrated, and the deficiencies which can be sensed by these tests were determined. All these examples show that even non-fitted models as well as models having several sets of chosen parameters can be helpful in studying robust systems.

4.2 Homogeneous Detailed Model of Reconstituted Coagulation Can be Modified and Used by Others Instead of the native plasma, prof. Mann’s group in 1994 studied the kinetics of coagulation in a reconstituted system prepared by controlled mixing of purified coagulation factors and phospholipid vesicles [48]; to describe these experiments, they constructed the mathematical model by explicitly accounting for all involved enzyme complexes (instead of proposing the Michaelis–Menten kinetics), and obtained good agreement with experimental kinetics [49]. They concluded that such model “assembly” from equations quantitatively describing each separate coagulation reaction gives a new useful research tool. Soon after that, this model was modified in DuPont Merck group to evaluate the effects of exogenous inhibitors of thrombin and other coagulation proteases for the antithrombotic therapy [50]. Later, Hockin and Mann included other sets of reactions into their model: TFPI pathway, ATIII action, VII/VIIa competition for TF, feedback FVII activation by FIXa, FXa and thrombin, etc., although not FXI activation [51]. The resulting model seems to be the most popular model of coagulation to date. This model was used to estimate the effect of “idling” coagulation enzymes in circulating blood in the TFindependent coagulation [34], to study the independence of advanced (following the lag-period) stages of TF-triggered coagulation on the TF activity [52], to study the impact of uncertainty of, i.e., model sensitivity to, the values of rate constants of coagulation reactions [53], to study the consequences of variations of coagulation zymogens’ concentrations in normal and pathological ranges [54], to differentiate the coagulation phenotypes in familial protein C deficiency [55], to investigate the very complex pathway of FVa inhibition by pCa [56], etc. Also, Mitrophanov’s group in the US Army Medical Research Command modified this model to estimate the effects of hypothermia [57] and diluted plasma supplementation with promising concentrates of coagulation factors [58] on thrombin generation.

4.3 Spatially Distributed Models: Blood Is an Active Media The first detailed model aimed to investigate the spatio-temporal dynamics of blood coagulation was proposed in prof. Ataullakhanov’s group in 1996 [59, 60]. The

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intrinsic pathway triggered by constant FXIa influx (from one of the boundaries) was considered. Similar to [43], the S-shaped [Ca2+]-dependence of tenase and prothrombinase complexes formation rate was used, and some unknown model constants were fitted. The model had three general properties: (1) solution in the form of a running thrombin peak, (2) the threshold response to FXIa influx, and (3) the ability of the long-range feedback loop of FXI activation by thrombin to sustain the autowave spatial propagation of thrombin peak. Later, this model was used in two different ways. First, to provide some biochemical ground to the previously proposed two-autowave phenomenological model [39] (see the Sect. 3.3 above), it was supplemented with hypothetical reaction block of thrombin switching from procoagulant into anticoagulant state, and severely reduced to three equations forming another 2-autowave model (considered in the Sect. 5.2 below). Second, it was supplemented with a more accurate description of the intrinsic pathway and with the extrinsic tenase influx to simulate the extrinsic pathway, and resulting model was intensively studied in both homogeneous and spatially distributed conditions [61]. This line of research was continued by Panteleev, who constructed the new spatio-temporal model of TF-triggered coagulation [62]. Instead of gathering a huge amount of reactions at once, this model was constructed in several sequential steps. First, some blocks were simulated separately (TFPI/extrinsic tenase; intrinsic tenase). Second, they were combined together, and other reaction routes were added one-by-one with simulations of the corresponding experiments. Model assumptions included: prothrombinase reactions proceed in analogy to intrinsic tenase ones; lipid-dependent reactions—by analogy to platelet-dependent ones; platelet activation by thrombin happen in one single step. This model complemented the experimental data of Ovanesov who graduated from the same laboratory and studied spatial dynamics of coagulation in details. Together, model and experiments showed that (1) spatial propagation of coagulation is regulated by the intrinsic tenase and, for example, hemophilias are diseases of impaired clot growth (with normal clot growth initiation by TF); (2) clot propagation is regulated by protein C pathway, and Tm action can terminate the clot growth. Panteleev’s model (modified accordingly to experimental conditions) was used in studies of other important questions of spatio-temporal dynamics of blood coagulation: regulation of coagulation initiation and propagation by the activity of FIX (in hemophilia B and its treatment) [18]; regulation of thrombin autowave propagation and termination (in the presence of added phospholipids) by the feedback loops of FXI, FVIII, and pC activation by thrombin [3]; regulation of coagulation sensitivity to shear flow by the feedback loop of TF-VII activation by FXa [63]; regulation of coagulation sensitivity to surface TF distribution pattern by the feedback loops of TF-VII activation by FXa and FV activation by thrombin [64]; regulation of coagulation threshold by fibrin polymerization and coagulation triggering by FV activation by thrombin [65]. The latter study concluded that coagulation network has several distinct tasks/properties: threshold (ability to not clot at low activation), triggering (narrow activation interval of fibrin gel formation), control by blood flow (switching-off above some shear rate), spatial propagation

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(ability to form a 3-dimensional clot of a nonzero size), and localization (prevention against clotting the entire blood)—and each of them is fulfilled/controlled by the corresponding sub-network; however, there exists partial overlapping of these subnetworks. Recently this model was supplemented by sub-network of fibrinolysis reactions to study spatial dynamics of clot lysis [4]. It is unclear why despite repeated citations, Panteleev’s model has not been used by any other research group.

4.4 Clotting in Flow: The Main Actors Are Platelets Completely different approach to study the hemostasis was used in prof. Fogelson’s group: from the beginning they aimed to study blood coagulation system combined with platelet activation and flow. They started with the platelet activation/aggregation model in flow [66] and extension of Jesty and Beltramy’s model by introducing membrane binding sites supporting enzymatic reactions [67]. In the latter case, they showed that variations in surface-binding site densities can serve as a “switch,” drastically altering the responsiveness of the system. Their model [68] accounted for influence on coagulation of both platelets (by introducing densities of platelet binding sites for coagulation factors) and flow (by introducing kf low · (cout − c) terms into ODEs, where cout is plasma concentration of a given protein or platelet), as well as simple submodel of platelet activation. The main conclusions of this work were: (1) existence of a threshold-like dependence of thrombin response to the surface densities of TF and binding sites for factors and (2) platelet adhesion to subendothelium can block TF-VIIa complex activity, and it can impact the reduction of thrombin production in hemophilias. Later they combined platelet and plasma coagulation submodels into the “complete” model of mixed platelet-fibrin thrombus formation in flow [69, 70] and performed its intensive sensitivity analysis [71]. Their model was used, for example, in Neeves’s lab to estimate the thrombin generation kinetics and relative impact of intrinsic/extrinsic tenases on FXa generation inside the growing thrombus as a function of [FVIII] [72], and in Mitrophanov’s group to investigate the effects of coagulation factors supplementation on thrombus resistance in flow [73]. A number of other groups have proposed their own detailed mathematical models of coagulation. For example, Jordan and Chaikof [74] studied the synergistic effect of multiple TF sites on coagulation activation in flow.

5 Reduction: Back to Low-Dimensional Models Complex models are hard to understand and study. Thus, reduction—lowering model dimension and simplification of the right-hand side (RHS) of equations—is a well-known way to treat complex models. Reduced models should be distinguished from the phenomenological ones as the latter are written to express some relation or

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even hypothesis and may not have a known mechanistic (biochemical) background in the “hypothetical” part. In contrast, reduced model should inherit the mechanistic (biochemical) background from the parent full model. A number of reduced models of blood coagulation were proposed to study qualitatively various aspects of regulation of this system. The most intriguing questions are mechanisms of spatial spreading of coagulation and its termination. As we will see below, these studies made impact not only to biophysics of coagulation, but also to the theory of active media.

5.1 Homogeneous Reduced Models: Easy Transition to the Spatial Case At least two attempts to reduce the dimension of the first detailed model of coagulation [43, 44] were proposed. The reductions were based on the separation of variables of different time scales and applying the Tikhonov’s theorem to the subsystem of fast variables. The fastest variables were the concentrations of complexes of intrinsic tenase and prothrombinase: the rate constants of the reactions of their formation and dissociation are two orders of magnitude higher than other rate constants of the system. Pokhilko considered thrombin concentration as a fast variable, too [44]. Introducing the new variable m = Z + y8 , where Z is the prothrombinase concentration and y8 is the concentration of free FVIIIa, suggesting the closeness of the kinetic constants of activation and inhibition/decay for FVa and VIIIa, and neglecting the inhibition of FVa by pCa, she obtained the following system of equations for FIXa, FXa, and FVIIa pool: ⎧ dx ⎪ = k9 C − h9 x, ⎪ ⎨ dt dy xz dt = k10 x + p1 .f (Ca ). 1+f (Ca )x − h10 y, ⎪ ⎪ ⎩ dz dt = y2 − h8 z.

(7)

Here, x = [IXa], y = [Xa], z = m/k8, y2 = p2 .y + p3 .f (Ca ). 1+fyz (Ca )y is the quasistationary concentration of thrombin, C is the constant concentration of FXIa [Ca ]a (which is an activator), f (Ca ) = b+[C a is the dependence of the assembly rate a] of the intrinsic tenase and prothrombinase complexes on the [Ca 2+ ], ki , hi are kinetic constants, pi are combinations of kinetic constants, a and b are parameters. This model describes well the experimental homogeneous kinetics of coagulation activated by the intrinsic pathway and the threshold on the concentration of calcium ions. Right away, this model was transferred to the spatially distributed conditions. Pokhilko [44] showed that taking into account the activation of FXI by thrombin does not affect the kinetics of the process at rate constant values ranged from 0 to 10−4 min−1 (the exact value of this constant was unknown, but literature data

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showed it was below 10−4 min−1 ). However, in a spatially distributed system, existence of this reaction should lead to the ability of coagulation to spread from the activating surface into the plasma in a self-sustaining way. Thus, the system (7) was supplemented by the kinetic term for FXI activation by thrombin and by diffusion terms of all three substances, and solved numerically; FXIa influx from the left wall was considered as a trigger. The threshold behavior of the system for FXIa influx and the FXIa activation constant by thrombin was obtained. Above these thresholds, thrombin peak velocity rapidly reached the steady value. These results were later reproduced in all qualitative models of Ataullakhanov’s lab (see the Sect. 4.3 above). Molchanova also performed the extensive study of time-scales separation in the model [43, 44] and reduced it (using Tikhonov’s theorem) to three variables corresponding to thrombin, pCa, and FXa concentration [75]. She found at least four types of dynamics regimes in the reduced system: bistable (nonzero bottom state), monostable, self-oscillating (limit cycle), and coexistence of the stable state with the limit cycle, and suggested their 3-dimensional model to be quasi-2-dimensional, i.e. that it exists a limiting surface containing all the steady states and the limit cycles. Panteleev also performed reduction of his quantitative model of coagulation in the limits of initial reaction time and low TF density in the full-stirred conditions [65]. His reduction was based on identification and removing of nonessential components followed by time-scale separation analysis, and resulted in 3-equation system for TF pool, thrombin, and FVa concentrations, respectively: ⎧ dx 3p ⎪ ⎪ ⎨ dt = −b2 x3p , ⎪ ⎪ ⎩

dx2 dt dx5 dt

= b3 x3p (a3 + a4 x5 ) − a5 x2 ,

(8)

= a6 x2 ,

which had the analytical solution. Phase-plane and time-dependent analysis of this system showed the necessity of FV feedback loop for the explosive fibrin formation.

5.2 Reduced Spatial Models: Variety of Autowave Regimes Taken together, discovery of variety of dynamic regimes in homogeneous model [75] and ability of coagulation to propagate in space in an autowave manner [44, 59, 60] raises the question: are there any other dynamic regimes possible for spatially distributed coagulation? The previously proposed two-autowave phenomenological model [39] (see the Sect. 3.3 above) illustrated the regime of coagulation propagation followed by termination. To provide some biochemical ground to that phenomenological model, and especially to the suggested inhibitor autocatalysis (responsible for the termination mode), Zarnitsina et al. [76] supplemented the detailed model [59, 60] with the hypothetical reaction block of thrombin switching from the procoagulant (T1) into the anticoagulant (T2) state (namely thrombin activates pC with cleaving the peptide P, which catalyzes the conversion T1 →

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T2 but not T2 → T1, and T2 activates pC more rapidly than T1), and then severely reduced the resulting model. The reduction was based on the suggestions: rapid equilibrium T1 ⇐⇒ T2, rapid inhibition and quasistationarity of P, quasistationarity of FIXa, Xa, Va, and VIIIa, and some simplification of the RHS of equations. The resulting model consisted in three equations for dimensionless concentrations of thrombin, pCa and FXIa: ⎧ ⎪ ⎨ ⎪ ⎩

du dt dv dt dw dt

2u = K1 uw(1 − u) 1+K 1+K3 v − u + Du,

= K5 u2 − K6 v + Dv,

(9)

= u − K4 w + Dw,

with all diffusion coefficients assumed to be equal. Extensive studies of model (9) followed. In the homogeneous case, this system has 1 to 3 positive steady states and various dynamic regimes (including chaotic oscillations), which however were not studied in details. In the spatial case, this model has solutions in the form of a running excitation pulse, and several regimes of this pulse were found: disappearing at some distance from the activator, stationary propagation with constant or oscillating amplitude, trigger wave, and stopping with transformation into the stationary stand-alone peak [76]. Lobanova conducted comprehensive studies of this model and found regimes of trigger waves with oscillating rear part and running replicating (dividing) trigger waves [77], multihumped pulses [78], and second regime of stand-alone peak formation [79].

5.3 Reduced Models in Flow: Scenarios of Coagulation Termination Ermakova et al. studied model (9) in the conditions of a steady parabolic flow in a two-dimensional channel, taking into account the wall-immobilized thrombomodulin which is a cofactor of protein C activation by thrombin. They found two possible scenarios of termination of the spatial propagation of coagulation: in the narrow channel, it can be achieved by the thrombomodulin activity, and in a wider channel it can be achieved by the flow [80].

6 Conclusions and Future Directions Four main types of mathematical models have been used to study blood coagulation both in homogeneous and spatially distributed conditions: phenomenological, simulating, detailed (quantitative), and reduced. Each of these types is extremely powerful for its own tasks and conditions, but limited for other tasks and conditions: (1) Phenomenological models are used to express mathematically and visualize some idea/hypothesis and show its principal possibility by itself or in some

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context. However, due to lack of mechanistic background, these models cannot be used to prove the hypothesis (only a combination of quantitative model with experimental data is able to confirm hypothesis). On the contrary, a hypothesis can be refused, depending on the specific problem and task, using any model— phenomenological, simulating, quantitative, or reduced. (2) Simulation models are needed just after the phenomenological ones—when all main processes in the system are known (for example, at least the skeleton of the biochemical reaction network), but without precise information about all parameter values (for example, kinetic constants, concentrations, etc.). These models are used to investigate rough model properties at a semi-quantitative level: existence of thresholds, possible solution types and their stability, etc. But inaccuracies of these models should be always kept in mind and underlined in the conclusions. (3) Detailed, quantitative models are ideal in the case all mechanisms are understood and their parameters can be determined. Along with obvious advantages, the desire for quantitative modeling entails a number of problems: (a) for a number of reasons (see the Sects. 4.1, 4.2), detailed knowledge of all mechanisms and parameters of the biological process of interest isn’t always possible; understanding of this uncertainty often requires serious immersion into the literature; therefore, some parts of the quantitative models remain simulating or even phenomenological, calling remarks of the previous paragraphs (2) and (1). (b) The natural patient-specificity precludes one-to-one comparison between experimental data and even the most detailed mathematical model, as well as immediate usage of this comparison in diagnostics. Special approaches to overcome this difficulty are urgently necessary to be created. For example, each mathematical model should be equipped with a “shell”— a module which extracts meaningful characteristics of the system (for example, response time, amplitude, thresholds, velocity of spatial propagation, etc.), analyzes the sensitivity of these characteristics to all model input parameters, and performs other tasks specific to the given model and system. (c) Despite the fact that increased computer performance practically removed previous restrictions on the number of model variables and parameters, a new difficulty has arisen: difficulty of sharing and interchange of complex models between research groups. This problem may be partially solved using special model exchange formats like SBML [81]. (4) Reduced model can be created if a detailed model has been built, and some parts of this detailed model turned out to be redundant on certain time/spatial scales: • there are slow variables that can be put to constants; • there are fast variables, and kinetics of their relaxation to the quasistationarity is not of interest;

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• there exist whole model blocks that do not participate in the processes of interest or have little effect on the results. In these cases, creation of the reduced model is extremely useful for: (a) qualitative analysis of the system behavior; (b) analysis of the role of certain system parts in its functioning (by comparing reduced and detailed models); (c) embedding into the model of a higher scale instead of the full model (for adjustment, or for simplicity). Regarding the point (4a), we must note that the set of regimes of a reduced model solution can be much richer than of the parental quantitative model. Therefore, this reduced model (and methods of its research) can appear to be interesting even separated from the parent system—for example, for neighboring (or far) research area. However, caution should be taken in transferring new regimes found in the reduced model to the real system. The only known successful example of this kind for coagulation is the prediction of its ability for spatial propagation in a selfsustained manner due to the activation of FXIa by thrombin and for control of this process by the pC/Tm subsystem. Apparently, this modeling pipeline is common to mathematical biology and biophysics. This review shows implementation of this pipeline in the fundamental studies of structure and regulation mechanisms, as well as in the development of experimental methods of diagnostics and correction of the blood coagulation system. Acknowledgements This work was partially supported by the “RUDN University Program 5100” to A.T. and V.V. and by the Dynasty Foundation Fellowship to A.T.

References 1. H.C. Hemker, S. Béguin, Thrombin generation in plasma: its assessment via the endogenous thrombin potential. Thromb. Haemost. 74, 134–138 (1995) 2. H.C. Hemker, P. Giesen, R. AlDieri, V. Regnault, E. De Smed, R. Wagenvoord, T. Lecompte, S. Béguin, The calibrated automated thrombogram (CAT): a universal routine test for hyperand hypocoagulability. Pathophysiol. Haemost. Thromb. 32, 249–253 (2002) 3. N.M.M. Dashkevich, M.V. V. Ovanesov, A.N.N. Balandina, S.S.S. Karamzin, P.I.I. Shestakov, N.P.P. Soshitova, A.A.A. Tokarev, M.A.A. Panteleev, F.I.I. Ataullakhanov, Thrombin activity propagates in space during blood coagulation as an excitation wave. Biophys. J. 103, 2233– 2240 (2012) 4. A.S. Zhalyalov, M.A. Panteleev, M.A. Gracheva, F.I. Ataullakhanov, A.M. Shibeko, Coordinated spatial propagation of blood plasma clotting and fibrinolytic fronts. PLoS One 12 (2017) 5. A.N. Balandina, I.I. Serebriyskiy, A.V. Poletaev, D.M. Polokhov, M.A. Gracheva, E.M. Koltsova, D.M. Vardanyan, I.A. Taranenko, A.Y. Krylov, E.S. Urnova, K.V. Lobastov, A.V. Chernyakov, E.M. Shulutko, A.P. Momot, A.M. Shulutko, F.I. Ataullakhanov, Thrombodynamics—a new global hemostasis assay for heparin monitoring in patients under the anticoagulant treatment. PLoS One 13, 1–18 (2018)

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Fig. 2. Union Railway Station, Springfield, Massachusetts The first roads that improved on the Indian trails were, of course, made for wagons. The gorge of the Westfield was so rugged that a hundred years ago it seemed almost impossible to make a good wagon road through it. There were some people, however, who thought that it could be done and who determined to do it. Their courage won, and before long there was a good highway all along the roaring river. The bowlders were rolled out of the way, the trees were cut, the roadbed was made, and people could go east and west in the stages without risk of losing their lives or even of breaking their bones. This was accomplished soon after 1825, but it did not solve all the problems of the Massachusetts people, for, as we shall soon learn fully, the Erie canal was finished in that year, and a long string of canal boats began to carry produce from the West to New York. The good people of Boston watched all this going on. Every load of grain was headed straight eastward as if it were coming to Massachusetts bay, thence to go by vessel to Europe. But when it reached the Hudson it was sure to turn off down that river to help

load ships at the piers of New York. And New England had only a wagon road across the mountains! A wagon road will never draw trade away from a tidal river, and thus we can understand why a prominent Massachusetts man, Charles Francis Adams, spoke of the Hudson as “a river so fatal to Boston.” Boston might have all the ships she wanted, but if she could not get cargoes for them they would be of no use. Shipowners, seeing that there was plenty of western freight in New York, sent their boats there. It was indeed time that Boston people began to ask themselves what they could do. They still had ships, but these were usually “down East” coasters, and the noble vessels from far eastern ports, laden with spices and teas, silks, and all the spoils of Europe and Asia, rarely came to Boston, but brought more and greater loads to New York and Baltimore, where they could lay in corn and wheat for the return voyage. Even the Cunards transferred most of their boats and finally all their mail steamers to New York.

Fig. 3. The Valley of Deerfield River at Charlemont, Massachusetts, on the Line of the Boston and Maine Railroad The people of Boston first said, “We will build another canal, up the Hoosick and down the Deerfield valley, and then the canal boats will keep on to the east.” As states often do, they appointed a commission to see if the canal could be built, and what it would cost. But what were they to do about Hoosac mountain, which stood a thousand feet high, of solid rock, between the Hoosick valley on the west and the Deerfield valley on the east? They decided that they would tunnel it for the water way. Rather strangely they thought it could be done for a little less than a million dollars. A wise engineer made the survey for the canal, and when he remarked, “It seems as if the finger of Providence had pointed out this route from the east to the west,” some one who stood near replied, “It’s a great pity that the same finger wasn’t thrust through the mountain.” The plans for the canal were finally given up, and though many years later such a tunnel was made, it was not for a canal, nor was the work done for a million dollars.

Every one was talking now of railways, but few thought that rails could be laid across the Berkshires. It was even said in a Boston paper that such a road could never be built to Albany; that it would cost as much to do it as all Massachusetts would sell for; and that if it should be finished, everybody with common sense knew it would be as useless as a railroad from Boston to the moon. We need not be too hard on this writer, for it was five years later when the De Witt Clinton train climbed the hill from Albany and carried its handful of passengers to Schenectady. One of the friends of the railway scheme was Abner Phelps. When he was a senior at Williams College, in 1806, he had thought of it, for he had heard about the tram cars in the English coal regions. In 1826 he became a member of the legislature of Massachusetts, and the second day he was there he proposed that the road should be built. In time the project went through, but at first it was planned to pull the cars with horses, and on the down grades to take the horses on the cars and let them ride. We do not know how it was intended that the cars should be held back, for it was long before the invention of air brakes. The line was built to its western end on the Hudson in 1842, and thus Boston, Worcester, Springfield, and Albany were bound together by iron rails. There was only a single track and the grades were heavy. The road brought little trade to Boston, and most of the goods from the West still went by way of the Hudson to New York. It was, however, a beginning, and it showed that the mountain wall could be crossed. The subject of a Hoosac tunnel now came up again. It would take a long time to tell how the tunnel was made; indeed, it was a long time in making. It was begun in 1850 or soon afterwards, and the work went slowly, with many stops and misfortunes, so that the hole through the mountain was not finished until November 27, 1873. On that day the last blast was set off, which made the opening from the east to the west side; and the first regular passenger train ran through July 8, 1875, fifty years after it had been planned to make a canal under the mountain.

In order to help on the work the engineers sunk a shaft a thousand feet deep from the top of the mountain to the level of the tunnel, and from the bottom worked east and west. This gave them four faces, or “headings,” on which to work, instead of two, and hastened the finishing. The whole cost was about fourteen million dollars.

Fig. 4. Eastern Portal of the Hoosac Tunnel, Boston and Maine Railroad It took great skill to sink the shaft on just the right line, and to make the parts of the tunnel exactly meet, as the men worked in from opposite directions. They brought the ends together under the mountain with a difference of only five sixteenths of an inch! You can measure this on a finger nail and see how much it is. The tremendous task was successfully accomplished, and Boston was no longer shut off from the rest of the country by the mountains.

Fig. 5. The South Station, Boston The end of it all is not that Boston has won all the ships away from New York, but that gradually she has been getting her share. Now she has great Cunarders, White Star Liners, and the Leyland boats,—all giant ships sailing for Liverpool,—and many other stately vessels bound for southern ports or foreign lands. Now you may see in Boston harbor not a forest of masts but great funnels painted to show the lines to which the boats belong, and marking a grander commerce than that which put out for the Indies long years ago; for to-day Boston is the second American port. The great freight yards of the railways are close upon the docks, and travelers from the West may come into either of two great stations, one of which is the largest railway terminal in the world. In and about Boston are more than a million people, reaching out with one hand for the riches of the great land to the west, and with the other passing them over the seas to the nations on the farther side. Man has taken a land of dense forests, stony hills, and wild valleys, and subdued it. It is dotted with cities, crossed by roads, and is one of the great gateways of North America.

CHAPTER II

PIONEERS OF THE MOHAWK AND THE HUDSON If a stranger from a distant land should come to New York, he might take an elevated train at the Battery and ride to the upper end of Harlem. He would then have seen Manhattan island, so named by the Indians, who but three hundred years ago built their wigwams there and paddled their canoes in the waters where great ships now wait for their cargoes. If the visitor should stay for a time, he might find that Harlem used to be spelled Haarlem, from a famous old town in Holland. He might walk through Bleecker street, or Cortlandt street, or see Stuyvesant square, and learn that these hard names belonged to old Dutch families; and if he studied history, he would find that the town was once called New Amsterdam and was settled by Dutchmen from Holland. They named the river on the west of the island the Great North river, to distinguish it from the Delaware, or Great South river, and they planned to keep all the land about these two streams and to call it New Netherland. Rocks and trees covered most of Manhattan island at that time, but the Dutch had a small village at its south end, where they built a fort and set up windmills, which ground the corn and made the place look like a town in Holland. The Indians did not like the windmills with their “big teeth biting the corn in pieces,” but they were usually friendly with the settlers, sometimes sitting before the fireplaces in the houses and eating supawn, or mush and milk, with their white friends. Little did the Indian dream what a bargain he offered to the white man when he consented to sell the whole island for a sum equal to twenty-four dollars; and the Dutchman, to do him justice, was equally ignorant.

All this came about because Henry Hudson with a Dutch vessel, the Half Moon, had sailed into the harbor in 1609, and had explored the river for a long distance from its mouth. Hudson was an Englishman, but with most people he has had to pass for a Dutchman. He has come down in stories as Hendrick instead of Henry, no doubt because he commanded a ship belonging to a Dutch company, and because a Dutch colony was soon planted at the mouth of the river which he discovered. Hudson spent a month of early autumn about Manhattan and on the river which afterwards took his name. Sailing was easy, for the channel is cut so deep into the land that the tides, which rise and fall on the ocean border by day and night, push far up the Hudson and make it like an inland sea. In what we call the Highlands Hudson found the river narrow, with rocky cliffs rising far above him. Beyond he saw lowlands covered with trees, and stretching west to the foot of the Catskill mountains. He went at least as far as the place where Albany now stands, but there he found the water shallow and turned his ship about, giving up the idea of reaching the Indies by going that way. He did not know that a few miles to the west a deep valley lies open through the mountains, a valley which is now full of busy people and is more important for travel and trade than a dozen northwest passages to China would be.

Fig. 6. Henry Hudson It was not long before this valley which leads to the west was found, and by a real Dutchman. Only five years after Hudson’s voyage Dutch traders built a fort near the spot where Albany now stands. Shortly afterwards, in 1624, the first settlers came and founded Fort Orange, which is now Albany. Arent Van Curler came over from Holland in 1630 and made his home near Fort Orange. He was an able man and became friendly with the Indians. They called him “Brother Corlear” and spoke of him as their “good friend.” A few years ago a diary kept by Van Curler was found in an old Dutch garret, where it had lain for two hundred and sixty years. It told the story of a journey that he made in 1634, only four years after he came to America. Setting out on December 11, he traveled up the valley of the Mohawk until he reached the home of the Oneida Indians in central New York. He stayed with them nearly two weeks, and then returned to Fort Orange, where he arrived on January 19. This is the earliest record of a white man’s journey through a region

which now contains large towns and is traversed by many railway trains every day in the year. No one knows how long there had been Indians and Indian trails in the Mohawk valley. These trails Van Curler followed, often coming upon some of the red men themselves, and visiting them in a friendly way. They, as well as the white settlers who followed them, chose the flat, rich lands along the river, for here it was easy to beat a path, and with their bark canoes they could travel and fish. The Indians entertained Van Curler with baked pumpkins, turkey, bear meat, and venison. As the turkey is an American bird, we may be sure that it was new to the Dutch explorer. These Indians, with whom Van Curler and all the New York colonists had much to do, were of several tribes,—the Mohawks, Oneidas, Onondagas, Cayugas, and Senecas. All together they were known as the Iroquois (ĭr-ṓ-kwoi´), or Iroquois Nation, a kind of confederation which met in council and went forth together to war. They called their five-fold league The Long House, from the style of dwelling which was common among them,—a long house in which as many as twenty families sometimes lived. The Iroquois built villages, cultivated plots of land, and sometimes planted apple orchards. They were often eloquent orators and always fierce fighters. Among the surrounding tribes they were greatly feared. They sailed on lake Ontario and lake Erie in their birch-bark canoes, and they followed the trails far eastward down the Mohawk valley. Before the white men came these fierce warriors occasionally invaded New England, to the terror of the weaker tribes. Sometimes they followed up their conquests by exacting a tribute of wampum. After Fort Orange was founded they went there with their packs of beaver skins and other furs to trade for clothes and trinkets. In fact the white man’s principal interest for many years was to barter for furs. The Dutch, and soon afterwards the English, bid for the trade from their settlements on the Hudson, and the French did the same from their forts on the St. Lawrence and the Great Lakes. Thus there was much letter writing and much fighting among the colonists, while each side tried to make friends of the Indians and get the whole of the fur trade. The result was that either in war or in

trade the white men and the savages were always going up and down the Mohawk valley, which thus was a well-traveled path long before there were turnpike roads, canals, or steam cars. When Van Curler made his journey into the Indian country, he did not reach the Mohawk river at once on leaving Fort Orange, but traveled for about sixteen miles across a sandy and half-barren stretch of scrubby pine woods. He came down to the river where its rich bottom lands spread out widely and where several large islands are inclosed by parts of the stream. South and east of these flats are the sand barrens, and on the west are high hills through which, by a deep, narrow gap, the Mohawk flows. The Indians called this place “Schonowe,” or “gateway.” It was well named, for entering by this gate one can go to the foot of the Rocky mountains without climbing any heights. A few years before his death Van Curler led a small band of colonists from Fort Orange, bought the “great flats” from the Mohawk Indians, and founded a town, calling it Schenectady, which is the old Indian name changed in its spelling. No easy time did these settlers have, for theirs was for many years the frontier town and they never knew when hostile savages might come down upon them to burn their houses and take their scalps. In 1690, twenty-eight years after the town was founded, a company of French and Indians from Montreal surprised Schenectady in the night, burned most of the houses, and killed about sixty of the people, taking others captive. But Dutchmen rarely give up an undertaking, and they soon rebuilt their town. It was an important place, for here was the end of the “carry” over the pine barrens from the Hudson, and here began the navigation of the river, which for a hundred years was the best means of carrying supplies up the valley and into central New York. The traveler of to-day on the New York Central Railway sees on Van Curler’s “great flats” the flourishing city of Schenectady, with its shops and houses, its college, and its vast factories for the manufacture of locomotives and electrical supplies.

Fig. 7. Sir William Johnson See Fort Johnson, Fig. 9

It is true that the Dutch pioneers played an important part in the early history of the state and are still widely represented by their descendants in the Mohawk valley, but the leading spirit of colonial days on the river was a native of Ireland who came when a young man to manage his uncle’s estates in America. This was in 1738, nearly fifty years after the Schenectady massacre. The young man, who was in the confidence of the governor of New York and of the king as well, is known to all readers of American history as Sir William Johnson. He built a fine stone mansion a short distance west of the present city of Amsterdam and lived there many years. He also founded Johnstown, a few miles to the north, now a thriving little city. He dealt honestly with the Indians, when many tried to get their lands by fraud, and he served as a high officer in the French and Indian wars. As the Dutch settled the lower Mohawk valley, so the upper parts were taken up and the forests cleared by Yankees from New England. One of these was Hugh White, a sturdy man with several grown children. He left Middletown, Connecticut, in 1784, and came

by water to Albany, sending one of his sons overland to drive two pair of oxen. Father and son met in Albany and went together across the sands to Schenectady, where they bought a boat to take some of the goods up the river. Four miles west of where Utica now stands they stopped, cut a few trees, and built a hut to shelter them until they could raise crops and have a better home. Thus the ancient village of Whitesboro was founded. White was one of many hardy and brave men who settled in central New York at that time, and they doubtless thought that they had gone a long way “out West.” Certainly their journey took more time than the emigrant would now need to reach California or Oregon. To cut the trees, build cabins, guard against the savages, and get enough to eat and wear gave the settlers plenty to do. Only the simplest ways of living were possible. Until a grist mill was built they often used samp mortars, such as the Indians made. They took a section of white ash log three feet long, and putting coals of fire on one end, kept them burning with a hand bellows until the hole was deep enough to hold the corn, which was then pounded for their meals of hominy. By and by a mill was built, and here settlers often came from a distance of many miles, sometimes carrying their grists on their backs. A dozen years after White came, General William Floyd set up another mill in the northern part of what is now Oneida county. He was one of the signers of the Declaration of Independence. One settler cleared several acres and planted corn with pumpkin seeds sprinkled in. The pigeons pulled up all the corn, but hundreds of great pumpkins grew and ripened. Since the crop was hardly enough, however, for either men or beasts, the latter had to be fed the next winter on the small top boughs of the elm, maple, and basswood. Much use was made of the river, for the only roads were Indian paths through the woods on the river flats. People and freight were carried in long, light boats suited to river traffic and known as bateaux (bȧ-tō̟ s´). These could be propelled with oars, but poles were necessary going upstream against a stiff current. It was impossible to go up the Mohawk from the Hudson above Albany, on

account of the great falls at Cohoes; hence the long carry to Schenectady. From that place, by hard work, the boatmen could make their way up to Little Falls, where the water descends forty feet in roaring rapids. Here the loads and the bateaux had to be carried along the banks to the still water above, where, with many windings and doublings on their course, the voyagers could reach the Oneida Carrying Place, or Fort Stanwix. There they unloaded again, and for a mile or more tramped across low ground to Wood creek, a little stream flowing into Oneida lake, and thence into Oswego river and lake Ontario. The city of Rome stands exactly on the road followed by the “carry.” This was an important place, and was called by the Dutch Trow Plat, while to the Indians it was De-o-wain-sta, “the place where canoes are carried across.” Several forts were built there, of which the most famous was Fort Stanwix. We should think Wood creek a difficult bit of navigation. It was a small stream, very crooked, and often interrupted by fallen trees. In times of low water the boats were dragged up and even down the creek by horses walking in the water.

Fig. 8. Genesee Street, Utica Part of the old Genesee road

The first merchant of old Fort Schuyler (Utica) was John Post, who had served his country well through the Revolution. In 1790 he brought hither his wife, three little children, and a carpenter from Schenectady, after a voyage of about nine days up the river. Near the long-used fording place he built a store, at the foot of what is now Genesee street. Here he supplied the simple needs of the few families in the new hamlet, and bought furs and ginseng of the Indians, giving in exchange paint, powder, shot, cloth, beads, mirrors, and, it must be added, rum also. Thus the fact that the river was shallow at this point and could be passed without a bridge or a boat led to the founding of the city of Utica. The first regular mail reached the settlement in 1793, the post rider being allowed twenty-eight hours to come up from Canajoharie, a distance of about forty miles, now traversed by many trains in much less than an hour. On one occasion the Fort Schuyler settlement received six letters in one mail. The people would hardly believe this astonishing fact until John Post, who had been made postmaster, assured them that it was true. Post established stages and lines of boats to Schenectady, and soon had a large business, for people were pouring into western New York to settle upon its fertile lands. All the boats did not go down to Oswego and lake Ontario. Some turned and entered the Seneca river, following its slow and winding waters to the country now lying between Syracuse and Rochester. But these boats were not equal to the traffic, for the new farms were producing grain to be transported, and the people needed many articles from the older towns on the Atlantic coast. Hence about a dozen years after Hugh White built his first cabin by the river, the state legislature took up the question of transportation and built a great road, a hundred miles long, from Fort Schuyler, or the future Utica, to Geneva, at the foot of Seneca lake. The road as laid out was six rods wide. It was improved for a width of four rods by the use of gravel and logs where the ground was soft and swampy, as much of it was in those days, being flat and shaded by trees. Over the famous Genesee road, as it was called, thousands of people went not only to the rich valley of the Genesee

in western New York but also on to Ohio, and even to the prairies of the Mississippi river. Genesee street in Utica and Genesee street in Syracuse are parts of this road. The historian tells of it as a triumph, for it was an Indian path in June, and before September was over a stage had started at Fort Schuyler, and on the afternoon of the third day had deposited its four passengers at the hotel in Geneva. After this wagons and stages began to run frequently between Albany and Geneva. A wagon could carry fourteen barrels of flour eastward, and in about a month could return to Geneva from Albany with a load of needed supplies. In five weeks, one winter, five hundred and seventy sleighs carrying families passed through Geneva to lands farther west. Geneva was quite a metropolis in those days, when there was nothing but woods where Syracuse and Rochester now are. Regular markets were held there, for there were fine farms and orchards about the beautiful shores of Seneca lake. It is recorded as remarkable that one settler had “dressed up” an old Indian orchard and made “one hundred barrels of cyder.” We might think that the founders of the city of Rochester would have come in by the Genesee road, but they did not. Far to the south, at Hagerstown in Maryland, a country already old, lived Colonel Rochester. He heard of the Genesee lands and at last bought, with his partners, a hundred acres by the falls, where the city now stands. When the little family procession passed down the street and entered upon the long journey up the Susquehanna valley to western New York, Rochester’s friends in Hagerstown wept to see him go. They thought that he had thrown his money away in buying swamp lands where only mosquitoes, rattlesnakes, and bears could live, but he saw farther than they did. If he had been unwilling to take any risk, he would never have laid out the first streets of the prosperous city which now bears his name.

Fig. 9. Old Fort Johnson, Amsterdam, New York Built by Sir William Johnson, 1742

Syracuse, like Utica and Rochester, had its own way of beginning. We can truly say that at first salt made the city. The beds of salt are not directly under Syracuse, but are in the hills not far away. The water from the rains and springs dissolves some of this salt, and as it flows down it fills the gravels in and around the town. While all was yet forest the Indian women had made salt from the brine which oozed up in the springs. So long ago as 1770, five years before the Revolution, the Delaware Indians went after Onondaga salt, and a little of it was now and then brought down to Albany. Sometimes it was sold far down the St. Lawrence in Quebec.

New York New York, Lake Erie and Western Railroad Delaware, Lackawanna and Western Railroad New York Central and Hudson River Railroad New York, Ontario and Western Railroad Delaware and Hudson Railroad Erie Canal (old location)

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The pioneers first made salt there in 1788. This was several years before the Genesee road was cut through the woods. One of these men, a Mr. Danforth, whose name a suburb of Syracuse now bears, used to put his coat on his head for a cushion and on that carry out a large kettle to the springs. He would put a pole across crotched sticks, hang up the kettle, and go to work to make salt. When he had made enough for the time he would hide his kettle in the bushes and bring home his salt. By and by so many hundreds of bushels were made by the settlers that the government of the state framed laws to regulate the making and selling of the salt, and as time went on a town arose and grew into a city. Many years later rock salt was found deep down under the surface farther west, and since that discovery the business of Syracuse has become more and more varied in character. The history of the state of New York shows well how the New World was settled along the whole Atlantic coast. The white men from Europe found first Manhattan island and the harbor. Then they followed the lead of a river and made a settlement that was to be Albany. Still they let a river guide them, this time the Mohawk, and it led them westward. They pushed their boats up the stream, and on land they widened the trails of the red men. Near its head the Mohawk valley led out into the wide, rich plains south and east of lake Ontario. Soon there were so many people that a good road became necessary. When the good road was made it brought more people, and thus the foundations of the Empire State were laid.

CHAPTER III

ORISKANY, A BATTLE OF THE REVOLUTION About halfway between old Fort Schuyler, or Utica, and Fort Stanwix, which is now Rome, is the village of Oriskany. A mile or two west of this small town, in a field south of the Mohawk river, stands a monument raised in memory of a fierce battle fought on that slope in the year following the Declaration of Independence. On the pedestal are four tablets in bronze, one of which shows a wounded general sitting on the ground in the woods, with his hand raised, giving orders to his men. The time was 1777, the strife was the battle of Oriskany, and the brave and suffering general was Nicholas Herkimer. On another of the tablets is this inscription: Here was fought

The battle of Oriskany

On the 6th day of August, 1777.

Here British invasion was checked and thwarted.

Here General Nicholas Herkimer,

Intrepid leader of the American forces,

Though mortally wounded kept his command of the fight

Till the enemy had fled.

The life blood of more than

Two hundred patriot heroes

Made this battle ground

Sacred forever. After the battle Herkimer was carried down the valley to his home, where a few days later he died. On the field he had calmly lighted his pipe and smoked it as he gave his orders, refusing to be